diff --git a/index.html b/index.html index 3ac25969..6a5afc30 100644 --- a/index.html +++ b/index.html @@ -1,3 +1,2 @@ <!--This file is automatically generated by Documenter.jl--> <meta http-equiv="refresh" content="0; url=./stable/"/> - diff --git a/previews/PR358/assets/documenter.js b/previews/PR358/assets/documenter.js new file mode 100644 index 00000000..6adfbbbf --- /dev/null +++ b/previews/PR358/assets/documenter.js @@ -0,0 +1,331 @@ +// Generated by Documenter.jl +requirejs.config({ + paths: { + 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/languages/julia.min', + 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', + 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.12.1/jquery-ui.min', + 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/contrib/auto-render.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.6.0/jquery.min', + 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', + 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/katex.min', + 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/highlight.min', + 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/languages/julia-repl.min', + }, + shim: { + "highlight-julia": { + "deps": [ + "highlight" + ] + }, + "katex-auto-render": { + "deps": [ + "katex" + ] + }, + "headroom-jquery": { + "deps": [ + "jquery", + "headroom" + ] + }, + "highlight-julia-repl": { + "deps": [ + "highlight" + ] + } +} +}); +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'katex', 'katex-auto-render'], function($, katex, renderMathInElement) { +$(document).ready(function() { + renderMathInElement( + document.body, + { + "delimiters": [ + { + "left": "$", + "right": "$", + "display": false + }, + { + "left": "$$", + "right": "$$", + "display": true + }, + { + "left": "\\[", + "right": "\\]", + "display": true + } + ] +} + + ); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) { +$(document).ready(function() { + hljs.highlightAll(); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require([], function() { +function addCopyButtonCallbacks() { + for (const el of document.getElementsByTagName("pre")) { + const button = document.createElement("button"); + button.classList.add("copy-button", "fas", "fa-copy"); + el.appendChild(button); + + const success = function () { + button.classList.add("success", "fa-check"); + button.classList.remove("fa-copy"); + }; + + const failure = function () { + button.classList.add("error", "fa-times"); + button.classList.remove("fa-copy"); + }; + + button.addEventListener("click", function () { + copyToClipboard(el.innerText).then(success, failure); + + setTimeout(function () { + button.classList.add("fa-copy"); + button.classList.remove("success", "fa-check", "fa-times"); + }, 5000); + }); + } +} + +function copyToClipboard(text) { + // clipboard API is only available in secure contexts + if (window.navigator && window.navigator.clipboard) { + return window.navigator.clipboard.writeText(text); + } else { + return new Promise(function (resolve, reject) { + try { + const el = document.createElement("textarea"); + el.textContent = text; + el.style.position = "fixed"; + el.style.opacity = 0; + document.body.appendChild(el); + el.select(); + document.execCommand("copy"); + + resolve(); + } catch (err) { + reject(err); + } finally { + document.body.removeChild(el); + } + }); + } +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", addCopyButtonCallbacks); +} else { + addCopyButtonCallbacks(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) { + +// Manages the top navigation bar (hides it when the user starts scrolling down on the +// mobile). +window.Headroom = Headroom; // work around buggy module loading? +$(document).ready(function() { + $('#documenter .docs-navbar').headroom({ + "tolerance": {"up": 10, "down": 10}, + }); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Modal settings dialog +$(document).ready(function() { + var settings = $('#documenter-settings'); + $('#documenter-settings-button').click(function(){ + settings.toggleClass('is-active'); + }); + // Close the dialog if X is clicked + $('#documenter-settings button.delete').click(function(){ + settings.removeClass('is-active'); + }); + // Close dialog if ESC is pressed + $(document).keyup(function(e) { + if (e.keyCode == 27) settings.removeClass('is-active'); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Manages the showing and hiding of the sidebar. +$(document).ready(function() { + var sidebar = $("#documenter > .docs-sidebar"); + var sidebar_button = $("#documenter-sidebar-button") + sidebar_button.click(function(ev) { + ev.preventDefault(); + sidebar.toggleClass('visible'); + if (sidebar.hasClass('visible')) { + // Makes sure that the current menu item is visible in the sidebar. + $("#documenter .docs-menu a.is-active").focus(); + } + }); + $("#documenter > .docs-main").bind('click', function(ev) { + if ($(ev.target).is(sidebar_button)) { + return; + } + if (sidebar.hasClass('visible')) { + sidebar.removeClass('visible'); + } + }); +}) + +// Resizes the package name / sitename in the sidebar if it is too wide. +// Inspired by: https://github.com/davatron5000/FitText.js +$(document).ready(function() { + e = $("#documenter .docs-autofit"); + function resize() { + var L = parseInt(e.css('max-width'), 10); + var L0 = e.width(); + if(L0 > L) { + var h0 = parseInt(e.css('font-size'), 10); + e.css('font-size', L * h0 / L0); + // TODO: make sure it survives resizes? + } + } + // call once and then register events + resize(); + $(window).resize(resize); + $(window).on('orientationchange', resize); +}); + +// Scroll the navigation bar to the currently selected menu item +$(document).ready(function() { + var sidebar = $("#documenter .docs-menu").get(0); + var active = $("#documenter .docs-menu .is-active").get(0); + if(typeof active !== 'undefined') { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +function set_theme(theme) { + var active = null; + var disabled = []; + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + var themename = ss.ownerNode.getAttribute("data-theme-name"); + if(themename === null) continue; // ignore non-theme stylesheets + // Find the active theme + if(themename === theme) active = ss; + else disabled.push(ss); + } + if(active !== null) { + active.disabled = false; + if(active.ownerNode.getAttribute("data-theme-primary") === null) { + document.getElementsByTagName('html')[0].className = "theme--" + theme; + } else { + document.getElementsByTagName('html')[0].className = ""; + } + disabled.forEach(function(ss){ + ss.disabled = true; + }); + } + + // Store the theme in localStorage + if(typeof(window.localStorage) !== "undefined") { + window.localStorage.setItem("documenter-theme", theme); + } else { + console.error("Browser does not support window.localStorage"); + } +} + +// Theme picker setup +$(document).ready(function() { + // onchange callback + $('#documenter-themepicker').change(function themepick_callback(ev){ + var themename = $('#documenter-themepicker option:selected').attr('value'); + set_theme(themename); + }); + + // Make sure that the themepicker displays the correct theme when the theme is retrieved + // from localStorage + if(typeof(window.localStorage) !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if(theme !== null) { + $('#documenter-themepicker option').each(function(i,e) { + e.selected = (e.value === theme); + }) + } else { + $('#documenter-themepicker option').each(function(i,e) { + e.selected = $("html").hasClass(`theme--${e.value}`); + }) + } + } +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function() { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if (typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === 'boolean' && DOCUMENTER_VERSION_SELECTOR_DISABLED) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function(x) { + target_href = version_selector_select.children("option:selected").get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if (typeof DOCUMENTER_CURRENT_VERSION !== 'undefined' && $('#version-selector > option').length == 0) { + var option = $("<option value='#' selected='selected'>" + DOCUMENTER_CURRENT_VERSION + "</option>"); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== 'undefined') { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = existing_versions.map(function(i,x){return x.text}); + DOC_VERSIONS.forEach(function(each) { + var version_url = documenterBaseURL + "/../" + each; + var existing_id = $.inArray(each, existing_versions_texts); + // if not already in the version selector, add it as a new option, + // otherwise update the old option with the URL and enable it + if (existing_id == -1) { + var option = $("<option value='" + version_url + "'>" + each + "</option>"); + version_selector_select.append(option); + } else { + var option = existing_versions[existing_id]; + option.value = version_url; + option.disabled = false; + } + }); + } + + // only show the version selector if the selector has been populated + if (version_selector_select.children("option").length > 0) { + version_selector.toggleClass("visible"); + } +}) + +}) diff --git a/previews/PR358/assets/search.js b/previews/PR358/assets/search.js new file mode 100644 index 00000000..c133f741 --- /dev/null +++ b/previews/PR358/assets/search.js @@ -0,0 +1,267 @@ +// Generated by Documenter.jl +requirejs.config({ + paths: { + 'lunr': 'https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min', + 'lodash': 'https://cdnjs.cloudflare.com/ajax/libs/lodash.js/4.17.21/lodash.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.6.0/jquery.min', + } +}); +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'lunr', 'lodash'], function($, lunr, _) { + +$(document).ready(function() { + // parseUri 1.2.2 + // (c) Steven Levithan <stevenlevithan.com> + // MIT License + function parseUri (str) { + var o = parseUri.options, + m = o.parser[o.strictMode ? "strict" : "loose"].exec(str), + uri = {}, + i = 14; + + while (i--) uri[o.key[i]] = m[i] || ""; + + uri[o.q.name] = {}; + uri[o.key[12]].replace(o.q.parser, function ($0, $1, $2) { + if ($1) uri[o.q.name][$1] = $2; + }); + + return uri; + }; + parseUri.options = { + strictMode: false, + key: ["source","protocol","authority","userInfo","user","password","host","port","relative","path","directory","file","query","anchor"], + q: { + name: "queryKey", + parser: /(?:^|&)([^&=]*)=?([^&]*)/g + }, + parser: { + strict: /^(?:([^:\/?#]+):)?(?:\/\/((?:(([^:@]*)(?::([^:@]*))?)?@)?([^:\/?#]*)(?::(\d*))?))?((((?:[^?#\/]*\/)*)([^?#]*))(?:\?([^#]*))?(?:#(.*))?)/, + loose: /^(?:(?![^:@]+:[^:@\/]*@)([^:\/?#.]+):)?(?:\/\/)?((?:(([^:@]*)(?::([^:@]*))?)?@)?([^:\/?#]*)(?::(\d*))?)(((\/(?:[^?#](?![^?#\/]*\.[^?#\/.]+(?:[?#]|$)))*\/?)?([^?#\/]*))(?:\?([^#]*))?(?:#(.*))?)/ + } + }; + + $("#search-form").submit(function(e) { + e.preventDefault() + }) + + // list below is the lunr 2.1.3 list minus the intersect with names(Base) + // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) + // ideally we'd just filter the original list but it's not available as a variable + lunr.stopWordFilter = lunr.generateStopWordFilter([ + 'a', + 'able', + 'about', + 'across', + 'after', + 'almost', + 'also', + 'am', + 'among', + 'an', + 'and', + 'are', + 'as', + 'at', + 'be', + 'because', + 'been', + 'but', + 'by', + 'can', + 'cannot', + 'could', + 'dear', + 'did', + 'does', + 'either', + 'ever', + 'every', + 'from', + 'got', + 'had', + 'has', + 'have', + 'he', + 'her', + 'hers', + 'him', + 'his', + 'how', + 'however', + 'i', + 'if', + 'into', + 'it', + 'its', + 'just', + 'least', + 'like', + 'likely', + 'may', + 'me', + 'might', + 'most', + 'must', + 'my', + 'neither', + 'no', + 'nor', + 'not', + 'of', + 'off', + 'often', + 'on', + 'or', + 'other', + 'our', + 'own', + 'rather', + 'said', + 'say', + 'says', + 'she', + 'should', + 'since', + 'so', + 'some', + 'than', + 'that', + 'the', + 'their', + 'them', + 'then', + 'there', + 'these', + 'they', + 'this', + 'tis', + 'to', + 'too', + 'twas', + 'us', + 'wants', + 'was', + 'we', + 'were', + 'what', + 'when', + 'who', + 'whom', + 'why', + 'will', + 'would', + 'yet', + 'you', + 'your' + ]) + + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!" + // would not find anything if searching for "add!", only for the entire qualification + lunr.tokenizer.separator = /[\s\-\.]+/ + + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + lunr.trimmer = function (token) { + return token.update(function (s) { + return s.replace(/^[^a-zA-Z0-9@!]+/, '').replace(/[^a-zA-Z0-9@!]+$/, '') + }) + } + + lunr.Pipeline.registerFunction(lunr.stopWordFilter, 'juliaStopWordFilter') + lunr.Pipeline.registerFunction(lunr.trimmer, 'juliaTrimmer') + + var index = lunr(function () { + this.ref('location') + this.field('title',{boost: 100}) + this.field('text') + documenterSearchIndex['docs'].forEach(function(e) { + this.add(e) + }, this) + }) + var store = {} + + documenterSearchIndex['docs'].forEach(function(e) { + store[e.location] = {title: e.title, category: e.category, page: e.page} + }) + + $(function(){ + searchresults = $('#documenter-search-results'); + searchinfo = $('#documenter-search-info'); + searchbox = $('#documenter-search-query'); + searchform = $('.docs-search'); + sidebar = $('.docs-sidebar'); + function update_search(querystring) { + tokens = lunr.tokenizer(querystring) + results = index.query(function (q) { + tokens.forEach(function (t) { + q.term(t.toString(), { + fields: ["title"], + boost: 100, + usePipeline: true, + editDistance: 0, + wildcard: lunr.Query.wildcard.NONE + }) + q.term(t.toString(), { + fields: ["title"], + boost: 10, + usePipeline: true, + editDistance: 2, + wildcard: lunr.Query.wildcard.NONE + }) + q.term(t.toString(), { + fields: ["text"], + boost: 1, + usePipeline: true, + editDistance: 0, + wildcard: lunr.Query.wildcard.NONE + }) + }) + }) + searchinfo.text("Number of results: " + results.length) + searchresults.empty() + results.forEach(function(result) { + data = store[result.ref] + link = $('<a class="docs-label">'+data.title+'</a>') + link.attr('href', documenterBaseURL+'/'+result.ref) + if (data.category != "page"){ + cat = $('<span class="docs-category">('+data.category+', '+data.page+')</span>') + } else { + cat = $('<span class="docs-category">('+data.category+')</span>') + } + li = $('<li>').append(link).append(" ").append(cat) + searchresults.append(li) + }) + } + + function update_search_box() { + querystring = searchbox.val() + update_search(querystring) + } + + searchbox.keyup(_.debounce(update_search_box, 250)) + searchbox.change(update_search_box) + + // Disable enter-key form submission for the searchbox on the search page + // and just re-run search rather than refresh the whole page. + searchform.keypress( + function(event){ + if (event.which == '13') { + if (sidebar.hasClass('visible')) { + sidebar.removeClass('visible'); + } + update_search_box(); + event.preventDefault(); + } + } + ); + + search_query_uri = parseUri(window.location).queryKey["q"] + if(search_query_uri !== undefined) { + search_query = decodeURIComponent(search_query_uri.replace(/\+/g, '%20')) + searchbox.val(search_query) + } + update_search_box(); + }) +}) + +}) diff --git a/previews/PR358/assets/themes/documenter-dark.css b/previews/PR358/assets/themes/documenter-dark.css new file mode 100644 index 00000000..c94a294d --- /dev/null +++ b/previews/PR358/assets/themes/documenter-dark.css @@ -0,0 +1,7 @@ +@keyframes spinAround{from{transform:rotate(0deg)}to{transform:rotate(359deg)}}html.theme--documenter-dark .tabs,html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark .pagination-next,html.theme--documenter-dark .pagination-link,html.theme--documenter-dark .pagination-ellipsis,html.theme--documenter-dark .breadcrumb,html.theme--documenter-dark .file,html.theme--documenter-dark .button,.is-unselectable,html.theme--documenter-dark .modal-close,html.theme--documenter-dark .delete{-webkit-touch-callout:none;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none}html.theme--documenter-dark .navbar-link:not(.is-arrowless)::after,html.theme--documenter-dark .select:not(.is-multiple):not(.is-loading)::after{border:3px solid rgba(0,0,0,0);border-radius:2px;border-right:0;border-top:0;content:" ";display:block;height:0.625em;margin-top:-0.4375em;pointer-events:none;position:absolute;top:50%;transform:rotate(-45deg);transform-origin:center;width:0.625em}html.theme--documenter-dark .admonition:not(:last-child),html.theme--documenter-dark .tabs:not(:last-child),html.theme--documenter-dark .message:not(:last-child),html.theme--documenter-dark .list:not(:last-child),html.theme--documenter-dark .level:not(:last-child),html.theme--documenter-dark .breadcrumb:not(:last-child),html.theme--documenter-dark 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kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-dark.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content 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.button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #282f2f #282f2f !important}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--documenter-dark .content 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Consider for example the one from above with <code>Beta(2, 2)</code>.</p><pre><code class="language-julia hljs">julia> using Random; + Random.seed!(42); + +julia> using Bijectors; + using Bijectors: Logit; + +julia> dist = Beta(2, 2) +Beta{Float64}(α=2.0, β=2.0) + +julia> x = rand(dist) +0.36888689965963756 + +julia> b = bijector(dist) # bijection (0, 1) → ℝ +Logit{Float64}(0.0, 1.0) + +julia> y = b(x) +-0.5369949942509267</code></pre><p>In this case we see that <code>bijector(d::Distribution)</code> returns the corresponding constrained-to-unconstrained bijection for <code>Beta</code>, which indeed is a <code>Logit</code> with <code>a = 0.0</code> and <code>b = 1.0</code>. The resulting <code>Logit <: Bijector</code> has a method <code>(b::Logit)(x)</code> defined, allowing us to call it just like any other function. Comparing with the above example, <code>b(x) ≈ link(dist, x)</code>. Just to convince ourselves:</p><pre><code class="language-julia hljs">julia> b(x) ≈ link(dist, x) +true</code></pre><h2 id="Transforming-distributions"><a class="docs-heading-anchor" href="#Transforming-distributions">Transforming distributions</a><a id="Transforming-distributions-1"></a><a class="docs-heading-anchor-permalink" href="#Transforming-distributions" title="Permalink"></a></h2><p>We can create a <em>transformed</em> <code>Distribution</code>, i.e. a <code>Distribution</code> defined by sampling from a given <code>Distribution</code> and then transforming using a given transformation:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia> dist = Beta(2, 2) # support on (0, 1)</code><code class="nohighlight hljs ansi" style="display:block;">Beta{Float64}(α=2.0, β=2.0)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> tdist = transformed(dist) # support on ℝ</code><code class="nohighlight hljs ansi" style="display:block;">UnivariateTransformed{Beta{Float64}, Bijectors.Logit{Float64, Float64}}( +dist: Beta{Float64}(α=2.0, β=2.0) +transform: Bijectors.Logit{Float64, Float64}(0.0, 1.0) +)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> tdist isa UnivariateDistribution</code><code class="nohighlight hljs ansi" style="display:block;">true</code></pre><p>We can the then compute the <code>logpdf</code> for the resulting distribution:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia> # Some example values + x = rand(dist)</code><code class="nohighlight hljs ansi" style="display:block;">0.49165590570439216</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> y = tdist.transform(x)</code><code class="nohighlight hljs ansi" style="display:block;">-0.03337947609209195</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> logpdf(tdist, y)</code><code class="nohighlight hljs ansi" style="display:block;">-0.9813863218628618</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../transforms/">« Transforms</a><a class="docs-footer-nextpage" href="../examples/">Examples »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Wednesday 8 January 2025 12:23">Wednesday 8 January 2025</span>. Using Julia version 1.11.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> diff --git a/previews/PR358/examples/index.html b/previews/PR358/examples/index.html new file mode 100644 index 00000000..f4e194ac --- /dev/null +++ b/previews/PR358/examples/index.html @@ -0,0 +1,78 @@ +<!DOCTYPE html> +<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Examples · Bijectors</title><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.045/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../">Bijectors</a></span></div><form class="docs-search" action="../search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li><a class="tocitem" href="../transforms/">Transforms</a></li><li><a class="tocitem" href="../distributions/">Distributions.jl integration</a></li><li class="is-active"><a class="tocitem" href>Examples</a><ul class="internal"><li><a class="tocitem" href="#Univariate-ADVI-example"><span>Univariate ADVI example</span></a></li><li><a class="tocitem" href="#Multivariate-ADVI-example"><span>Multivariate ADVI example</span></a></li><li><a class="tocitem" href="#Normalizing-flows"><span>Normalizing flows</span></a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Examples</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Examples</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/TuringLang/Bijectors.jl/blob/master/docs/src/examples.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h2 id="Univariate-ADVI-example"><a class="docs-heading-anchor" href="#Univariate-ADVI-example">Univariate ADVI example</a><a id="Univariate-ADVI-example-1"></a><a class="docs-heading-anchor-permalink" href="#Univariate-ADVI-example" title="Permalink"></a></h2><p>But the real utility of <code>TransformedDistribution</code> becomes more apparent when using <code>transformed(dist, b)</code> for any bijector <code>b</code>. To get the transformed distribution corresponding to the <code>Beta(2, 2)</code>, we called <code>transformed(dist)</code> before. This is simply an alias for <code>transformed(dist, bijector(dist))</code>. Remember <code>bijector(dist)</code> returns the constrained-to-constrained bijector for that particular <code>Distribution</code>. But we can of course construct a <code>TransformedDistribution</code> using different bijectors with the same <code>dist</code>. This is particularly useful in something called <em>Automatic Differentiation Variational Inference (ADVI)</em>.[2] An important part of ADVI is to approximate a constrained distribution, e.g. <code>Beta</code>, as follows:</p><ol><li>Sample <code>x</code> from a <code>Normal</code> with parameters <code>μ</code> and <code>σ</code>, i.e. <code>x ~ Normal(μ, σ)</code>.</li><li>Transform <code>x</code> to <code>y</code> s.t. <code>y ∈ support(Beta)</code>, with the transform being a differentiable bijection with a differentiable inverse (a "bijector")</li></ol><p>This then defines a probability density with same <em>support</em> as <code>Beta</code>! Of course, it's unlikely that it will be the same density, but it's an <em>approximation</em>. Creating such a distribution becomes trivial with <code>Bijector</code> and <code>TransformedDistribution</code>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia> using StableRNGs: StableRNG</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> rng = StableRNG(42);</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> dist = Beta(2, 2)</code><code class="nohighlight hljs ansi" style="display:block;">Beta{Float64}(α=2.0, β=2.0)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> b = bijector(dist) # (0, 1) → ℝ</code><code class="nohighlight hljs ansi" style="display:block;">Bijectors.Logit{Float64, Float64}(0.0, 1.0)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> b⁻¹ = inverse(b) # ℝ → (0, 1)</code><code class="nohighlight hljs ansi" style="display:block;">Inverse{Bijectors.Logit{Float64, Float64}}(Bijectors.Logit{Float64, Float64}(0.0, 1.0))</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> td = transformed(Normal(), b⁻¹) # x ∼ 𝓝(0, 1) then b(x) ∈ (0, 1)</code><code class="nohighlight hljs ansi" style="display:block;">UnivariateTransformed{Normal{Float64}, Inverse{Bijectors.Logit{Float64, Float64}}}( +dist: Normal{Float64}(μ=0.0, σ=1.0) +transform: Inverse{Bijectors.Logit{Float64, Float64}}(Bijectors.Logit{Float64, Float64}(0.0, 1.0)) +)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> x = rand(rng, td) # ∈ (0, 1)</code><code class="nohighlight hljs ansi" style="display:block;">0.3384404850130036</code></pre><p>It's worth noting that <code>support(Beta)</code> is the <em>closed</em> interval <code>[0, 1]</code>, while the constrained-to-unconstrained bijection, <code>Logit</code> in this case, is only well-defined as a map <code>(0, 1) → ℝ</code> for the <em>open</em> interval <code>(0, 1)</code>. This is of course not an implementation detail. <code>ℝ</code> is itself open, thus no continuous bijection exists from a <em>closed</em> interval to <code>ℝ</code>. But since the boundaries of a closed interval has what's known as measure zero, this doesn't end up affecting the resulting density with support on the entire real line. In practice, this means that</p><pre><code class="language-julia-repl hljs" style="display:block;">julia> td = transformed(Beta())</code><code class="nohighlight hljs ansi" style="display:block;">UnivariateTransformed{Beta{Float64}, Bijectors.Logit{Float64, Float64}}( +dist: Beta{Float64}(α=1.0, β=1.0) +transform: Bijectors.Logit{Float64, Float64}(0.0, 1.0) +)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> inverse(td.transform)(rand(rng, td))</code><code class="nohighlight hljs ansi" style="display:block;">0.8130302707446476</code></pre><p>will never result in <code>0</code> or <code>1</code> though any sample arbitrarily close to either <code>0</code> or <code>1</code> is possible. <em>Disclaimer: numerical accuracy is limited, so you might still see <code>0</code> and <code>1</code> if you're lucky.</em></p><h2 id="Multivariate-ADVI-example"><a class="docs-heading-anchor" href="#Multivariate-ADVI-example">Multivariate ADVI example</a><a id="Multivariate-ADVI-example-1"></a><a class="docs-heading-anchor-permalink" href="#Multivariate-ADVI-example" title="Permalink"></a></h2><p>We can also do <em>multivariate</em> ADVI using the <code>Stacked</code> bijector. <code>Stacked</code> gives us a way to combine univariate and/or multivariate bijectors into a singe multivariate bijector. Say you have a vector <code>x</code> of length 2 and you want to transform the first entry using <code>Exp</code> and the second entry using <code>Log</code>. <code>Stacked</code> gives you an easy and efficient way of representing such a bijector.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia> using Bijectors: SimplexBijector + + # Original distributions</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> dists = (Beta(), InverseGamma(), Dirichlet(2, 3)); + + # Construct the corresponding ranges</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> ranges = [];</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> idx = 1;</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> for i in 1:length(dists) + d = dists[i] + push!(ranges, idx:(idx + length(d) - 1)) + + global idx + idx += length(d) + end;</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> ranges + + # Base distribution; mean-field normal</code><code class="nohighlight hljs ansi" style="display:block;">3-element Vector{Any}: + 1:1 + 2:2 + 3:4</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> num_params = ranges[end][end]</code><code class="nohighlight hljs ansi" style="display:block;">4</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> d = MvNormal(zeros(num_params), ones(num_params)); + + # Construct the transform</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> bs = bijector.(dists); # constrained-to-unconstrained bijectors for dists</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> ibs = inverse.(bs); # invert, so we get unconstrained-to-constrained</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> sb = Stacked(ibs, ranges) # => Stacked <: Bijector + + # Mean-field normal with unconstrained-to-constrained stacked bijector</code><code class="nohighlight hljs ansi" style="display:block;">Stacked(Any[Inverse{Bijectors.Logit{Float64, Float64}}(Bijectors.Logit{Float64, Float64}(0.0, 1.0)), Base.Fix1{typeof(broadcast), typeof(exp)}(broadcast, exp), Inverse{Bijectors.SimplexBijector}(Bijectors.SimplexBijector())], Any[1:1, 2:2, 3:4], Any[1:1, 2:2, 3:5])</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> td = transformed(d, sb);</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> y = rand(td)</code><code class="nohighlight hljs ansi" style="display:block;">5-element Vector{Float64}: + 0.32311287605143296 + 0.5037012025416184 + 0.7704792724267555 + 0.16710798430605445 + 0.06241274326719004</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> 0.0 ≤ y[1] ≤ 1.0</code><code class="nohighlight hljs ansi" style="display:block;">true</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> 0.0 < y[2]</code><code class="nohighlight hljs ansi" style="display:block;">true</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> sum(y[3:4]) ≈ 1.0</code><code class="nohighlight hljs ansi" style="display:block;">false</code></pre><h2 id="Normalizing-flows"><a class="docs-heading-anchor" href="#Normalizing-flows">Normalizing flows</a><a id="Normalizing-flows-1"></a><a class="docs-heading-anchor-permalink" href="#Normalizing-flows" title="Permalink"></a></h2><p>A very interesting application is that of <em>normalizing flows</em>.[1] Usually this is done by sampling from a multivariate normal distribution, and then transforming this to a target distribution using invertible neural networks. Currently there are two such transforms available in Bijectors.jl: <code>PlanarLayer</code> and <code>RadialLayer</code>. Let's create a flow with a single <code>PlanarLayer</code>:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia> d = MvNormal(zeros(2), ones(2));</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> b = PlanarLayer(2)</code><code class="nohighlight hljs ansi" style="display:block;">PlanarLayer(w = [0.5890388815974733, -0.1812320016016404], u = [-0.6648710511255393, 0.8124262111038535], b = [0.2008048790720398])</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> flow = transformed(d, b)</code><code class="nohighlight hljs ansi" style="display:block;">MultivariateTransformed{DiagNormal, PlanarLayer{Vector{Float64}, Vector{Float64}}}( +dist: DiagNormal( +dim: 2 +μ: [0.0, 0.0] +Σ: [1.0 0.0; 0.0 1.0] +) + +transform: PlanarLayer(w = [0.5890388815974733, -0.1812320016016404], u = [-0.6648710511255393, 0.8124262111038535], b = [0.2008048790720398]) +)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> flow isa MultivariateDistribution</code><code class="nohighlight hljs ansi" style="display:block;">true</code></pre><p>That's it. Now we can sample from it using <code>rand</code> and compute the <code>logpdf</code>, like any other <code>Distribution</code>.</p><pre><code class="language-julia-repl hljs" style="display:block;">julia> y = rand(rng, flow)</code><code class="nohighlight hljs ansi" style="display:block;">2-element Vector{Float64}: + -0.4912213393668016 + 0.22894864670382697</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> logpdf(flow, y) # uses inverse of `b`</code><code class="nohighlight hljs ansi" style="display:block;">-1.4662639662622385</code></pre><p>Similarily to the multivariate ADVI example, we could use <code>Stacked</code> to get a <em>bounded</em> flow:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia> d = MvNormal(zeros(2), ones(2));</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> ibs = inverse.(bijector.((InverseGamma(2, 3), Beta())));</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> sb = Stacked(ibs) # == Stacked(ibs, [i:i for i = 1:length(ibs)]</code><code class="nohighlight hljs ansi" style="display:block;">Stacked((Base.Fix1{typeof(broadcast), typeof(exp)}(broadcast, exp), Inverse{Bijectors.Logit{Float64, Float64}}(Bijectors.Logit{Float64, Float64}(0.0, 1.0))), (1:1, 2:2), (1:1, 2:2))</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> b = sb ∘ PlanarLayer(2)</code><code class="nohighlight hljs ansi" style="display:block;">Stacked((Base.Fix1{typeof(broadcast), typeof(exp)}(broadcast, exp), Inverse{Bijectors.Logit{Float64, Float64}}(Bijectors.Logit{Float64, Float64}(0.0, 1.0))), (1:1, 2:2), (1:1, 2:2)) ∘ PlanarLayer(w = [-0.47796686421986656, 0.6063420702660066], u = [-0.47131965879318943, 0.455586956707409], b = [-0.9852945411381964])</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> td = transformed(d, b);</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> y = rand(rng, td)</code><code class="nohighlight hljs ansi" style="display:block;">2-element Vector{Float64}: + 4.08593799269824 + 0.7965225716038887</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> 0 < y[1]</code><code class="nohighlight hljs ansi" style="display:block;">true</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> 0 ≤ y[2] ≤ 1</code><code class="nohighlight hljs ansi" style="display:block;">true</code></pre><p>Want to fit the flow?</p><pre><code class="language-julia-repl hljs" style="display:block;">julia> using Zygote + + # Construct the flow.</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> b = PlanarLayer(2) + + # Convenient for extracting parameters and reconstructing the flow.</code><code class="nohighlight hljs ansi" style="display:block;">PlanarLayer(w = [1.1385218112571092, 1.5355226884562456], u = [0.6267044784437085, 1.7435020548820541], b = [-2.759090750583682])</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> using Functors</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> θs, reconstruct = Functors.functor(b); + + # Make the objective a `struct` to avoid capturing global variables.</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> struct NLLObjective{R,D,T} + reconstruct::R + basedist::D + data::T + end</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> function (obj::NLLObjective)(θs) + transformed_dist = transformed(obj.basedist, obj.reconstruct(θs)) + return -sum(Base.Fix1(logpdf, transformed_dist), eachcol(obj.data)) + end + + # Some random data to estimate the density of.</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> xs = randn(2, 1000); + + # Construct the objective.</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> f = NLLObjective(reconstruct, MvNormal(2, 1), xs); + + # Initial loss.</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> @info "Initial loss: $(f(θs))" + + # Train using gradient descent.</code><code class="nohighlight hljs ansi" style="display:block;">[ Info: Initial loss: 3598.201625352348</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> ε = 1e-3;</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> for i in 1:100 + (∇s,) = Zygote.gradient(f, θs) + θs = fmap(θs, ∇s) do θ, ∇ + θ - ε .* ∇ + end + end + + # Final loss</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> @info "Final loss: $(f(θs))" + + # Very simple check to see if we learned something useful.</code><code class="nohighlight hljs ansi" style="display:block;">[ Info: Final loss: 2807.0942318268967</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> samples = rand(transformed(f.basedist, f.reconstruct(θs)), 1000);</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> mean(eachcol(samples)) # ≈ [0, 0]</code><code class="nohighlight hljs ansi" style="display:block;">2-element Vector{Float64}: + 0.019718854385928804 + 0.055629487376851375</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> cov(samples; dims=2) # ≈ I</code><code class="nohighlight hljs ansi" style="display:block;">2×2 Matrix{Float64}: + 1.01148 -0.0223856 + -0.0223856 0.965967</code></pre><p>We can easily create more complex flows by simply doing <code>PlanarLayer(10) ∘ PlanarLayer(10) ∘ RadialLayer(10)</code> and so on.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../distributions/">« Distributions.jl integration</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Wednesday 8 January 2025 12:23">Wednesday 8 January 2025</span>. Using Julia version 1.11.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> diff --git a/previews/PR358/index.html b/previews/PR358/index.html new file mode 100644 index 00000000..aa8ddfba --- /dev/null +++ b/previews/PR358/index.html @@ -0,0 +1,40 @@ +<!DOCTYPE html> +<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Home · Bijectors</title><script data-outdated-warner src="assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.045/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="assets/documenter.js"></script><script src="siteinfo.js"></script><script src="../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href>Bijectors</a></span></div><form class="docs-search" action="search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li class="is-active"><a class="tocitem" href>Home</a></li><li><a class="tocitem" href="transforms/">Transforms</a></li><li><a class="tocitem" href="distributions/">Distributions.jl integration</a></li><li><a class="tocitem" href="examples/">Examples</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Home</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Home</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/TuringLang/Bijectors.jl/blob/master/docs/src/index.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="Bijectors.jl"><a class="docs-heading-anchor" href="#Bijectors.jl">Bijectors.jl</a><a id="Bijectors.jl-1"></a><a class="docs-heading-anchor-permalink" href="#Bijectors.jl" title="Permalink"></a></h1><p>This package implements a set of functions for transforming constrained random variables (e.g. simplexes, intervals) to Euclidean space. The 3 main functions implemented in this package are the <code>link</code>, <code>invlink</code> and <code>logpdf_with_trans</code> for a number of distributions.</p><article class="docstring"><header><a class="docstring-binding" id="Bijectors.link" href="#Bijectors.link"><code>Bijectors.link</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">link(d::Distribution, x)</code></pre><p>Transforms the input <code>x</code> using the constrained-to-unconstrained bijector for distribution <code>d</code>.</p><p>See also: <a href="#Bijectors.invlink"><code>invlink</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia> using Bijectors + +julia> d = LogNormal() # support is (0, Inf) +LogNormal{Float64}(μ=0.0, σ=1.0) + +julia> b = bijector(d) # log function transforms to unconstrained space +(::Base.Fix1{typeof(broadcast), typeof(log)}) (generic function with 1 method) + +julia> b(1.0) +0.0 + +julia> link(LogNormal(), 1.0) +0.0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/TuringLang/Bijectors.jl/blob/22713b3cb349f7be40e2243f127cc9f0c6608cce/src/Bijectors.jl#L123-L148">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="Bijectors.invlink" href="#Bijectors.invlink"><code>Bijectors.invlink</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">invlink(d::Distribution, y)</code></pre><p>Performs the inverse transform on a value <code>y</code> that was transformed using the constrained-to-unconstrained bijector for distribution <code>d</code>.</p><p>It should hold that <code>invlink(d, link(d, x)) = x</code>.</p><p>See also: <a href="#Bijectors.link"><code>link</code></a>.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia> using Bijectors + +julia> d = LogNormal() # support is (0, Inf) +LogNormal{Float64}(μ=0.0, σ=1.0) + +julia> link(LogNormal(), 1.0) # uses a log transform +0.0 + +julia> invlink(LogNormal(), 0.0) +1.0</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/TuringLang/Bijectors.jl/blob/22713b3cb349f7be40e2243f127cc9f0c6608cce/src/Bijectors.jl#L151-L175">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="Bijectors.logpdf_with_trans" href="#Bijectors.logpdf_with_trans"><code>Bijectors.logpdf_with_trans</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">logpdf_with_trans(d::Distribution, x, transform::Bool)</code></pre><p>If <code>transform</code> is <code>false</code>, <code>logpdf_with_trans</code> calculates the log probability density function (logpdf) of distribution <code>d</code> at <code>x</code>.