Optionally parameterize one observation error scale relative to the others.

[dylanhmorris]

It should be (optionally) possible to parameterize the observation error scale of one signal relative to the observation error scale of another. Something like this:

Let $\bar{E}_t$ and $E_t$ be predicted and observed ED visits at time $t$. Let $\bar{H}_t$ / $H_t$ and $\bar{W}_t$ / $W_t$ be the same but for observed admissions and wastewater concentrations, respectively.

$$E_t \sim \mathrm{NegBin}(\bar{E}_t, k_e)$$

$$H_t \sim \mathrm{NegBin}(\bar{H}_t, k_h)$$

$$k_e \sim \mathrm{Some Prior}$$

$$k_h = a_{he} k_e$$

$$a_{he} \sim \mathrm{Some Prior}$$

[dylanhmorris]

From discussions with @SamuelBrand1. CC @damonbayer @sbidari @AFg6K7h4fhy2 for awareness.

[SamuelBrand1]

For the specific case of the Negative binomial I think this works best with the parameterisation such that:

$$\text{variance} = \mu + \alpha^2 \mu^2$$

Where $\alpha$ is then roughly $\alpha \approx \frac{\text{std}}{\mu}$, with the approx being good when the mean is largish. Then the prior on the H can be reasoned quite nicely I think e.g. $a_{he} = 1.5$ means that the standard fluctuation is 50% higher in the H compared to E.

[damonbayer]

Do you expect the posterior concentrations to be correlated? For the parameterization @SamuelBrand1 suggests, it is somewhat easy to reason about setting a prior in this way, but, overall, I don't see any benefit over specifying the priors independently.