irreversible Markov chains

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , on November 20, 2018 by xi'an

Werner Krauth (ENS, Paris) was in Dauphine today to present his papers on irreversible Markov chains at the probability seminar. He went back to the 1953 Metropolis et al. paper. And mentioned a 1962 paper I had never heard of by Alder and Wainwright demonstrating phase transition can occur, via simulation. The whole talk was about simulating the stationary distribution of a large number of hard spheres on a one-dimensional ring, which made it hard for me to understand. (Maybe the triathlon before did not help.) And even to realise a part was about PDMPs… His slides included this interesting entry on factorised MCMC which reminded me of delayed acceptance and thinning and prefetching. Plus a notion of lifted Metropolis that could have applications in a general setting, if it differs from delayed rejection.

spacings on a torus

Posted in Books, Kids, R, Statistics, University life with tags , , , , , , , , on March 22, 2018 by xi'an

While in Brussels last week I noticed an interesting question on X validated that I considered in the train back home and then more over the weekend. This is a question about spacings, namely how long on average does it take to cover an interval of length L when drawing unit intervals at random (with a torus handling of the endpoints)? Which immediately reminded me of Wilfrid Kendall (Warwick) famous gif animation of coupling from the past via leaves covering a square region, from the top (forward) and from the bottom (backward)…

The problem is rather easily expressed in terms of uniform spacings, more specifically on the maximum spacing being less than 1 (or 1/L depending on the parameterisation). Except for the additional constraint at the boundary, which is not independent of the other spacings. Replacing this extra event with an independent spacing, there exists a direct formula for the expected stopping time, which can be checked rather easily by simulation. But the exact case appears to be add a few more steps to the draws, 3/2 apparently. The following graph displays the regression of the Monte Carlo number of steps over 10⁴ replicas against the exact values: