Four Metropolis chains sample the posterior of a dimensionless rate
PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Computational Physics · Data analysis and model fitting in the laboratory
Error, Chance and the Machine, Ch Markov-chain Monte Carlo
Problem
Four Metropolis chains sample the posterior of a dimensionless rate θ. They start from θ=0.5, 1.5, 1.0 and 3.0, and each keeps n=2000 steps after discarding the first 200. The chain means of θ are 1.02, 0.99, 1.03 and 1.21, and every chain has sample variance 0.040. The integrated autocorrelation time of θ, estimated from chains 1 to 3, is τint≈8.
((a)) Compute W, B/n and R^ from Eq. (11.22).
((b)) For chains that have converged, B/n is about 2τintvar(θ)/n. Find what B/n and R^ would be for converged chains here, take var(θ)≈W, and compare with part (a). Give a verdict.
((c)) Suppose only chain 3 had been run. Find its blocked error from σ2=2τintW/n and say how many such errors separate its mean from that of chain 4. What would the single run have reported?
((d)) Compute R^ from chains 1 to 3 alone. Does the result show that they have converged? What does the comparison with part (a) say about the choice of starts?