Particle filter

Definition

Consider a state-space model: a Markov chain $(X_t)_{t \ge 0}$ with initial density $\mu$ and transition density $f(x_t \mid x_{t-1})$. The observations $Y_t$ are conditionally independent given the states, with density $g(y_t \mid x_t)$. Fix a number $N \ge 1$ of particles and observed values $y_1, y_2, \dots$. The bootstrap particle filter (sequential importance resampling) runs as follows.

1. Draw $x_0^{(1)}, \dots, x_0^{(N)}$ independently from $\mu$.
2. At time $t \ge 1$, draw $\tilde x_t^{(i)}$ from $f(\cdot \mid x_{t-1}^{(i)})$ for each $i$.
3. Compute the weights $w_t^{(i)} = g(y_t \mid \tilde x_t^{(i)}) / \sum_{k=1}^{N} g(y_t \mid \tilde x_t^{(k)})$.
4. Draw $x_t^{(1)}, \dots, x_t^{(N)}$ from the weighted points, choosing $\tilde x_t^{(i)}$ with probability $w_t^{(i)}$.

The filtering distribution of $X_t$ given $y_1, \dots, y_t$ is approximated by the weighted empirical measure $\sum_{i=1}^{N} w_t^{(i)} \delta_{\tilde x_t^{(i)}}$.