</p><p>If <code>transform</code> is <code>true</code>, <code>x</code> is transformed using the constrained-to-unconstrained bijector for distribution <code>d</code>, and then the logpdf of the resulting value is calculated with respect to the unconstrained (transformed) distribution. Equivalently, if <code>x</code> is distributed according to <code>d</code> and <code>y = link(d, x)</code> is distributed according to <code>td = transformed(d)</code>, then <code>logpdf_with_trans(d, x, true) = logpdf(td, y)</code>. This is accomplished by subtracting the log Jacobian of the transformation.</p><p><strong>Example</strong></p><pre><code class="language-julia-repl hljs">julia> using Bijectors + +julia> logpdf_with_trans(LogNormal(), ℯ, false) +-2.4189385332046727 + +julia> logpdf(LogNormal(), ℯ) # Same as above +-2.4189385332046727 + +julia> logpdf_with_trans(LogNormal(), ℯ, true) +-1.4189385332046727 + +julia> # If x ~ LogNormal(), then log(x) ~ Normal() + logpdf(Normal(), 1.0) +-1.4189385332046727 + +julia> # The difference between the two is due to the Jacobian + logabsdetjac(bijector(LogNormal()), ℯ) +-1</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/TuringLang/Bijectors.jl/blob/22713b3cb349f7be40e2243f127cc9f0c6608cce/src/Bijectors.jl#L205-L241">source</a></section></article><p>The distributions supported are:</p><ol><li><code>RealDistribution</code>: <code>Union{Cauchy, Gumbel, Laplace, Logistic, NoncentralT, Normal, NormalCanon, TDist}</code>,</li><li><code>PositiveDistribution</code>: <code>Union{BetaPrime, Chi, Chisq, Erlang, Exponential, FDist, Frechet, Gamma, InverseGamma, InverseGaussian, Kolmogorov, LogNormal, NoncentralChisq, NoncentralF, Rayleigh, Weibull}</code>,</li><li><code>UnitDistribution</code>: <code>Union{Beta, KSOneSided, NoncentralBeta}</code>,</li><li><code>SimplexDistribution</code>: <code>Union{Dirichlet}</code>,</li><li><code>PDMatDistribution</code>: <code>Union{InverseWishart, Wishart}</code>, and</li><li><code>TransformDistribution</code>: <code>Union{T, Truncated{T}} where T<:ContinuousUnivariateDistribution</code>.</li></ol><p>All exported names from the <a href="https://juliastats.org/Distributions.jl/stable/">Distributions.jl</a> package are reexported from <code>Bijectors</code>.</p><p>Bijectors.jl also provides a nice interface for working with these maps: composition, inversion, etc. The following table lists mathematical operations for a bijector and the corresponding code in Bijectors.jl.</p><table><tr><th style="text-align: center">Operation</th><th style="text-align: center">Method</th><th style="text-align: center">Automatic</th></tr><tr><td style="text-align: center"><code>b ↦ b⁻¹</code></td><td style="text-align: center"><code>inverse(b)</code></td><td style="text-align: center">✓</td></tr><tr><td style="text-align: center"><code>(b₁, b₂) ↦ (b₁ ∘ b₂)</code></td><td style="text-align: center"><code>b₁ ∘ b₂</code></td><td style="text-align: center">✓</td></tr><tr><td style="text-align: center"><code>(b₁, b₂) ↦ [b₁, b₂]</code></td><td style="text-align: center"><code>stack(b₁, b₂)</code></td><td style="text-align: center">✓</td></tr><tr><td style="text-align: center"><code>x ↦ b(x)</code></td><td style="text-align: center"><code>b(x)</code></td><td style="text-align: center">×</td></tr><tr><td style="text-align: center"><code>y ↦ b⁻¹(y)</code></td><td style="text-align: center"><code>inverse(b)(y)</code></td><td style="text-align: center">×</td></tr><tr><td style="text-align: center"><code>x ↦ log|det J(b, x)|</code></td><td style="text-align: center"><code>logabsdetjac(b, x)</code></td><td style="text-align: center">AD</td></tr><tr><td style="text-align: center"><code>x ↦ b(x), log|det J(b, x)|</code></td><td style="text-align: center"><code>with_logabsdet_jacobian(b, x)</code></td><td style="text-align: center">✓</td></tr><tr><td style="text-align: center"><code>p ↦ q := b_* p</code></td><td style="text-align: center"><code>q = transformed(p, b)</code></td><td style="text-align: center">✓</td></tr><tr><td style="text-align: center"><code>y ∼ q</code></td><td style="text-align: center"><code>y = rand(q)</code></td><td style="text-align: center">✓</td></tr><tr><td style="text-align: center"><code>p ↦ b</code> such that <code>support(b_* p) = ℝᵈ</code></td><td style="text-align: center"><code>bijector(p)</code></td><td style="text-align: center">✓</td></tr><tr><td style="text-align: center"><code>(x ∼ p, b(x), log|det J(b, x)|, log q(y))</code></td><td style="text-align: center"><code>forward(q)</code></td><td style="text-align: center">✓</td></tr></table><p>In this table, <code>b</code> denotes a <code>Bijector</code>, <code>J(b, x)</code> denotes the Jacobian of <code>b</code> evaluated at <code>x</code>, <code>b_*</code> denotes the <a href="https://www.wikiwand.com/en/Pushforward_measure">push-forward</a> of <code>p</code> by <code>b</code>, and <code>x ∼ p</code> denotes <code>x</code> sampled from the distribution with density <code>p</code>.</p><p>The "Automatic" column in the table refers to whether or not you are required to implement the feature for a custom <code>Bijector</code>. "AD" refers to the fact that it can be implemented "automatically" using automatic differentiation.</p></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="transforms/">Transforms »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Wednesday 8 January 2025 12:23">Wednesday 8 January 2025</span>. Using Julia version 1.11.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> diff --git a/previews/PR358/search/index.html b/previews/PR358/search/index.html new file mode 100644 index 00000000..f42a4e51 --- /dev/null +++ b/previews/PR358/search/index.html @@ -0,0 +1,2 @@ +<!DOCTYPE html> +<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Search · Bijectors</title><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.045/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../">Bijectors</a></span></div><form class="docs-search" action><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li><a class="tocitem" href="../transforms/">Transforms</a></li><li><a class="tocitem" href="../distributions/">Distributions.jl integration</a></li><li><a class="tocitem" href="../examples/">Examples</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Search</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Search</a></li></ul></nav><div class="docs-right"><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article><p id="documenter-search-info">Loading search...</p><ul id="documenter-search-results"></ul></article><nav class="docs-footer"><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Wednesday 8 January 2025 12:23">Wednesday 8 January 2025</span>. Using Julia version 1.11.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body><script src="../search_index.js"></script><script src="../assets/search.js"></script></html> diff --git a/previews/PR358/search_index.js b/previews/PR358/search_index.js new file mode 100644 index 00000000..9822814c --- /dev/null +++ b/previews/PR358/search_index.js @@ -0,0 +1,3 @@ +var documenterSearchIndex = {"docs": +[{"location":"transforms/#Usage","page":"Transforms","title":"Usage","text":"","category":"section"},{"location":"transforms/","page":"Transforms","title":"Transforms","text":"A very simple example of a \"bijector\"/diffeomorphism, i.e. a differentiable transformation with a differentiable inverse, is the exp function:","category":"page"},{"location":"transforms/","page":"Transforms","title":"Transforms","text":"The inverse of exp is log.\nThe derivative of exp at an input x is simply exp(x), hence logabsdetjac is simply x.","category":"page"},{"location":"transforms/","page":"Transforms","title":"Transforms","text":"using Bijectors\ntransform(exp, 1.0)\nlogabsdetjac(exp, 1.0)\nwith_logabsdet_jacobian(exp, 1.0)","category":"page"},{"location":"transforms/","page":"Transforms","title":"Transforms","text":"Some transformations are well-defined for different types of inputs, e.g. exp can also act elementwise on an N-dimensional Array{<:Real,N}. To specify that a transformation should act elementwise, we use the elementwise method:","category":"page"},{"location":"transforms/","page":"Transforms","title":"Transforms","text":"x = ones(2, 2)\ntransform(elementwise(exp), x)\nlogabsdetjac(elementwise(exp), x)\nwith_logabsdet_jacobian(elementwise(exp), x)","category":"page"},{"location":"transforms/","page":"Transforms","title":"Transforms","text":"These methods also work nicely for compositions of transformations:","category":"page"},{"location":"transforms/","page":"Transforms","title":"Transforms","text":"transform(elementwise(log ∘ exp), x)","category":"page"},{"location":"transforms/","page":"Transforms","title":"Transforms","text":"Unlike exp, some transformations have parameters affecting the resulting transformation they represent, e.g. Logit has two parameters a and b representing the lower- and upper-bound, respectively, of its domain:","category":"page"},{"location":"transforms/","page":"Transforms","title":"Transforms","text":"using Bijectors: Logit\n\nf = Logit(0.0, 1.0)\nf(rand()) # takes us from `(0, 1)` to `(-∞, ∞)`","category":"page"},{"location":"transforms/#User-facing-methods","page":"Transforms","title":"User-facing methods","text":"","category":"section"},{"location":"transforms/","page":"Transforms","title":"Transforms","text":"Without mutation:","category":"page"},{"location":"transforms/","page":"Transforms","title":"Transforms","text":"transform\nlogabsdetjac","category":"page"},{"location":"transforms/#Bijectors.transform","page":"Transforms","title":"Bijectors.transform","text":"transform(b, x)\n\nTransform x using b, treating x as a single input.\n\n\n\n\n\n","category":"function"},{"location":"transforms/#Bijectors.logabsdetjac","page":"Transforms","title":"Bijectors.logabsdetjac","text":"logabsdetjac(b, x)\n\nReturn log(abs(det(J(b, x)))), where J(b, x) is the jacobian of b at x.\n\n\n\n\n\n","category":"function"},{"location":"transforms/","page":"Transforms","title":"Transforms","text":"with_logabsdet_jacobian","category":"page"},{"location":"transforms/","page":"Transforms","title":"Transforms","text":"With mutation:","category":"page"},{"location":"transforms/","page":"Transforms","title":"Transforms","text":"transform!\nlogabsdetjac!\nwith_logabsdet_jacobian!","category":"page"},{"location":"transforms/#Bijectors.transform!","page":"Transforms","title":"Bijectors.transform!","text":"transform!(b, x[, y])\n\nTransform x using b, storing the result in y.\n\nIf y is not provided, x is used as the output.\n\n\n\n\n\n","category":"function"},{"location":"transforms/#Bijectors.logabsdetjac!","page":"Transforms","title":"Bijectors.logabsdetjac!","text":"logabsdetjac!(b, x[, logjac])\n\nCompute log(abs(det(J(b, x)))) and store the result in logjac, where J(b, x) is the jacobian of b at x.\n\n\n\n\n\n","category":"function"},{"location":"transforms/#Bijectors.with_logabsdet_jacobian!","page":"Transforms","title":"Bijectors.with_logabsdet_jacobian!","text":"with_logabsdet_jacobian!(b, x[, y, logjac])\n\nCompute transform(b, x) and logabsdetjac(b, x), storing the result in y and logjac, respetively.\n\nIf y is not provided, then x will be used in its place.\n\nDefaults to calling with_logabsdet_jacobian(b, x) and updating y and logjac with the result.\n\n\n\n\n\n","category":"function"},{"location":"transforms/#Implementing-a-transformation","page":"Transforms","title":"Implementing a transformation","text":"","category":"section"},{"location":"transforms/","page":"Transforms","title":"Transforms","text":"Any callable can be made into a bijector by providing an implementation of ChangeOfVariables.with_logabsdet_jacobian(b, x).","category":"page"},{"location":"transforms/","page":"Transforms","title":"Transforms","text":"You can also optionally implement transform and logabsdetjac to avoid redundant computations. This is usually only worth it if you expect transform or logabsdetjac to be used heavily without the other.","category":"page"},{"location":"transforms/","page":"Transforms","title":"Transforms","text":"Similarly with the mutable versions with_logabsdet_jacobian!, transform!, and logabsdetjac!.","category":"page"},{"location":"transforms/#Working-with-Distributions.jl","page":"Transforms","title":"Working with Distributions.jl","text":"","category":"section"},{"location":"transforms/","page":"Transforms","title":"Transforms","text":"Bijectors.bijector\nBijectors.transformed(d::Distribution, b::Bijector)","category":"page"},{"location":"transforms/#Bijectors.bijector","page":"Transforms","title":"Bijectors.bijector","text":"bijector(d::Distribution)\n\nReturns the constrained-to-unconstrained bijector for distribution d.\n\n\n\n\n\n","category":"function"},{"location":"transforms/#Bijectors.transformed-Tuple{Distribution, Bijector}","page":"Transforms","title":"Bijectors.transformed","text":"transformed(d::Distribution)\ntransformed(d::Distribution, b::Bijector)\n\nCouples distribution d with the bijector b by returning a TransformedDistribution.\n\nIf no bijector is provided, i.e. transformed(d) is called, then transformed(d, bijector(d)) is returned.\n\n\n\n\n\n","category":"method"},{"location":"transforms/#Utilities","page":"Transforms","title":"Utilities","text":"","category":"section"},{"location":"transforms/","page":"Transforms","title":"Transforms","text":"Bijectors.elementwise\nBijectors.isinvertible\nBijectors.isclosedform(t::Bijectors.Transform)","category":"page"},{"location":"transforms/#Bijectors.elementwise","page":"Transforms","title":"Bijectors.elementwise","text":"elementwise(f)\n\nAlias for Base.Fix1(broadcast, f).