Fastest mixing Markov chain on a path [3 citations — 2 self]
Abstract:
Simulation using Markov chain Monte Carlo is a mainstay of scientific computing; see, e.g., [4, 5] for pointers to the literature. Thus the analysis and design of fast mixing Markov PSfrag chains, replacements with given stationary distribution, has become a research area. In [2], we show how to numerically find the fastest mixing Markov chain (i.e., the one with smallest secondlargest eigenvalue modulus) on a given underlying graph using tools of convex optimization, in particular, semidefinite programming [8, 3]. The present note presents a simple, self contained example where the optimal Markov chain can be identified analytically. We consider a random walk on a path with n ≥ 2 nodes, labeled 0, 1,..., n − 1. There are n − 1 edges connecting pairs of adjacent nodes; in addition, we allow a loop at each node, as shown in figure 1. Let Pij denote the transition probability from node i to node j. We consider symmetric transition probabilities, i.e., those that satisfy Pij = Pji. Since P is symmetric, the uniform distribution is stationary. The requirement that transitions can only occur along an edge or loop of the path is equivalent to Pij = 0 for |i − j |> 1, i.e., P is tridiagonal. Thus, P is a symmetric, stochastic, tridiagonal matrix.
Citations
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