Where we have got to

Model fitting and inference for infectious disease dynamics

Have the chains converged?

Straight, plump at both ends, several chains on top of each other. R̂=V̂/W\hat{R} = \sqrt{\hat{V}/W} compares the variance between chains with the variance within them. We want this to be below 1.01.

Before you see the data

θ(i)∼p(θ),y(i)∼p(y∣θ(i))\theta^{(i)} \sim p(\theta), \qquad y^{(i)} \sim p(y \mid \theta^{(i)})

Two hundred draws from our priors. Most burn through the population in a week.

After fitting

θ(i)∼p(θ∣y),yrep,(i)∼p(y∣θ(i))\theta^{(i)} \sim p(\theta \mid y), \qquad y^{\text{rep},(i)} \sim p(y \mid \theta^{(i)})

The fit misses both waves: a flatter, later, smaller epidemic that splits the difference between them.

A stochastic model makes the likelihood an integral

p(data∣θ)=∫Xp(data∣X,θ)p(X∣θ)p(\text{data} \mid \theta) = \int_{X} p(\text{data} \mid X, \theta)\, p(X \mid \theta)

Run a stochastic model twice and you get two different trajectories. One θ\theta no longer gives one curve to hold against the data; it gives a distribution of them, and the likelihood is the average over that distribution. Estimating that average, for SEIT4L, is where today starts.

Questions?

Anything from yesterday worth revisiting.

MCMC diagnostics · Model checking · SEITL