\n\nIn the case where f::ComposedFunction, the result is Base.Fix1(broadcast, f.outer) ∘ Base.Fix1(broadcast, f.inner) rather than Base.Fix1(broadcast, f).\n\n\n\n\n\n","category":"function"},{"location":"transforms/#Bijectors.isinvertible","page":"Transforms","title":"Bijectors.isinvertible","text":"isinvertible(t)\n\nReturn true if t is invertible, and false otherwise.\n\n\n\n\n\n","category":"function"},{"location":"transforms/#Bijectors.isclosedform-Tuple{Bijectors.Transform}","page":"Transforms","title":"Bijectors.isclosedform","text":"isclosedform(b::Transform)::bool\nisclosedform(b⁻¹::Inverse{<:Transform})::bool\n\nReturns true or false depending on whether or not evaluation of b has a closed-form implementation.\n\nMost transformations have closed-form evaluations, but there are cases where this is not the case. For example the inverse evaluation of PlanarLayer requires an iterative procedure to evaluate.\n\n\n\n\n\n","category":"method"},{"location":"transforms/#API","page":"Transforms","title":"API","text":"","category":"section"},{"location":"transforms/","page":"Transforms","title":"Transforms","text":"Bijectors.Transform\nBijectors.Bijector\nBijectors.Inverse","category":"page"},{"location":"transforms/#Bijectors.Transform","page":"Transforms","title":"Bijectors.Transform","text":"Abstract type for a transformation.\n\nImplementing\n\nA subtype of Transform of should at least implement transform(b, x).\n\nIf the Transform is also invertible:\n\nRequired:\nEither of the following:\ntransform(::Inverse{<:MyTransform}, x): the transform for its inverse.\nInverseFunctions.inverse(b::MyTransform): returns an existing Transform.\nlogabsdetjac: computes the log-abs-det jacobian factor.\nOptional:\nwith_logabsdet_jacobian: transform and logabsdetjac combined. Useful in cases where we can exploit shared computation in the two.\n\nFor the above methods, there are mutating versions which can optionally be implemented:\n\nwith_logabsdet_jacobian!\nlogabsdetjac!\nwith_logabsdet_jacobian!\n\n\n\n\n\n","category":"type"},{"location":"transforms/#Bijectors.Bijector","page":"Transforms","title":"Bijectors.Bijector","text":"Abstract type of a bijector, i.e. differentiable bijection with differentiable inverse.\n\n\n\n\n\n","category":"type"},{"location":"transforms/#Bijectors.Inverse","page":"Transforms","title":"Bijectors.Inverse","text":"inverse(b::Transform)\nInverse(b::Transform)\n\nA Transform representing the inverse transform of b.\n\n\n\n\n\n","category":"type"},{"location":"transforms/#Bijectors","page":"Transforms","title":"Bijectors","text":"","category":"section"},{"location":"transforms/","page":"Transforms","title":"Transforms","text":"Bijectors.CorrBijector\nBijectors.LeakyReLU\nBijectors.Stacked\nBijectors.RationalQuadraticSpline\nBijectors.Coupling\nBijectors.OrderedBijector\nBijectors.NamedTransform\nBijectors.NamedCoupling","category":"page"},{"location":"transforms/#Bijectors.CorrBijector","page":"Transforms","title":"Bijectors.CorrBijector","text":"CorrBijector <: Bijector\n\nA bijector implementation of Stan's parametrization method for Correlation matrix: https://mc-stan.org/docs/2_23/reference-manual/correlation-matrix-transform-section.html\n\nBasically, a unconstrained strictly upper triangular matrix y is transformed to a correlation matrix by following readable but not that efficient form:\n\nK = size(y, 1)\nz = tanh.(y)\n\nfor j=1:K, i=1:K\n if i>j\n w[i,j] = 0\n elseif 1==i==j\n w[i,j] = 1\n elseif 1<i==j\n w[i,j] = prod(sqrt(1 .- z[1:i-1, j].^2))\n elseif 1==i<j\n w[i,j] = z[i,j]\n elseif 1<i<j\n w[i,j] = z[i,j] * prod(sqrt(1 .- z[1:i-1, j].^2))\n end\nend\n\nIt is easy to see that every column is a unit vector, for example:\n\nw3' w3 ==\nw[1,3]^2 + w[2,3]^2 + w[3,3]^2 ==\nz[1,3]^2 + (z[2,3] * sqrt(1 - z[1,3]^2))^2 + (sqrt(1-z[1,3]^2) * sqrt(1-z[2,3]^2))^2 ==\nz[1,3]^2 + z[2,3]^2 * (1-z[1,3]^2) + (1-z[1,3]^2) * (1-z[2,3]^2) ==\nz[1,3]^2 + z[2,3]^2 - z[2,3]^2 * z[1,3]^2 + 1 -z[1,3]^2 - z[2,3]^2 + z[1,3]^2 * z[2,3]^2 ==\n1\n\nAnd diagonal elements are positive, so w is a cholesky factor for a positive matrix.\n\nx = w' * w\n\nConsider block matrix representation for x\n\nx = [w1'; w2'; ... wn'] * [w1 w2 ... wn] == \n[w1'w1 w1'w2 ... w1'wn;\n w2'w1 w2'w2 ... w2'wn;\n ...\n]\n\nThe diagonal elements are given by wk'wk = 1, thus x is a correlation matrix.\n\nEvery step is invertible, so this is a bijection(bijector).\n\nNote: The implementation doesn't follow their \"manageable expression\" directly, because their equation seems wrong (7/30/2020). Insteadly it follows definition above the \"manageable expression\" directly, which is also described in above doc.\n\n\n\n\n\n","category":"type"},{"location":"transforms/#Bijectors.LeakyReLU","page":"Transforms","title":"Bijectors.LeakyReLU","text":"LeakyReLU{T}(α::T) <: Bijector\n\nDefines the invertible mapping\n\nx ↦ x if x ≥ 0 else αx\n\nwhere α > 0.\n\n\n\n\n\n","category":"type"},{"location":"transforms/#Bijectors.Stacked","page":"Transforms","title":"Bijectors.Stacked","text":"Stacked(bs)\nStacked(bs, ranges)\nstack(bs::Bijector...)\n\nA Bijector which stacks bijectors together which can then be applied to a vector where bs[i]::Bijector is applied to x[ranges[i]]::UnitRange{Int}.\n\nArguments\n\nbs can be either a Tuple or an AbstractArray of 0- and/or 1-dimensional bijectors\nIf bs is a Tuple, implementations are type-stable using generated functions\nIf bs is an AbstractArray, implementations are not type-stable and use iterative methods\nranges needs to be an iterable consisting of UnitRange{Int}\nlength(bs) == length(ranges) needs to be true.\n\nExamples\n\nb1 = Logit(0.0, 1.0)\nb2 = identity\nb = stack(b1, b2)\nb([0.0, 1.0]) == [b1(0.0), 1.0] # => true\n\n\n\n\n\n","category":"type"},{"location":"transforms/#Bijectors.RationalQuadraticSpline","page":"Transforms","title":"Bijectors.RationalQuadraticSpline","text":"RationalQuadraticSpline{T} <: Bijector\n\nImplementation of the Rational Quadratic Spline flow [1].\n\nOutside of the interval [minimum(widths), maximum(widths)], this mapping is given by the identity map. \nInside the interval it's given by a monotonic spline (i.e. monotonic polynomials connected at intermediate points) with endpoints fixed so as to continuously transform into the identity map.\n\nFor the sake of efficiency, there are separate implementations for 0-dimensional and 1-dimensional inputs.\n\nNotes\n\nThere are two constructors for RationalQuadraticSpline:\n\nRationalQuadraticSpline(widths, heights, derivatives): it is assumed that widths, \n\nheights, and derivatives satisfy the constraints that makes this a valid bijector, i.e.\n\nwidths: monotonically increasing and length(widths) == K,\nheights: monotonically increasing and length(heights) == K,\nderivatives: non-negative and derivatives[1] == derivatives[end] == 1.\nRationalQuadraticSpline(widths, heights, derivatives, B): other than than the lengths, no assumptions are made on parameters. Therefore we will transform the parameters s.t.:\nwidths_new ∈ [-B, B]ᴷ⁺¹, where K == length(widths),\nheights_new ∈ [-B, B]ᴷ⁺¹, where K == length(heights),\nderivatives_new ∈ (0, ∞)ᴷ⁺¹ with derivatives_new[1] == derivates_new[end] == 1, where (K - 1) == length(derivatives).\n\nExamples\n\nUnivariate\n\njulia> using StableRNGs: StableRNG; rng = StableRNG(42); # For reproducibility.\n\njulia> using Bijectors: RationalQuadraticSpline\n\njulia> K = 3; B = 2;\n\njulia> # Monotonic spline on '[-B, B]' with `K` intermediate knots/\"connection points\".\n b = RationalQuadraticSpline(randn(rng, K), randn(rng, K), randn(rng, K - 1), B);\n\njulia> b(0.5) # inside of `[-B, B]` → transformed\n1.1943325397834206\n\njulia> b(5.) # outside of `[-B, B]` → not transformed\n5.0\n\njulia> b = RationalQuadraticSpline(b.widths, b.heights, b.derivatives);\n\njulia> b(0.5) # inside of `[-B, B]` → transformed\n1.1943325397834206\n\njulia> d = 2; K = 3; B = 2;\n\njulia> b = RationalQuadraticSpline(randn(rng, d, K), randn(rng, d, K), randn(rng, d, K - 1), B);\n\njulia> b([-1., 1.])\n2-element Vector{Float64}:\n -1.5660106244288925\n 0.5384702734738573\n\njulia> b([-5., 5.])\n2-element Vector{Float64}:\n -5.0\n 5.0\n\njulia> b([-1., 5.])\n2-element Vector{Float64}:\n -1.5660106244288925\n 5.0\n\nReferences\n\n[1] Durkan, C., Bekasov, A., Murray, I., & Papamakarios, G., Neural Spline Flows, CoRR, arXiv:1906.04032 [stat.ML], (2019). \n\n\n\n\n\n","category":"type"},{"location":"transforms/#Bijectors.Coupling","page":"Transforms","title":"Bijectors.Coupling","text":"Coupling{F, M}(θ::F, mask::M)\n\nImplements a coupling-layer as defined in [1].\n\nExamples\n\njulia> using Bijectors: Shift, Coupling, PartitionMask, coupling, couple\n\njulia> m = PartitionMask(3, [1], [2]); # <= going to use x[2] to parameterize transform of x[1]\n\njulia> cl = Coupling(Shift, m); # <= will do `y[1:1] = x[1:1] + x[2:2]`;\n\njulia> x = [1., 2., 3.];\n\njulia> cl(x)\n3-element Vector{Float64}:\n 3.0\n 2.0\n 3.0\n\njulia> inverse(cl)(cl(x))\n3-element Vector{Float64}:\n 1.0\n 2.0\n 3.0\n\njulia> coupling(cl) # get the `Bijector` map `θ -> b(⋅, θ)`\nShift\n\njulia> couple(cl, x) # get the `Bijector` resulting from `x`\nShift{Vector{Float64}}([2.0])\n\njulia> with_logabsdet_jacobian(cl, x)\n([3.0, 2.0, 3.0], 0.0)\n\nReferences\n\n[1] Kobyzev, I., Prince, S., & Brubaker, M. A., Normalizing flows: introduction and ideas, CoRR, (), (2019). \n\n\n\n\n\n","category":"type"},{"location":"transforms/#Bijectors.OrderedBijector","page":"Transforms","title":"Bijectors.OrderedBijector","text":"OrderedBijector()\n\nA bijector mapping unordered vectors in ℝᵈ to ordered vectors in ℝᵈ.\n\nSee also\n\nStan's documentation\nNote that this transformation and its inverse are the opposite of in this reference.\n\n\n\n\n\n","category":"type"},{"location":"transforms/#Bijectors.NamedTransform","page":"Transforms","title":"Bijectors.NamedTransform","text":"NamedTransform <: AbstractNamedTransform\n\nWraps a NamedTuple of key -> Bijector pairs, implementing evaluation, inversion, etc.\n\nExamples\n\njulia> using Bijectors: NamedTransform, Scale\n\njulia> b = NamedTransform((a = Scale(2.0), b = exp));\n\njulia> x = (a = 1., b = 0., c = 42.);\n\njulia> b(x)\n(a = 2.0, b = 1.0, c = 42.0)\n\njulia> (a = 2 * x.a, b = exp(x.b), c = x.c)\n(a = 2.0, b = 1.0, c = 42.0)\n\n\n\n\n\n","category":"type"},{"location":"transforms/#Bijectors.NamedCoupling","page":"Transforms","title":"Bijectors.NamedCoupling","text":"NamedCoupling{target, deps, F} <: AbstractNamedTransform\n\nImplements a coupling layer for named bijectors.\n\nSee also: Coupling\n\nExamples\n\njulia> using Bijectors: NamedCoupling, Scale\n\njulia> b = NamedCoupling(:b, (:a, :c), (a, c) -> Scale(a + c));\n\njulia> x = (a = 1., b = 2., c = 3.);\n\njulia> b(x)\n(a = 1.0, b = 8.0, c = 3.0)\n\njulia> (a = x.a, b = (x.a + x.c) * x.b, c = x.c)\n(a = 1.0, b = 8.0, c = 3.0)\n\n\n\n\n\n","category":"type"},{"location":"examples/","page":"Examples","title":"Examples","text":"using Bijectors","category":"page"},{"location":"examples/#Univariate-ADVI-example","page":"Examples","title":"Univariate ADVI example","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"But the real utility of TransformedDistribution becomes more apparent when using transformed(dist, b) for any bijector b. To get the transformed distribution corresponding to the Beta(2, 2), we called transformed(dist) before. This is simply an alias for transformed(dist, bijector(dist)). Remember bijector(dist) returns the constrained-to-constrained bijector for that particular Distribution. But we can of course construct a TransformedDistribution using different bijectors with the same dist. This is particularly useful in something called Automatic Differentiation Variational Inference (ADVI).[2] An important part of ADVI is to approximate a constrained distribution, e.g. Beta, as follows:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Sample x from a Normal with parameters μ and σ, i.e. x ~ Normal(μ, σ).\nTransform x to y s.t. y ∈ support(Beta), with the transform being a differentiable bijection with a differentiable inverse (a \"bijector\")","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"This then defines a probability density with same support as Beta! Of course, it's unlikely that it will be the same density, but it's an approximation. Creating such a distribution becomes trivial with Bijector and TransformedDistribution:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"using StableRNGs: StableRNG\nrng = StableRNG(42);\ndist = Beta(2, 2)\nb = bijector(dist) # (0, 1) → ℝ\nb⁻¹ = inverse(b) # ℝ → (0, 1)\ntd = transformed(Normal(), b⁻¹) # x ∼ 𝓝(0, 1) then b(x) ∈ (0, 1)\nx = rand(rng, td) # ∈ (0, 1)","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"It's worth noting that support(Beta) is the closed interval [0, 1], while the constrained-to-unconstrained bijection, Logit in this case, is only well-defined as a map (0, 1) → ℝ for the open interval (0, 1). This is of course not an implementation detail. ℝ is itself open, thus no continuous bijection exists from a closed interval to ℝ. But since the boundaries of a closed interval has what's known as measure zero, this doesn't end up affecting the resulting density with support on the entire real line. In practice, this means that","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"td = transformed(Beta())\ninverse(td.transform)(rand(rng, td))","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"will never result in 0 or 1 though any sample arbitrarily close to either 0 or 1 is possible. Disclaimer: numerical accuracy is limited, so you might still see 0 and 1 if you're lucky.","category":"page"},{"location":"examples/#Multivariate-ADVI-example","page":"Examples","title":"Multivariate ADVI example","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"We can also do multivariate ADVI using the Stacked bijector. Stacked gives us a way to combine univariate and/or multivariate bijectors into a singe multivariate bijector. Say you have a vector x of length 2 and you want to transform the first entry using Exp and the second entry using Log. Stacked gives you an easy and efficient way of representing such a bijector.","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"using Bijectors: SimplexBijector\n\n# Original distributions\ndists = (Beta(), InverseGamma(), Dirichlet(2, 3));\n\n# Construct the corresponding ranges\nranges = [];\nidx = 1;\n\nfor i in 1:length(dists)\n d = dists[i]\n push!