Interactive demo · Bayesian and Probabilistic Machine Learning · Chapter 13

Particle filters

When the state-space model is not linear-Gaussian, the posterior p(z_n|X_n) has no closed form. A particle filter represents it by L samples with weights w_n^(l) ∝ p(x_n|z_n^(l)). To move to the next step it draws L new particles from the mixture Σ_l w_n^(l) p(z_n+1|z_n^(l)) (pick a parent by its weight, then move it by the dynamics) and weights them by the new observation. The lower left panel shows one such step, as in Figure 13.23 of Bishop (2006).

Model

Particles L

Measurement noise sd

Step shown below, n to n + 1

resample at every step (the book's algorithm) exact Kalman filter (linear model)

Try these

Linear-Gaussian model: z_n = 0.95 z_n−1 + N(0, 0.5), x_n = z_n + noise, z₁ ~ N(0, 1); here the Kalman filter is exact. Nonlinear growth model (Kitagawa 1996; Gordon, Salmond and Smith 1993): z_n = z/2 + 25z/(1 + z²) + 8 cos(1.2n) + N(0, 10) with z = z_n−1, x_n = z_n²/20 + noise, z₁ ~ N(0, 5); since x depends on z², p(x|z) has two bumps at ±√(20x) (green ticks in the top panel). 50 time steps; seeded random numbers, so a class sees the same pictures.

In the lecture notes

Chapter 13 notes: Section 13.3.4 (particle filters: the weights (13.118), the mixture predictive distribution (13.119) and the algorithm), with Section 13.3.1 for the Kalman filter used in the comparison.
Importance sampling and sampling-importance-resampling, on which the particle filter is built: Chapter 11, Section 11.1.4 and Section 11.1.5. The linear-Gaussian case in two dimensions: the demo Kalman filter and smoother.
Python code: code/ch13_sequential_data.py (functions pf_simulate, particle_filter, pf_kalman; Figure 13.23) and the checks in code/checks_ch13.py. Passing JSRandom(seed) reproduces this page's data and particles; preset 1 is panel (a) of our Figure 13.23.