Where we have got to

Model fitting and inference for infectious disease dynamics

The story so far

Deterministic models

$$p(\theta \mid \text{data}) \propto p(\text{data} \mid \theta)\, p(\theta)$$

Use MCMC to get samples from it: $\theta_1, \theta_2, \theta_3, \ldots$

Stochastic models

• What is $p(\text{data} \mid \theta)$?

• We need to marginalise:

  $$p(\text{data} \mid \theta) = \int_{X} p(\text{data} \mid X, \theta)\, p(X \mid \theta)$$

The particle filter

Red diamonds are the data, yellow circles the particles, sized by weight.

The estimator

Average the weights each day, before resampling:

$$\hat{p}(y \mid \theta) = \prod_{t=1}^{T} \frac{1}{J}\sum_{j=1}^{J} w_{t,j}$$

A Monte Carlo estimate of the integral, built one day at a time.

Particle MCMC

The estimate goes inside Metropolis-Hastings

$$r = \min\left(1, \frac{\hat{p}(y \mid \theta^*)\, p(\theta^*)}{\hat{p}(y \mid \theta)\, p(\theta)}\right)$$

Unbiased $\hat{p}$, so the chain still targets the right posterior.

Deterministic vs. stochastic

Same SEIT4L model, priors and data.

Filtered vs. resimulated trajectories

Filtered paths are conditioned on data, resimulated just drawn from parameters that are.

Questions?

Anything from the first three days worth revisiting before the last session.

Deterministic fitting · Stochastic models · Particle filters · Particle MCMC