26.1 Metropolis Method 26.1.1 Mcmc Review 26.1.2 Metropolis Filter

But how do we actually construct a Markov chain with a stationary distribution equal to our target distribution? Also, we want this method to have a good (that is, small) mixing time. The Metropolis method allows us achieve these goals by defining our Markov chain as a random walk over a suitably defined graph. We define the approach as follows. Say we which to sample values i ∈ Ω from a distribution Q(i). Then we define an undirected d-regular graph G on Ω, picking this graph in such a way that it has high conductance. Then from node v, pick the next node u uniformly from the d neighbors. Then: