Parallel Tempering
Last updated
Last updated
There is an important probability distribution used in applications (Liu, 2008):
where . This type of model exhibits a so-called "Markovian structure" because
We can simulate from by an exact method. Note that
The simulating procedure is as follows:
For a one-dimensional Ising Model,
Liu (2008) also introduced the parallel tempering strategy:
I wrote the following Julia program to implement this procedure and reproduce the simulation example of Liu (2008):
Finally, I reproduce the Figure 10.1 as follows:
and the corresponding autocorrelation plots:
where . And thus the simulating procedure is much easy. Note that for ,
and .
We can use the following Julia program to simulate from :
It is important to note that we should use logpdf
to avoid too big values, in which the rejection part doesn't work well. Refer to for complete source code.