When discussing the time taken by the Metropolis algorithm to generate independent samples we considered a distribution with longest spatial length scale L being explored using a proposal distribution...

When discussing the time taken by the Metropolis algorithm

to generate independent samples we considered a distribution with longest spatial length scale L being explored using a proposal distribution with step size . Another dimension that a MCMC method must explore is the range of possible values of the log probability ln P _(x).Assuming that the state x contains a number of independent random variables proportional to N, when samples are drawn from P(x), the `asymptotic equipartition" principle tell us that the value of ln P(x) is likely to be close to the entropy of x, varying either side with a standard deviation that scales as. Consider a Metropolis method with a symmetrical proposal density, that is, one that satisfiesAssuming that accepted jumps either increase ln P (x) by some amount or decrease it by a small amount, (is this a reasonable

assumption?), discuss how long it must take to generate roughly independent samples from P(x). Discuss whether Gibbs sampling has similar properties.