(ranges, idx:(idx + length(d) - 1))\n\n global idx\n idx += length(d)\nend;\n\nranges\n\n# Base distribution; mean-field normal\nnum_params = ranges[end][end]\n\nd = MvNormal(zeros(num_params), ones(num_params));\n\n# Construct the transform\nbs = bijector.(dists); # constrained-to-unconstrained bijectors for dists\nibs = inverse.(bs); # invert, so we get unconstrained-to-constrained\nsb = Stacked(ibs, ranges) # => Stacked <: Bijector\n\n# Mean-field normal with unconstrained-to-constrained stacked bijector\ntd = transformed(d, sb);\ny = rand(td)\n0.0 ≤ y[1] ≤ 1.0\n0.0 < y[2]\nsum(y[3:4]) ≈ 1.0","category":"page"},{"location":"examples/#Normalizing-flows","page":"Examples","title":"Normalizing flows","text":"","category":"section"},{"location":"examples/","page":"Examples","title":"Examples","text":"A very interesting application is that of normalizing flows.[1] Usually this is done by sampling from a multivariate normal distribution, and then transforming this to a target distribution using invertible neural networks. Currently there are two such transforms available in Bijectors.jl: PlanarLayer and RadialLayer. Let's create a flow with a single PlanarLayer:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"using Bijectors\nusing StableRNGs: StableRNG\nrng = StableRNG(42);","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"d = MvNormal(zeros(2), ones(2));\nb = PlanarLayer(2)\nflow = transformed(d, b)\nflow isa MultivariateDistribution","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"That's it. Now we can sample from it using rand and compute the logpdf, like any other Distribution.","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"y = rand(rng, flow)\nlogpdf(flow, y) # uses inverse of `b`","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Similarily to the multivariate ADVI example, we could use Stacked to get a bounded flow:","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"d = MvNormal(zeros(2), ones(2));\nibs = inverse.(bijector.((InverseGamma(2, 3), Beta())));\nsb = Stacked(ibs) # == Stacked(ibs, [i:i for i = 1:length(ibs)]\nb = sb ∘ PlanarLayer(2)\ntd = transformed(d, b);\ny = rand(rng, td)\n0 < y[1]\n0 ≤ y[2] ≤ 1","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"Want to fit the flow?","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"using Zygote\n\n# Construct the flow.\nb = PlanarLayer(2)\n\n# Convenient for extracting parameters and reconstructing the flow.\nusing Functors\nθs, reconstruct = Functors.functor(b);\n\n# Make the objective a `struct` to avoid capturing global variables.\nstruct NLLObjective{R,D,T}\n reconstruct::R\n basedist::D\n data::T\nend\n\nfunction (obj::NLLObjective)(θs)\n transformed_dist = transformed(obj.basedist, obj.reconstruct(θs))\n return -sum(Base.Fix1(logpdf, transformed_dist), eachcol(obj.data))\nend\n\n# Some random data to estimate the density of.\nxs = randn(2, 1000);\n\n# Construct the objective.\nf = NLLObjective(reconstruct, MvNormal(2, 1), xs);\n\n# Initial loss.\n@info \"Initial loss: $(f(θs))\"\n\n# Train using gradient descent.\nε = 1e-3;\nfor i in 1:100\n (∇s,) = Zygote.gradient(f, θs)\n θs = fmap(θs, ∇s) do θ, ∇\n θ - ε .* ∇\n end\nend\n\n# Final loss\n@info \"Final loss: $(f(θs))\"\n\n# Very simple check to see if we learned something useful.\nsamples = rand(transformed(f.basedist, f.reconstruct(θs)), 1000);\nmean(eachcol(samples)) # ≈ [0, 0]\ncov(samples; dims=2) # ≈ I","category":"page"},{"location":"examples/","page":"Examples","title":"Examples","text":"We can easily create more complex flows by simply doing PlanarLayer(10) ∘ PlanarLayer(10) ∘ RadialLayer(10) and so on.","category":"page"},{"location":"distributions/#Basic-usage","page":"Distributions.jl integration","title":"Basic usage","text":"","category":"section"},{"location":"distributions/","page":"Distributions.jl integration","title":"Distributions.jl integration","text":"Other than the logpdf_with_trans methods, the package also provides a more composable interface through the Bijector types. Consider for example the one from above with Beta(2, 2).","category":"page"},{"location":"distributions/","page":"Distributions.jl integration","title":"Distributions.jl integration","text":"julia> using Random;\n Random.seed!(42);\n\njulia> using Bijectors;\n using Bijectors: Logit;\n\njulia> dist = Beta(2, 2)\nBeta{Float64}(α=2.0, β=2.0)\n\njulia> x = rand(dist)\n0.36888689965963756\n\njulia> b = bijector(dist) # bijection (0, 1) → ℝ\nLogit{Float64}(0.0, 1.0)\n\njulia> y = b(x)\n-0.5369949942509267","category":"page"},{"location":"distributions/","page":"Distributions.jl integration","title":"Distributions.jl integration","text":"In this case we see that bijector(d::Distribution) returns the corresponding constrained-to-unconstrained bijection for Beta, which indeed is a Logit with a = 0.0 and b = 1.0. The resulting Logit <: Bijector has a method (b::Logit)(x) defined, allowing us to call it just like any other function. Comparing with the above example, b(x) ≈ link(dist, x). Just to convince ourselves:","category":"page"},{"location":"distributions/","page":"Distributions.jl integration","title":"Distributions.jl integration","text":"julia> b(x) ≈ link(dist, x)\ntrue","category":"page"},{"location":"distributions/#Transforming-distributions","page":"Distributions.jl integration","title":"Transforming distributions","text":"","category":"section"},{"location":"distributions/","page":"Distributions.jl integration","title":"Distributions.jl integration","text":"using Bijectors","category":"page"},{"location":"distributions/","page":"Distributions.jl integration","title":"Distributions.jl integration","text":"We can create a transformed Distribution, i.e. a Distribution defined by sampling from a given Distribution and then transforming using a given transformation:","category":"page"},{"location":"distributions/","page":"Distributions.jl integration","title":"Distributions.jl integration","text":"dist = Beta(2, 2) # support on (0, 1)\ntdist = transformed(dist) # support on ℝ\n\ntdist isa UnivariateDistribution","category":"page"},{"location":"distributions/","page":"Distributions.jl integration","title":"Distributions.jl integration","text":"We can the then compute the logpdf for the resulting distribution:","category":"page"},{"location":"distributions/","page":"Distributions.jl integration","title":"Distributions.jl integration","text":"# Some example values\nx = rand(dist)\ny = tdist.transform(x)\n\nlogpdf(tdist, y)","category":"page"},{"location":"#Bijectors.jl","page":"Home","title":"Bijectors.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"This package implements a set of functions for transforming constrained random variables (e.g. simplexes, intervals) to Euclidean space. The 3 main functions implemented in this package are the link, invlink and logpdf_with_trans for a number of distributions.","category":"page"},{"location":"","page":"Home","title":"Home","text":"Bijectors.link\nBijectors.invlink\nBijectors.logpdf_with_trans","category":"page"},{"location":"#Bijectors.link","page":"Home","title":"Bijectors.link","text":"link(d::Distribution, x)\n\nTransforms the input x using the constrained-to-unconstrained bijector for distribution d.\n\nSee also: invlink.\n\nExample\n\njulia> using Bijectors\n\njulia> d = LogNormal() # support is (0, Inf)\nLogNormal{Float64}(μ=0.0, σ=1.0)\n\njulia> b = bijector(d) # log function transforms to unconstrained space\n(::Base.Fix1{typeof(broadcast), typeof(log)}) (generic function with 1 method)\n\njulia> b(1.0)\n0.0\n\njulia> link(LogNormal(), 1.0)\n0.0\n\n\n\n\n\n","category":"function"},{"location":"#Bijectors.invlink","page":"Home","title":"Bijectors.invlink","text":"invlink(d::Distribution, y)\n\nPerforms the inverse transform on a value y that was transformed using the constrained-to-unconstrained bijector for distribution d.\n\nIt should hold that invlink(d, link(d, x)) = x.\n\nSee also: link.\n\nExample\n\njulia> using Bijectors\n\njulia> d = LogNormal() # support is (0, Inf)\nLogNormal{Float64}(μ=0.0, σ=1.0)\n\njulia> link(LogNormal(), 1.0) # uses a log transform\n0.0\n\njulia> invlink(LogNormal(), 0.0)\n1.0\n\n\n\n\n\n","category":"function"},{"location":"#Bijectors.logpdf_with_trans","page":"Home","title":"Bijectors.logpdf_with_trans","text":"logpdf_with_trans(d::Distribution, x, transform::Bool)\n\nIf transform is false, logpdf_with_trans calculates the log probability density function (logpdf) of distribution d at x.\n\nIf transform is true, x is transformed using the constrained-to-unconstrained bijector for distribution d, and then the logpdf of the resulting value is calculated with respect to the unconstrained (transformed) distribution. Equivalently, if x is distributed according to d and y = link(d, x) is distributed according to td = transformed(d), then logpdf_with_trans(d, x, true) = logpdf(td, y). This is accomplished by subtracting the log Jacobian of the transformation.\n\nExample\n\njulia> using Bijectors\n\njulia> logpdf_with_trans(LogNormal(), ℯ, false)\n-2.4189385332046727\n\njulia> logpdf(LogNormal(), ℯ) # Same as above\n-2.4189385332046727\n\njulia> logpdf_with_trans(LogNormal(), ℯ, true)\n-1.4189385332046727\n\njulia> # If x ~ LogNormal(), then log(x) ~ Normal()\n logpdf(Normal(), 1.0) \n-1.4189385332046727\n\njulia> # The difference between the two is due to the Jacobian\n logabsdetjac(bijector(LogNormal()), ℯ)\n-1\n\n\n\n\n\n","category":"function"},{"location":"","page":"Home","title":"Home","text":"The distributions supported are:","category":"page"},{"location":"","page":"Home","title":"Home","text":"RealDistribution: Union{Cauchy, Gumbel, Laplace, Logistic, NoncentralT, Normal, NormalCanon, TDist},\nPositiveDistribution: Union{BetaPrime, Chi, Chisq, Erlang, Exponential, FDist, Frechet, Gamma, InverseGamma, InverseGaussian, Kolmogorov, LogNormal, NoncentralChisq, NoncentralF, Rayleigh, Weibull},\nUnitDistribution: Union{Beta, KSOneSided, NoncentralBeta},\nSimplexDistribution: Union{Dirichlet},\nPDMatDistribution: Union{InverseWishart, Wishart}, and\nTransformDistribution: Union{T, Truncated{T}} where T<:ContinuousUnivariateDistribution.","category":"page"},{"location":"","page":"Home","title":"Home","text":"All exported names from the Distributions.jl package are reexported from Bijectors.","category":"page"},{"location":"","page":"Home","title":"Home","text":"Bijectors.jl also provides a nice interface for working with these maps: composition, inversion, etc. The following table lists mathematical operations for a bijector and the corresponding code in Bijectors.jl.","category":"page"},{"location":"","page":"Home","title":"Home","text":"Operation Method Automatic\nb ↦ b⁻¹ inverse(b) ✓\n(b₁, b₂) ↦ (b₁ ∘ b₂) b₁ ∘ b₂ ✓\n(b₁, b₂) ↦ [b₁, b₂] stack(b₁, b₂) ✓\nx ↦ b(x) b(x) ×\ny ↦ b⁻¹(y) inverse(b)(y) ×\nx ↦ log|det J(b, x)| logabsdetjac(b, x) AD\nx ↦ b(x), log|det J(b, x)| with_logabsdet_jacobian(b, x) ✓\np ↦ q := b_* p q = transformed(p, b) ✓\ny ∼ q y = rand(q) ✓\np ↦ b such that support(b_* p) = ℝᵈ bijector(p) ✓\n(x ∼ p, b(x), log|det J(b, x)|, log q(y)) forward(q) ✓","category":"page"},{"location":"","page":"Home","title":"Home","text":"In this table, b denotes a Bijector, J(b, x) denotes the Jacobian of b evaluated at x, b_* denotes the push-forward of p by b, and x ∼ p denotes x sampled from the distribution with density p.","category":"page"},{"location":"","page":"Home","title":"Home","text":"The \"Automatic\" column in the table refers to whether or not you are required to implement the feature for a custom Bijector. \"AD\" refers to the fact that it can be implemented \"automatically\" using automatic differentiation.","category":"page"}] +} diff --git a/previews/PR358/siteinfo.js b/previews/PR358/siteinfo.js new file mode 100644 index 00000000..971958be --- /dev/null +++ b/previews/PR358/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "previews/PR358"; diff --git a/previews/PR358/transforms/index.html b/previews/PR358/transforms/index.html new file mode 100644 index 00000000..36141f45 --- /dev/null +++ b/previews/PR358/transforms/index.html @@ -0,0 +1,124 @@ +<!DOCTYPE html> +<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Transforms · Bijectors</title><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.045/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/solid.min.css" rel="stylesheet" type="text/css"/><link 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href="../">Bijectors</a></span></div><form class="docs-search" action="../search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li class="is-active"><a class="tocitem" href>Transforms</a><ul class="internal"><li><a class="tocitem" href="#Usage"><span>Usage</span></a></li><li><a class="tocitem" href="#User-facing-methods"><span>User-facing methods</span></a></li><li><a class="tocitem" href="#Implementing-a-transformation"><span>Implementing a transformation</span></a></li><li><a class="tocitem" href="#Working-with-Distributions.jl"><span>Working with Distributions.jl</span></a></li><li><a class="tocitem" href="#Utilities"><span>Utilities</span></a></li><li><a class="tocitem" href="#API"><span>API</span></a></li><li><a class="tocitem" href="#Bijectors"><span>Bijectors</span></a></li></ul></li><li><a class="tocitem" href="../distributions/">Distributions.jl integration</a></li><li><a class="tocitem" href="../examples/">Examples</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Transforms</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Transforms</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/TuringLang/Bijectors.jl/blob/master/docs/src/transforms.