Posterior predictions for count proportions/percentages

[SamuelBrand1]

Goal

The goal of this issue is to add to existing observation processes to include proportion (or %) of some underlying total count are from a particular pathogen of interest.

Context

CDC releases emergency dept visits dues to specific pathogen infection as %s of total visits, for example: proportion of ED visits due to Covid. Therefore, it makes sense to be capable of observing, and nowcasting/predicting this target.

My first view on modelling this is that the lowest difficulty appropriate modelling approach is to have:

• A pyrenew model for count incidence of the target pathogen $E(t)$. Functionality for this exists already.
• An independent model to predict total counts less the target pathogen $N(t)$.

Then the proportion of interest is:

$$\text{perc. total counts are target pathogen} = 100 \times \frac{E(t)}{E(t) + N(t)}.$$

Which we can sample from the posterior distribution of the model, either with an independent sample of $N(t)$ or a point est of $N(t)$.

Alternative approaches to consider

Main alternative is to treat the proportion of interest as the target, rather than the approach above which treats the inference problem as a transformation on currently existing inference dependent on the sampling/point est of an independent model.

Required features

• A model to generate $N(t)$. #53
• An observation transformation approach which converts count predictions in to $N(t)$ dependent proportion predictions. #69

Specifications

Potentially to be added after investigation.

[SamuelBrand1]

@damonbayer is #68 aimed at the completing the second task here?