Multilevel model tutorial with dynamical systems

[SamWitty]

This tutorial demonstrates how ChiRho's dynamical systems module composes with existing techniques in Bayesian hierarchical modeling. Specifically, this tutorial shows how to represent joint dependencies between parameters of non-interacting SIR models for different towns, and how ordinary probabilistic inference reflects reasonable intuition about how observations from one stratum inform our predictions for another.

Subsumes #351 .

TODO:

• Render graphical model
• Add basic intervention on stratum without observations
• Clean up figures
• Add explainer text throughout

[djinnome]

When initializing the cadence of observations, you say:

obs_sample_rate = torch.tensor(1/7) # Take measurements one per week

However, since the timescale is months, not days, it should be

obs_sample_rate = torch.tensor(1/4) # Take measurements one per week

If you meant for the timestep to be days, then it should be:

obs_sample_rate = torch.tensor(7.0) # Take measurements one per week

[djinnome]

It looks like the obs_sample_rate is still 1/7, not 1/4.

[djinnome]

Since time is measured in units of months, a weekly obs_sample rate should be 1/4, not 1/7.

[djinnome]

Why is sir_observation_model a recursive function?

Edit: Nevermind. I now see that single_observation_model calls sir_observation_model with one less dimension.

[djinnome]

In bayesian_multilevel_sir_prior the variable called plate should be called strata

[djinnome]

The prior marginal and conditional distributions of the parameters should be a different color for each town. Currently, they are all the same color. This could be fixed by introducing:

town_colors = ["cyan", "magenta", "chartreuse"]
...
for i in range(3):
    sns.kdeplot(
        parameter_prior_samples['beta'][:, i],
        ax=ax[1,0],
        alpha=0.4,
        color=town_colors[i],
        label=f"Town {i} $\\beta$ - Marginal Probability",
        linewidth=1.5,
        bw_adjust=.6,
        clip=(0, 1),
    )

[djinnome]

Just a couple minor plotting issues, but otherwise, it looks ready to go.

[djinnome]

Since time is measured in units of months, a weekly obs_sample rate should be 1/4, not 1/7.