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

Replaces the integral by a sum over Monte Carlo samples of trajectories.

propagate \[p(X_1 \mid \theta)\]

resample \[p(\text{data}_1 \mid X_1, \theta)\, p(X_1 \mid \theta)\]

propagate \[p(X_2 \mid X_1, \theta)\, p(\text{data}_1 \mid X_1, \theta)\, p(X_1 \mid \theta)\]

resample \[p(\text{data}_2 \mid X_2, \theta)\, p(X_2 \mid X_1, \theta)\, p(\text{data}_1 \mid X_1, \theta)\, p(X_1 \mid \theta)\]

\[\ldots\]

evaluate \[p(\text{data}_N \mid X_N, \theta)\, p(X_N \mid X_{1:N-1}, \theta) \ldots p(\text{data}_1 \mid X_1, \theta)\, p(X_1 \mid \theta)\]

average \[\sum p(\text{data}_N \mid X_N, \theta) \ldots p(\text{data}_1 \mid X_1, \theta)\, p(X \mid \theta)\]

Result: \[\sum p(\text{data} \mid X)\, p(X \mid \theta)\]

Questions?

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

Deterministic fitting · Stochastic models · Particle filters · Particle MCMC