As mentioned so far, the discrete distributions can be estimated by
using samples. In fact, it turns out that the Voronoi regions over
the samples do not even need to be carefully considered. One can work
directly with a collection of samples drawn randomly from the initial
probability density, . The general method is referred to as
*particle filtering* and has yielded good performance in
applications to experimental mobile robotics. Recall Figure
1.7 and see Section 12.2.3.

Let denote a finite collection of samples. A probability distribution is defined over . The collection of samples, together with its probability distribution, is considered as an approximation of a probability density over . Since is used to represent probabilistic I-states, let denote the probability distribution over , which is computed at stage using the history I-state . Thus, at every stage, there is a new sample set, , and probability distribution, .

The general method to compute the probabilistic I-state update proceeds as follows. For some large number, , of iterations, perform the following:

- Select a state according to the distribution .
- Generate a new sample, , for by generating a single sample according to the density .
- Assign the weight, .

Steven M LaValle 2012-04-20