Rao-Blackwellization
The Rao-Blackwell theorem says that if we want an estimator with small MSE we can confine our search to estimators which are functions of the sufficient statistic. - Richard Weber, Statistics (Part IB) Lecture Notes
To give a simple example:
If we want to approximate the joint distribution $p(a,b)$ with samples, a sampler that samples a from $p(a)$ and then determines b analytically conditioned on a, $p(b \vert a)$, is no worse than a sampler that samples directly from the joint distribution p(a, b).
The former process is sometimes called a Rao-Blackwellization process.
Rao-Blackwellization for SLAM
Rao-Blackwellization provides a powerful tool for particle filter in high dimensional space. More specifically speaking, particle filter becomes inefficient when the state is high-dimensional. That is because it requires more samples to sufficiently approximate the joint distribution of all the state variables. With Rao-Blackwellization, we can factor the joint distribution to exploit dependencies between variables.
Example in robot mapping:
The joint distribution robot path $x_{0:t}$ and the landmarks $m_{1:M}$ can be represented by individual sample of the robot paths and the corresponding conditional probability of the landmarks given the robot path:
The state of the particle filter in FastSLAM becomes the robot path only $x_{1:t}$, rather than the higher dimensional counterpart, $x_{1:t}, m1, \dots, m_M$. This way the particle filtering becomes more efficient.