Improve the prior interface for make_epi_model_inference

[seabbs]

As the title says we should be able to improve the prior interface. The simplest improvement would be to generalise the argument name for the latent process priors to reflect that it is used across all submodules. An additional improvement would be to think more deeply about the coupling of priors to model submodules.

This issue was discussed in #44

[SamuelBrand1]

The simplest improvement would be to generalise the argument name for the latent process priors to reflect that it is used across all submodules.

Yes, atm I just use merge on the NameTuples of each set of prior distributions.

An additional improvement would be to think more deeply about the coupling of priors to model submodules.

I'm going to try and prototype this with the easy fix as back up. It shouldn't be very hard. Basic idea is to wrap the function that makes a Turing model for each process with its priors in a struct. If you think thats cleaner than having a NamedTuple of priors lets do that.

The essential question is:

• Should all the priors "live" together in code as one object?
• Should the priors of a particular process "live" with the process definition as one object?

[seabbs]

I'm going to try and prototype this with the easy fix as back up. It shouldn't be very hard. Basic idea is to wrap the function that makes a Turing model for each process with its priors in a struct. If you think thats cleaner than having a NamedTuple of priors lets do that.

Yes I think this is a nice idea.

Should the priors of a particular process "live" with the process definition as one object?

This seems cleanest to me and I think likely to have fewer failure states.

[SamuelBrand1]

My proposal is to create a struct to contain both the Turing model and its priors; this will get passed to the overall Turing model constructor.

Proposed new model diagram

flowchart LR

    A["Underlying dists.
and specify length of sims
---------------------
EpiData"]

    B["Choice of target
for latent process
---------------------
DirectInfections
    ExpGrowthRate
    Renewal"]

C["Observational Data
---------------------
Obs. cases y_t"]
D["Latent processes
---------------------
random_walk"]
E[Turing model constructor]
F["Latent Process priors"]
G[Posterior draws]
H[Posterior checking]
I[Post-processing]
DataW[Data wrangling and QC]
J["Observation models
---------------------
delay_observations"]
K["Observation model priors"]

A --> EpiModel
B --> EpiModel
EpiModel -->E
C-->E
D-->LatentProcess
F-->|default_rw_priors|LatentProcess
J-->ObservationModel
K-->|default_delay_obs_priors|ObservationModel
LatentProcess-->E
ObservationModel-->E
E-->|sample...NUTS...| G
G-.->H
H-.->I
DataW-.->C

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