State-Dependent Mixing

Green (1995) proposed an algorithm that involves state-dependent mixing having mixing probabilities that depend on the current state. There are a finite or infinite set of transition kernels , , and state-dependent mixing probabilities . The overall transition kernel is

To make a move when at x, we choose kernel with probability and then simulate the next state with probability .

No nice properties are transferred from the to P, so we introduce the substochastic kernels . Then , so if all of the are reversible with respect to , then so is P, and if P is stochastic, then is a stationary distribution of P. Note that each determines and through

 

and

 

so we may consider that we have been given the to specify the algorithm.

A simple trick allows us to use the same argument when we are given a set of substochastic kernels , , that sum to a substochastic kernel, that is

 

Define the defect

and a new kernel

where I is the identity kernel, defined by

Then is reversible with respect to any distribution since

is trivially symmetric under the interchange of u and v. If we add to our set of kernels, then the sum is stochastic.

Thus we have the following formulation of state-dependent mixing. Suppose we are given a set of substochastic kernels satisfying (1.7). Then the following combined update

1. Choose with i probability defined by (1.5). With probability , skip step 2 and stay at the current position.
2. Simulate a new value of x from the probability distribution defined by (1.6).

has the stochastic transition kernel and is reversible with respect to if each of the is reversible with respect to .