So far, we have discussed various techniques to sample using full instantiations over the variables. However, the problem with full instantiations is that they can only cover a very small region of the space, as the space is exponential to the number of variables. The solution to this is to have partial instantiations of the variables and use a closed-form representation of a distribution over the rest. Collapsed particles are also known as Rao-Blackwellized particles.

So, considering as the set of variables over which we will do the assignments and which the particle will depend on, and as the set of variables over which we define a closed-form distribution, if we want to estimate the expectation of some function relative to our posterior distribution we have the following:

Also, we are assuming that the internal expectation can be computed easily. So essentially, we are using a hybrid approach in the case of collapsed particles. We generate particles for the variables...