Particle filters are general algorithms for inferring the state of a system with noisy dynamics and noisy measurements. Here's an example with a robot in a circular room. Red=true robot, blue=guesses, occasional red line=noisy range sensor measurement. Details in thread 1/
A particle filter (PF) does the same job as a Kalman filter (KF). Generally: you have a system in an unknown state, evolving over time according to some known dynamics + noise, and you occasionally get noisy sensor data. The task is to infer the current state 2/
In a KF your current uncertainty of the system is represented by a Gaussian, the dynamics are linear + Gaussian noise, and measurements have Gaussian noise. All those assumptions together mean that at any time your uncertainty is captured by a MV Gaussian 3/
In a PF, your uncertainty of the system is represented by a discrete collection of possible states ('particles'). Each particle evolves according to the dynamics of the system (+ sampling whatever noise is involved) 4/
Initially, the 'particles' are equally weighted. When you make a measurement, you reweight each particle according to the likelihood of the measurement. When the weights become sufficiently unbalanced, you resample with replacement according to weight 5/
It's essentially Bayesian inference with discrete blobs of probability mass. (In the simulation, I've done the re-sampling step after every measurement, because it makes it clearer what's going on) 6/
Here's what happens to the state of the particle filter if the robot never makes a measurement 7/
Advantages of PFs: 1) The dynamics don't have to be linear; they can be anything at all. 2) The noise in the dynamics and the measurement can follow any distribution whatsoever. 3) They're really simple to code up 8/
Disadvantages: 1) being sample-based, they won't work well in very high dimensions. 2) if you end up with no samples near the true state of the system, you can't really recover 9/
Sometimes you can model systems with a hybrid KF/PF approach. E.g., you have a variable x, and you model p(x) sampled via a PF, and you have another variable y and you model p(y|x) as a Gaussian with a KF (so you have 1 KF per particle) 10/
This can be especially useful in very high dimensional systems where a very large collection of variables can be modelled by a multivariate Gaussian, conditionally on a small number of variables that you handle with a PF 11/11
Err, 12/11. Here's a kernel density estimate of the posterior distribution over time, computed from the particle positions. It's fun to watch the distribution spreading out between measurements before dramatically collapsing
