## Hastings at 50, from a Metropolis

Posted in Kids, pictures, Running, Travel with tags , , , , , , , , , , , , , , , , , , , , , , on January 4, 2020 by xi'an

A weekend trip to the quaint seaside city of Le Touquet Paris-Plage, facing the city of Hastings on the other side of the Channel, 50 miles away (and invisible on the pictures!), during and after a storm that made for a fantastic watch from our beach-side rental, if less for running! The town is far from being a metropolis, actually, but it got its added surname “Paris-Plage” from British investors who wanted to attract their countrymen in the late 1800s. The writers H.G. Wells and P.G. Wodehouse lived there for a while. (Another type of tourist, William the Conqueror, left for Hastings in 1066 from a wee farther south, near Saint-Valéry-sur-Somme.)

And the coincidental on-line publication in Biometrika of a 50 year anniversary paper, The Hastings algorithm at fifty by David Dunson and James Johndrow. More of a celebration than a comprehensive review, with focus on scalable MCMC, gradient based algorithms, Hamiltonian Monte Carlo, nonreversible Markov chains, and interesting forays into approximate Bayes. Which makes for a great read for graduate students and seasoned researchers alike!

## single variable transformation approach to MCMC

Posted in Books, Statistics, Travel with tags , , , , on September 9, 2014 by xi'an

I read the newly arXived paper “On Single Variable Transformation Approach to Markov Chain Monte Carlo” by Dey and Bhattacharya on the pleasant train ride between Bristol and Coventry last weekend. The paper actually follows several earlier papers by the authors that I have not read in detail. The notion of single variable transform is to add plus or minus the same random noise to all components of the current value of the Markov chain, instead of the standard d-dimensional random walk proposal of the reference Metropolis-Hastings algorithm, namely all proposals are of the form

$x_i'=x_i\pm \epsilon\ i=1,\cdots,d$

meaning the chain proceeds [after acceptance] along one and only one of the d diagonals. The authors’ arguments are that (a) the proposal is cheaper and (b) the acceptance rate is higher. What I find questionable in this argument is that this does not directly matter in the evaluation of the performances of the algorithm. For instance, higher acceptance in a Metropolis-Hasting algorithm does not imply faster convergence and smaller asymptotic variance. (This goes without mentioning the fact that the comparative Figure 1 is so variable with the dimension as to be of limited worth. Figure 1 and 2 are also found in an earlier arXived paper of the authors.) For instance, restricting the moves along the diagonals of the Euclidean space implies that there is a positive probability to make two successive proposals along the same diagonal, which is a waste of time. When considering the two-dimensional case, joining two arbitrary points using an everywhere positive density g upon ε means generating two successive values from g, which is equivalent cost-wise to generating a single noise from a two-dimensional proposal. Without the intermediate step of checking the one-dimensional move along one diagonal. So much for a gain. In fine, the proposal found in this paper sums up as being a one-at-a-time version of a standard random walk Metropolis-Hastings algorithm.

## austerity in MCMC land (#2)

Posted in R, Statistics with tags , , , on April 29, 2013 by xi'an

After reading the arXiv paper by Korattikara, Chen and Welling, I wondered about the expression of the acceptance step of the Metropolis-Hastings algorithm as a mean of log-likelihoods over the sample. More specifically the long sleepless nights at the hospital led me to ponder the rather silly question of the impact of replacing mean by median. I thus tried running a Metropolis-Hastings algorithm with the substitute and it (of course!) let to a nonsensical answer, as shown by the above graph. The true posterior is the one for a normal model and the histogram indicates a lack of convergence of the Markov chain to this posterior even though it does converge to some posterior. Here is the R code for this tiny experiment:

#data generation
N=100
x=rnorm(N)

#HM steps
T=10^5
theta=rep(0,T)
curlike=dnorm(x,log=TRUE)
for (t in 2:T){

prop=theta[t-1]+.1*rnorm(1)
proplike=dnorm(x,mean=prop,log=TRUE)
u=runif(1)
bound=log(u)-dnorm(prop,sd=10,log=TRUE)+
dnorm(theta[t-1],sd=10,log=TRUE)
if (median(proplike)-median(curlike)>bound/N){
theta[t]=prop;curlike=proplike
} else { theta[t]=theta[t-1]}
}


## making a random walk geometrically ergodic

Posted in R, Statistics with tags , , , , , , , , , on March 2, 2013 by xi'an

While a random walk Metropolis-Hastings algorithm cannot be uniformly ergodic in a general setting (Mengersen and Tweedie, AoS, 1996), because it needs more energy to leave far away starting points, it can be geometrically ergodic depending on the target (and the proposal). In a recent Annals of Statistics paper, Leif Johnson and Charlie Geyer designed a trick to turn a random walk Metropolis-Hastings algorithm into a geometrically ergodic random walk Metropolis-Hastings algorithm by virtue of an isotropic transform (under the provision that the original target density has a moment generating function). This theoretical result is complemented by an R package called mcmc. (I have not tested it so far, having read the paper in the métro.) The examples included in the paper are however fairly academic and I wonder how the method performs in practice, on truly complex models, in particular because the change of variables relies on (a) an origin and (b) changing the curvature of space uniformly in all dimensions. Nonetheless, the idea is attractive and reminds me of a project of ours with Randal Douc,  started thanks to the ‘Og and still under completion.