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From what I've read so far, it seems that a Rao-Blackwellized particle filter is just a normal particle filter used after marginalizing a variable from:

$$p(r_t,s_t | y^t)$$

I'm not really sure about that conclusion, so I would like to know the precise differences between these two types of filters. Thanks in advance.

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  • $\begingroup$ are you asking how Rao-Blackwellized particle filter works in slam ? what do you mean by regular filters ? $\endgroup$
    – nayab
    Commented Feb 20, 2014 at 9:53

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The Rao-Blackwellized Particle Filter (RBPF) as you say in your question performs a marginalization of the probability distribution of your state space.

The particle filter uses sampling to represent the multivariate probability distribution of your state space. Using samples to represent a distribution is firstly only an approximation, and secondly not very efficient in most cases. The higher the dimension of the state, the more particles you require. One trick introduced by Doucet et al. is to marginalize out a subset of the state space, which can be handled in a more efficient way by using a Gaussian representation.

These figures from my thesis might help to visualize the concept. Let's say you have a joint distribution over $X$ and $Y$

enter image description here

instead of sampling jointly over $X$ and $Y$, we can marginalize $Y$ and represent it as a Gaussian distribution. In this way we just need to sample over $X$,

enter image description here

which is much more efficient. Note that each sample now represent a distribution over $Y$.

This marginalization is very popular in SLAM. The reason is that jointly sampling over position and map is impractical. The RBPF idea has been made popular in FastSLAM, realizing that marginalizing the maps from the joint distribution makes the problem tractable. As in the example above, each particle there represents a pose, correspondences and a map. So there is one map per particle.

So the difference between an RBPF and a regular particle filter is that the RBPF samples over a subspace of the probability distribution of the state, and represents the rest using a different statistic. I have another related question on the math background of the Rao-Blackwellization part.

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  • $\begingroup$ That's a nice explanation of why marginalization is important, but doesn't really go into the mechanism of marginalization. $\endgroup$
    – Madmartigan
    Commented Aug 20, 2022 at 4:19
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Page 151 should help,

https://users.aalto.fi/~ssarkka/pub/cup_book_online_20131111.pdf

where u is a sample (particle without weight)

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  • $\begingroup$ Nice explanation in the book. For the answer it is always good to at least summarize the content of the link, as it may not be available forever. $\endgroup$
    – Jakob
    Commented Apr 27, 2017 at 9:31

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