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
$$ W_{i\rightarrow j}=A_{i\rightarrow j}T_{i\rightarrow j} $$