10.5 Sequential Monte Carlo
One of the caveats of Metropolis-Hastings and NUTS (and other Hamiltonian Monte Carlo variants) is that if the posterior has multiple peaks and these peaks are separated by regions of very low probability, these methods can get stuck in a single mode and miss the others!
Many of the methods developed to overcome this multiple minima problem are based on the idea of tempering. This idea, once again, is borrowed from statistical mechanics. The number of states a physical system can populate depends on the temperature of the system; at 0 Kelvin (the lowest possible temperature), every system is stuck in a single state. On the other extreme, for an infinite temperature, all possible states are equally likely. Generally, we are interested in systems at some intermediate temperature. For Bayesian models, there is a very intuitive way to adapt this tempering idea by writing Bayes’ theorem with a twist.
The parameter β is known as the inverse temperature...