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h2 id="Usage"><a class="docs-heading-anchor" href="#Usage">Usage</a><a id="Usage-1"></a><a class="docs-heading-anchor-permalink" href="#Usage" title="Permalink"></a></h2><p>A very simple example of a "bijector"/diffeomorphism, i.e. a differentiable transformation with a differentiable inverse, is the <code>exp</code> function:</p><ul><li>The inverse of <code>exp</code> is <code>log</code>.</li><li>The derivative of <code>exp</code> at an input <code>x</code> is simply <code>exp(x)</code>, hence <code>logabsdetjac</code> is simply <code>x</code>.</li></ul><pre><code class="language-julia-repl hljs" style="display:block;">julia> using Bijectors</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> transform(exp, 1.0)</code><code class="nohighlight hljs ansi" style="display:block;">2.718281828459045</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> logabsdetjac(exp, 1.0)</code><code class="nohighlight hljs ansi" style="display:block;">1.0</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> with_logabsdet_jacobian(exp, 1.0)</code><code class="nohighlight hljs ansi" style="display:block;">(2.718281828459045, 1.0)</code></pre><p>Some transformations are well-defined for different types of inputs, e.g. <code>exp</code> can also act elementwise on an <code>N</code>-dimensional <code>Array{<:Real,N}</code>. To specify that a transformation should act elementwise, we use the <a href="#Bijectors.elementwise"><code>elementwise</code></a> method:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia> x = ones(2, 2)</code><code class="nohighlight hljs ansi" style="display:block;">2×2 Matrix{Float64}: + 1.0 1.0 + 1.0 1.0</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> transform(elementwise(exp), x)</code><code class="nohighlight hljs ansi" style="display:block;">2×2 Matrix{Float64}: + 2.71828 2.71828 + 2.71828 2.71828</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> logabsdetjac(elementwise(exp), x)</code><code class="nohighlight hljs ansi" style="display:block;">4.0</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> with_logabsdet_jacobian(elementwise(exp), x)</code><code class="nohighlight hljs ansi" style="display:block;">([2.718281828459045 2.718281828459045; 2.718281828459045 2.718281828459045], 4.0)</code></pre><p>These methods also work nicely for compositions of transformations:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia> transform(elementwise(log ∘ exp), x)</code><code class="nohighlight hljs ansi" style="display:block;">2×2 Matrix{Float64}: + 1.0 1.0 + 1.0 1.0</code></pre><p>Unlike <code>exp</code>, some transformations have parameters affecting the resulting transformation they represent, e.g. <code>Logit</code> has two parameters <code>a</code> and <code>b</code> representing the lower- and upper-bound, respectively, of its domain:</p><pre><code class="language-julia-repl hljs" style="display:block;">julia> using Bijectors: Logit</code><code class="nohighlight hljs ansi" style="display:block;"></code><br/><code class="language-julia-repl hljs" style="display:block;">julia> f = Logit(0.0, 1.0)</code><code class="nohighlight hljs ansi" style="display:block;">Bijectors.Logit{Float64, Float64}(0.0, 1.0)</code><br/><code class="language-julia-repl hljs" style="display:block;">julia> f(rand()) # takes us from `(0, 1)` to `(-∞, ∞)`</code><code class="nohighlight hljs ansi" style="display:block;">-4.078605265672502</code></pre><h2 id="User-facing-methods"><a class="docs-heading-anchor" href="#User-facing-methods">User-facing methods</a><a id="User-facing-methods-1"></a><a class="docs-heading-anchor-permalink" href="#User-facing-methods" title="Permalink"></a></h2><p>Without mutation:</p><article class="docstring"><header><a class="docstring-binding" id="Bijectors.transform" href="#Bijectors.transform"><code>Bijectors.transform</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">transform(b, x)</code></pre><p>Transform <code>x</code> using <code>b</code>, treating <code>x</code> as a single input.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/TuringLang/Bijectors.jl/blob/22713b3cb349f7be40e2243f127cc9f0c6608cce/src/interface.jl#L86-L90">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="Bijectors.logabsdetjac" href="#Bijectors.logabsdetjac"><code>Bijectors.logabsdetjac</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">logabsdetjac(b, x)</code></pre><p>Return <code>log(abs(det(J(b, x))))</code>, where <code>J(b, x)</code> is the jacobian of <code>b</code> at <code>x</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/TuringLang/Bijectors.jl/blob/22713b3cb349f7be40e2243f127cc9f0c6608cce/src/interface.jl#L113-L117">source</a></section></article><pre><code class="language-julia hljs">with_logabsdet_jacobian</code></pre><p>With mutation:</p><article class="docstring"><header><a class="docstring-binding" id="Bijectors.transform!" href="#Bijectors.transform!"><code>Bijectors.transform!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">transform!(b, x[, y])</code></pre><p>Transform <code>x</code> using <code>b</code>, storing the result in <code>y</code>.</p><p>If <code>y</code> is not provided, <code>x</code> is used as the output.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/TuringLang/Bijectors.jl/blob/22713b3cb349f7be40e2243f127cc9f0c6608cce/src/interface.jl#L103-L109">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="Bijectors.logabsdetjac!" href="#Bijectors.logabsdetjac!"><code>Bijectors.logabsdetjac!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">logabsdetjac!(b, x[, logjac])</code></pre><p>Compute <code>log(abs(det(J(b, x))))</code> and store the result in <code>logjac</code>, where <code>J(b, x)</code> is the jacobian of <code>b</code> at <code>x</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/TuringLang/Bijectors.jl/blob/22713b3cb349f7be40e2243f127cc9f0c6608cce/src/interface.jl#L129-L133">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="Bijectors.with_logabsdet_jacobian!" href="#Bijectors.with_logabsdet_jacobian!"><code>Bijectors.with_logabsdet_jacobian!</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">with_logabsdet_jacobian!(b, x[, y, logjac])</code></pre><p>Compute <code>transform(b, x)</code> and <code>logabsdetjac(b, x)</code>, storing the result in <code>y</code> and <code>logjac</code>, respetively.</p><p>If <code>y</code> is not provided, then <code>x</code> will be used in its place.</p><p>Defaults to calling <code>with_logabsdet_jacobian(b, x)</code> and updating <code>y</code> and <code>logjac</code> with the result.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/TuringLang/Bijectors.jl/blob/22713b3cb349f7be40e2243f127cc9f0c6608cce/src/interface.jl#L137-L146">source</a></section></article><h2 id="Implementing-a-transformation"><a class="docs-heading-anchor" href="#Implementing-a-transformation">Implementing a transformation</a><a id="Implementing-a-transformation-1"></a><a class="docs-heading-anchor-permalink" href="#Implementing-a-transformation" title="Permalink"></a></h2><p>Any callable can be made into a bijector by providing an implementation of <code>ChangeOfVariables.with_logabsdet_jacobian(b, x)</code>.</p><p>You can also optionally implement <a href="#Bijectors.transform"><code>transform</code></a> and <a href="#Bijectors.logabsdetjac"><code>logabsdetjac</code></a> to avoid redundant computations. This is usually only worth it if you expect <code>transform</code> or <code>logabsdetjac</code> to be used heavily without the other.</p><p>Similarly with the mutable versions <a href="#Bijectors.with_logabsdet_jacobian!"><code>with_logabsdet_jacobian!</code></a>, <a href="#Bijectors.transform!"><code>transform!</code></a>, and <a href="#Bijectors.logabsdetjac!"><code>logabsdetjac!</code></a>.</p><h2 id="Working-with-Distributions.jl"><a class="docs-heading-anchor" href="#Working-with-Distributions.jl">Working with Distributions.jl</a><a id="Working-with-Distributions.jl-1"></a><a class="docs-heading-anchor-permalink" href="#Working-with-Distributions.jl" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-binding" id="Bijectors.bijector" href="#Bijectors.bijector"><code>Bijectors.bijector</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">bijector(d::Distribution)</code></pre><p>Returns the constrained-to-unconstrained bijector for distribution <code>d</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/TuringLang/Bijectors.jl/blob/22713b3cb349f7be40e2243f127cc9f0c6608cce/src/transformed_distribution.jl#L41-L45">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="Bijectors.transformed-Tuple{Distribution, Bijector}" href="#Bijectors.transformed-Tuple{Distribution, Bijector}"><code>Bijectors.transformed</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">transformed(d::Distribution) +transformed(d::Distribution, b::Bijector)</code></pre><p>Couples distribution <code>d</code> with the bijector <code>b</code> by returning a <code>TransformedDistribution</code>.</p><p>If no bijector is provided, i.e. <code>transformed(d)</code> is called, then <code>transformed(d, bijector(d))</code> is returned.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/TuringLang/Bijectors.jl/blob/22713b3cb349f7be40e2243f127cc9f0c6608cce/src/transformed_distribution.jl#L29-L37">source</a></section></article><h2 id="Utilities"><a class="docs-heading-anchor" href="#Utilities">Utilities</a><a id="Utilities-1"></a><a class="docs-heading-anchor-permalink" href="#Utilities" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-binding" id="Bijectors.elementwise" href="#Bijectors.elementwise"><code>Bijectors.elementwise</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">elementwise(f)</code></pre><p>Alias for <code>Base.Fix1(broadcast, f)</code>.</p><p>In the case where <code>f::ComposedFunction</code>, the result is <code>Base.Fix1(broadcast, f.outer) ∘ Base.Fix1(broadcast, f.inner)</code> rather than <code>Base.Fix1(broadcast, f)</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/TuringLang/Bijectors.jl/blob/22713b3cb349f7be40e2243f127cc9f0c6608cce/src/interface.jl#L7-L15">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="Bijectors.isinvertible" href="#Bijectors.isinvertible"><code>Bijectors.isinvertible</code></a> — <span class="docstring-category">Function</span></header><section><div><pre><code class="language-julia hljs">isinvertible(t)</code></pre><p>Return <code>true</code> if <code>t</code> is invertible, and <code>false</code> otherwise.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/TuringLang/Bijectors.jl/blob/22713b3cb349f7be40e2243f127cc9f0c6608cce/src/interface.jl#L168-L172">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="Bijectors.isclosedform-Tuple{Bijectors.Transform}" href="#Bijectors.isclosedform-Tuple{Bijectors.Transform}"><code>Bijectors.isclosedform</code></a> — <span class="docstring-category">Method</span></header><section><div><pre><code class="language-julia hljs">isclosedform(b::Transform)::bool +isclosedform(b⁻¹::Inverse{<:Transform})::bool</code></pre><p>Returns <code>true</code> or <code>false</code> depending on whether or not evaluation of <code>b</code> has a closed-form implementation.</p><p>Most transformations have closed-form evaluations, but there are cases where this is not the case. For example the <em>inverse</em> evaluation of <code>PlanarLayer</code> requires an iterative procedure to evaluate.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/TuringLang/Bijectors.jl/blob/22713b3cb349f7be40e2243f127cc9f0c6608cce/src/interface.jl#L155-L165">source</a></section></article><h2 id="API"><a class="docs-heading-anchor" href="#API">API</a><a id="API-1"></a><a class="docs-heading-anchor-permalink" href="#API" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-binding" id="Bijectors.Transform" href="#Bijectors.Transform"><code>Bijectors.Transform</code></a> — <span class="docstring-category">Type</span></header><section><div><p>Abstract type for a transformation.</p><p><strong>Implementing</strong></p><p>A subtype of <code>Transform</code> of should at least implement <a href="#Bijectors.transform"><code>transform(b, x)</code></a>.</p><p>If the <code>Transform</code> is also invertible:</p><ul><li>Required:<ul><li><em>Either</em> of the following:<ul><li><code>transform(::Inverse{<:MyTransform}, x)</code>: the <code>transform</code> for its inverse.</li><li><code>InverseFunctions.inverse(b::MyTransform)</code>: returns an existing <code>Transform</code>.</li></ul></li><li><a href="#Bijectors.logabsdetjac"><code>logabsdetjac</code></a>: computes the log-abs-det jacobian factor.