The Metropolis algorithm is a method to sample a normalized probability distribution by a stochastic process. We define {\cal w}_i^{(n)} to be the probability for finding the system in the state i at step n .
In the simulations, our assumption is that we have a model for {\cal w}_i^{(n)} , but we do not know W . We will hence model W in terms of a likelihood for making transition T and a likelihood for accepting a transition. That is
W_{i\rightarrow j}=A_{i\rightarrow j}T_{i\rightarrow j}