This is a nonparametric non-linear model that is an implementation of the Bayes filter

(Bayes filter is parametric as it represents the Gaussian by its definition (in exponents))

Particle filter: How to sample more efficiently in Bayes filter?

—> Represent the posterior bel(xt) by a set of random state samples drawn from this posterior

It is not as efficient as a Kalman filter

We want to use Monte Carlo Estimation

This is because we often want to get the expected value of some function $f(x)$

$E_{x\sim Bel(x_t)}[f] = \int_x f(x)bel(x)dx \text{ (cont.)}= \frac1N\sum^N_i f(x^i) \text{ (dis.)}$

e.g. Our mean position, probability of collision, mean value f(x)

We wish to sample from the belief distribution $x_2, x_2, …x_N$

Importance sampling

1. we sample from the proposal (simple) distribution, we do get bad estimate but that is ok
2. we reweigh it by scaling it by k = p(x) (no need to sample, just get p(x)!)

The weights become the expected value