</li></ul></li><li>Optional:<ul><li><code>with_logabsdet_jacobian</code>: <code>transform</code> and <code>logabsdetjac</code> combined. Useful in cases where we can exploit shared computation in the two.</li></ul></li></ul><p>For the above methods, there are mutating versions which can <em>optionally</em> be implemented:</p><ul><li><a href="#Bijectors.with_logabsdet_jacobian!"><code>with_logabsdet_jacobian!</code></a></li><li><a href="#Bijectors.logabsdetjac!"><code>logabsdetjac!</code></a></li><li><a href="#Bijectors.with_logabsdet_jacobian!"><code>with_logabsdet_jacobian!</code></a></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/TuringLang/Bijectors.jl/blob/22713b3cb349f7be40e2243f127cc9f0c6608cce/src/interface.jl#L57-L79">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="Bijectors.Bijector" href="#Bijectors.Bijector"><code>Bijectors.Bijector</code></a> — <span class="docstring-category">Type</span></header><section><div><p>Abstract type of a bijector, i.e. differentiable bijection with differentiable inverse.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/TuringLang/Bijectors.jl/blob/22713b3cb349f7be40e2243f127cc9f0c6608cce/src/interface.jl#L205">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="Bijectors.Inverse" href="#Bijectors.Inverse"><code>Bijectors.Inverse</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">inverse(b::Transform) +Inverse(b::Transform)</code></pre><p>A <code>Transform</code> representing the inverse transform of <code>b</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/TuringLang/Bijectors.jl/blob/22713b3cb349f7be40e2243f127cc9f0c6608cce/src/interface.jl#L175-L180">source</a></section></article><h2 id="Bijectors"><a class="docs-heading-anchor" href="#Bijectors">Bijectors</a><a id="Bijectors-1"></a><a class="docs-heading-anchor-permalink" href="#Bijectors" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-binding" id="Bijectors.CorrBijector" href="#Bijectors.CorrBijector"><code>Bijectors.CorrBijector</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">CorrBijector <: Bijector</code></pre><p>A bijector implementation of Stan's parametrization method for Correlation matrix: https://mc-stan.org/docs/2_23/reference-manual/correlation-matrix-transform-section.html</p><p>Basically, a unconstrained strictly upper triangular matrix <code>y</code> is transformed to a correlation matrix by following readable but not that efficient form:</p><pre><code class="nohighlight hljs">K = size(y, 1) +z = tanh.(y) + +for j=1:K, i=1:K + if i>j + w[i,j] = 0 + elseif 1==i==j + w[i,j] = 1 + elseif 1<i==j + w[i,j] = prod(sqrt(1 .- z[1:i-1, j].^2)) + elseif 1==i<j + w[i,j] = z[i,j] + elseif 1<i<j + w[i,j] = z[i,j] * prod(sqrt(1 .- z[1:i-1, j].^2)) + end +end</code></pre><p>It is easy to see that every column is a unit vector, for example:</p><pre><code class="nohighlight hljs">w3' w3 == +w[1,3]^2 + w[2,3]^2 + w[3,3]^2 == +z[1,3]^2 + (z[2,3] * sqrt(1 - z[1,3]^2))^2 + (sqrt(1-z[1,3]^2) * sqrt(1-z[2,3]^2))^2 == +z[1,3]^2 + z[2,3]^2 * (1-z[1,3]^2) + (1-z[1,3]^2) * (1-z[2,3]^2) == +z[1,3]^2 + z[2,3]^2 - z[2,3]^2 * z[1,3]^2 + 1 -z[1,3]^2 - z[2,3]^2 + z[1,3]^2 * z[2,3]^2 == +1</code></pre><p>And diagonal elements are positive, so <code>w</code> is a cholesky factor for a positive matrix.</p><pre><code class="nohighlight hljs">x = w' * w</code></pre><p>Consider block matrix representation for <code>x</code></p><pre><code class="nohighlight hljs">x = [w1'; w2'; ... wn'] * [w1 w2 ... wn] == +[w1'w1 w1'w2 ... w1'wn; + w2'w1 w2'w2 ... w2'wn; + ... +]</code></pre><p>The diagonal elements are given by <code>wk'wk = 1</code>, thus <code>x</code> is a correlation matrix.</p><p>Every step is invertible, so this is a bijection(bijector).</p><p>Note: The implementation doesn't follow their "manageable expression" directly, because their equation seems wrong (7/30/2020). Insteadly it follows definition above the "manageable expression" directly, which is also described in above doc.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/TuringLang/Bijectors.jl/blob/22713b3cb349f7be40e2243f127cc9f0c6608cce/src/bijectors/corr.jl#L1-L63">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="Bijectors.LeakyReLU" href="#Bijectors.LeakyReLU"><code>Bijectors.LeakyReLU</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">LeakyReLU{T}(α::T) <: Bijector</code></pre><p>Defines the invertible mapping</p><pre><code class="nohighlight hljs">x ↦ x if x ≥ 0 else αx</code></pre><p>where α > 0.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/TuringLang/Bijectors.jl/blob/22713b3cb349f7be40e2243f127cc9f0c6608cce/src/bijectors/leaky_relu.jl#L1-L9">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="Bijectors.Stacked" href="#Bijectors.Stacked"><code>Bijectors.Stacked</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Stacked(bs) +Stacked(bs, ranges) +stack(bs::Bijector...)</code></pre><p>A <code>Bijector</code> which stacks bijectors together which can then be applied to a vector where <code>bs[i]::Bijector</code> is applied to <code>x[ranges[i]]::UnitRange{Int}</code>.</p><p><strong>Arguments</strong></p><ul><li><code>bs</code> can be either a <code>Tuple</code> or an <code>AbstractArray</code> of 0- and/or 1-dimensional bijectors<ul><li>If <code>bs</code> is a <code>Tuple</code>, implementations are type-stable using generated functions</li><li>If <code>bs</code> is an <code>AbstractArray</code>, implementations are <em>not</em> type-stable and use iterative methods</li></ul></li><li><code>ranges</code> needs to be an iterable consisting of <code>UnitRange{Int}</code><ul><li><code>length(bs) == length(ranges)</code> needs to be true.</li></ul></li></ul><p><strong>Examples</strong></p><pre><code class="nohighlight hljs">b1 = Logit(0.0, 1.0) +b2 = identity +b = stack(b1, b2) +b([0.0, 1.0]) == [b1(0.0), 1.0] # => true</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/TuringLang/Bijectors.jl/blob/22713b3cb349f7be40e2243f127cc9f0c6608cce/src/bijectors/stacked.jl#L1-L23">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="Bijectors.RationalQuadraticSpline" href="#Bijectors.RationalQuadraticSpline"><code>Bijectors.RationalQuadraticSpline</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">RationalQuadraticSpline{T} <: Bijector</code></pre><p>Implementation of the Rational Quadratic Spline flow [1].</p><ul><li>Outside of the interval <code>[minimum(widths), maximum(widths)]</code>, this mapping is given by the identity map. </li><li>Inside the interval it's given by a monotonic spline (i.e. monotonic polynomials connected at intermediate points) with endpoints fixed so as to continuously transform into the identity map.</li></ul><p>For the sake of efficiency, there are separate implementations for 0-dimensional and 1-dimensional inputs.</p><p><strong>Notes</strong></p><p>There are two constructors for <code>RationalQuadraticSpline</code>:</p><ul><li><code>RationalQuadraticSpline(widths, heights, derivatives)</code>: it is assumed that <code>widths</code>, </li></ul><p><code>heights</code>, and <code>derivatives</code> satisfy the constraints that makes this a valid bijector, i.e.</p><ul><li><code>widths</code>: monotonically increasing and <code>length(widths) == K</code>,</li><li><code>heights</code>: monotonically increasing and <code>length(heights) == K</code>,</li><li><code>derivatives</code>: non-negative and <code>derivatives[1] == derivatives[end] == 1</code>.</li><li><code>RationalQuadraticSpline(widths, heights, derivatives, B)</code>: other than than the lengths, no assumptions are made on parameters. Therefore we will transform the parameters s.t.:</li><li><code>widths_new</code> ∈ [-B, B]ᴷ⁺¹, where <code>K == length(widths)</code>,</li><li><code>heights_new</code> ∈ [-B, B]ᴷ⁺¹, where <code>K == length(heights)</code>,</li><li><code>derivatives_new</code> ∈ (0, ∞)ᴷ⁺¹ with <code>derivatives_new[1] == derivates_new[end] == 1</code>, where <code>(K - 1) == length(derivatives)</code>.</li></ul><p><strong>Examples</strong></p><p><strong>Univariate</strong></p><pre><code class="language-julia-repl hljs">julia> using StableRNGs: StableRNG; rng = StableRNG(42); # For reproducibility. + +julia> using Bijectors: RationalQuadraticSpline + +julia> K = 3; B = 2; + +julia> # Monotonic spline on '[-B, B]' with `K` intermediate knots/"connection points". + b = RationalQuadraticSpline(randn(rng, K), randn(rng, K), randn(rng, K - 1), B); + +julia> b(0.5) # inside of `[-B, B]` → transformed +1.1943325397834206 + +julia> b(5.) # outside of `[-B, B]` → not transformed +5.0 + +julia> b = RationalQuadraticSpline(b.widths, b.heights, b.derivatives); + +julia> b(0.5) # inside of `[-B, B]` → transformed +1.1943325397834206 + +julia> d = 2; K = 3; B = 2; + +julia> b = RationalQuadraticSpline(randn(rng, d, K), randn(rng, d, K), randn(rng, d, K - 1), B); + +julia> b([-1., 1.]) +2-element Vector{Float64}: + -1.5660106244288925 + 0.5384702734738573 + +julia> b([-5., 5.]) +2-element Vector{Float64}: + -5.0 + 5.0 + +julia> b([-1., 5.]) +2-element Vector{Float64}: + -1.5660106244288925 + 5.0</code></pre><p><strong>References</strong></p><p>[1] Durkan, C., Bekasov, A., Murray, I., & Papamakarios, G., Neural Spline Flows, CoRR, arXiv:1906.04032 [stat.ML], (2019). </p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/TuringLang/Bijectors.jl/blob/22713b3cb349f7be40e2243f127cc9f0c6608cce/src/bijectors/rational_quadratic_spline.jl#L1-L74">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="Bijectors.Coupling" href="#Bijectors.Coupling"><code>Bijectors.Coupling</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">Coupling{F, M}(θ::F, mask::M)</code></pre><p>Implements a coupling-layer as defined in [1].</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia> using Bijectors: Shift, Coupling, PartitionMask, coupling, couple + +julia> m = PartitionMask(3, [1], [2]); # <= going to use x[2] to parameterize transform of x[1] + +julia> cl = Coupling(Shift, m); # <= will do `y[1:1] = x[1:1] + x[2:2]`; + +julia> x = [1., 2., 3.]; + +julia> cl(x) +3-element Vector{Float64}: + 3.0 + 2.0 + 3.0 + +julia> inverse(cl)(cl(x)) +3-element Vector{Float64}: + 1.0 + 2.0 + 3.0 + +julia> coupling(cl) # get the `Bijector` map `θ -> b(⋅, θ)` +Shift + +julia> couple(cl, x) # get the `Bijector` resulting from `x` +Shift{Vector{Float64}}([2.0]) + +julia> with_logabsdet_jacobian(cl, x) +([3.0, 2.0, 3.0], 0.0)</code></pre><p><strong>References</strong></p><p>[1] Kobyzev, I., Prince, S., & Brubaker, M. A., Normalizing flows: introduction and ideas, CoRR, (), (2019). </p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/TuringLang/Bijectors.jl/blob/22713b3cb349f7be40e2243f127cc9f0c6608cce/src/bijectors/coupling.jl#L138-L177">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="Bijectors.OrderedBijector" href="#Bijectors.OrderedBijector"><code>Bijectors.OrderedBijector</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">OrderedBijector()</code></pre><p>A bijector mapping unordered vectors in ℝᵈ to ordered vectors in ℝᵈ.</p><p><strong>See also</strong></p><ul><li><a href="https://mc-stan.org/docs/2_27/reference-manual/ordered-vector.html">Stan's documentation</a><ul><li>Note that this transformation and its inverse are the <em>opposite</em> of in this reference.</li></ul></li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/TuringLang/Bijectors.jl/blob/22713b3cb349f7be40e2243f127cc9f0c6608cce/src/bijectors/ordered.jl#L9-L17">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="Bijectors.NamedTransform" href="#Bijectors.NamedTransform"><code>Bijectors.NamedTransform</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NamedTransform <: AbstractNamedTransform</code></pre><p>Wraps a <code>NamedTuple</code> of key -> <code>Bijector</code> pairs, implementing evaluation, inversion, etc.</p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia> using Bijectors: NamedTransform, Scale + +julia> b = NamedTransform((a = Scale(2.0), b = exp)); + +julia> x = (a = 1., b = 0., c = 42.); + +julia> b(x) +(a = 2.0, b = 1.0, c = 42.0) + +julia> (a = 2 * x.a, b = exp(x.b), c = x.c) +(a = 2.0, b = 1.0, c = 42.0)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/TuringLang/Bijectors.jl/blob/22713b3cb349f7be40e2243f127cc9f0c6608cce/src/bijectors/named_bijector.jl#L7-L26">source</a></section></article><article class="docstring"><header><a class="docstring-binding" id="Bijectors.NamedCoupling" href="#Bijectors.NamedCoupling"><code>Bijectors.NamedCoupling</code></a> — <span class="docstring-category">Type</span></header><section><div><pre><code class="language-julia hljs">NamedCoupling{target, deps, F} <: AbstractNamedTransform</code></pre><p>Implements a coupling layer for named bijectors.</p><p>See also: <a href="#Bijectors.Coupling"><code>Coupling</code></a></p><p><strong>Examples</strong></p><pre><code class="language-julia-repl hljs">julia> using Bijectors: NamedCoupling, Scale + +julia> b = NamedCoupling(:b, (:a, :c), (a, c) -> Scale(a + c)); + +julia> x = (a = 1., b = 2., c = 3.); + +julia> b(x) +(a = 1.0, b = 8.0, c = 3.0) + +julia> (a = x.a, b = (x.a + x.c) * x.b, c = x.c) +(a = 1.0, b = 8.0, c = 3.0)</code></pre></div><a class="docs-sourcelink" target="_blank" href="https://github.com/TuringLang/Bijectors.jl/blob/22713b3cb349f7be40e2243f127cc9f0c6608cce/src/bijectors/named_bijector.jl#L97-L118">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../distributions/">Distributions.jl integration »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Wednesday 8 January 2025 12:23">Wednesday 8 January 2025</span>. 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