In her plenary talk this morning, Christine Lemieux discussed connections between quasi-Monte Carlo and copulas, covering a question I have been considering for a while. Namely, when provided with a (multivariate) joint cdf F, is there a generic way to invert a vector of uniforms [or quasi-uniforms] into a simulation from F? For Archimedian copulas (as we always can get back to copulas), there is a resolution by the Marshall-Olkin representation, but this puts a restriction on the distributions F that can be considered. The session on synthetic likelihoods [as introduced by Simon Wood in 2010] put together by Scott Sisson was completely focussed on using normal approximations for the distribution of the vector of summary statistics, rather than the standard ABC non-parametric approximation. While there is a clear (?) advantage in using a normal pseudo-likelihood, since it stabilises with much less simulations than a non-parametric version, I find it difficult to compare both approaches, as they lead to different posterior distributions. In particular, I wonder at the impact of the dimension of the summary statistics on the approximation, in the sense that it is less and less likely that the joint is normal as this dimension increases. Whether this is damaging for the resulting inference is another issue, possibly handled by a supplementary ABC step that would take the first-step estimate as summary statistic. (As a side remark, I am intrigued at everyone being so concerned with unbiasedness of methods that are approximations with no assessment of the amount of approximation!) The last session of the day was about multimodality and MCMC solutions, with talks by Hyungsuk Tak, Pierre Jacob and Babak Shababa, plus mine. Hunsuk presented the RAM algorithm I discussed earlier under the title of “love-hate” algorithm, which was a kind reference to my post! (I remain puzzled by the ability of the algorithm to jump to another mode, given that the intermediary step aims at a low or even zero probability region with an infinite mass target.) And Pierre talked about using SMC for Wang-Landau algorithms, with a twist to the classical stochastic optimisation schedule that preserves convergence. And a terrific illustration on a distribution inspired from the Golden Gate Bridge that reminded me of my recent crossing! The discussion around my folded Markov chain talk focussed on the extension of the partition to more than two sets, the difficulty being in generating automated projections, with comments about connections with computer graphic tools. (Too bad that the parallel session saw talks by Mark Huber and Rémi Bardenet that I missed! Enjoying a terrific Burmese dinner with Rémi, Pierre and other friends also meant I could not post this entry on time for the customary 00:16. Not that it matters in the least…)

## Archive for The night of the hunter

## MCqMC 2016 [#2]

Posted in pictures, Running, Statistics, Travel, University life with tags California, Cambodia, conference, copulas, folded Markov chain, Golden Gate Bridge, Laplace-Stieljes transform, love-hate algorithm, Marshall-Olkin algorithm, MCMC, MCqMC 2016, Monte Carlo Statistical Methods, quadrangle, quasi-Monte Carlo methods, simulation, SMC, Stanford University, summary statistics, synthetic likelihood, The night of the hunter, Wang-Landau algorithm on August 17, 2016 by xi'an## even dogs in the wild

Posted in Books, Mountains, Travel with tags book review, Edinburgh, Glasgow, Ian Rankin, Rebus, Scotland, The Associates, The night of the hunter on August 10, 2016 by xi'an**A** new Rankin, a new Rebus! (New as in 2015 since I waited to buy the paperback version.) Sounds like Ian Rankin cannot let his favourite character rest for his retirement and hence set in back into action, along with the new Malcom Fox [working in the Complaints] and most major characters of the Rebus series. Including the unbreakable villain, Big Ger Cafferty. This as classical as you get, borrows from half a dozen former Rebus novels, not to mention this neo-Holmes novel I reviewed a while ago. But it is gritty, deadly efficient and captivating. I read the book within a few days from returning from Warwick.

About the title, this is a song by The Associates that plays a role in the book. I did not this band, but looking for it got me to a clip that used an excerpt from the Night of the Hunter. Fantastic movie, one of my favourites.

## love-hate Metropolis algorithm

Posted in Books, pictures, R, Statistics, Travel with tags auxiliary variable, doubly intractable problems, Metropolis-Hastings algorithm, Monte Carlo Statistical Methods, multimodality, normalising constant, parallel tempering, pseudo-marginal MCMC, The night of the hunter, unbiased estimation on January 28, 2016 by xi'an**H**yungsuk Tak, Xiao-Li Meng and David van Dyk just arXived a paper on a multiple choice proposal in Metropolis-Hastings algorithms towards dealing with multimodal targets. Called “A repulsive-attractive Metropolis algorithm for multimodality” *[although I wonder why XXL did not jump at the opportunity to use the “love-hate” denomination!]*. The proposal distribution includes a [forced] downward Metropolis-Hastings move that uses the inverse of the target density π as its own target, namely 1/{π(x)+ε}. Followed by a [forced] Metropolis-Hastings upward move which target is {π(x)+ε}. The +ε is just there to avoid handling ratios of zeroes (although I wonder why using the convention 0/0=1 would not work). And chosen as 10⁻³²³ by default in connection with R smallest positive number. Whether or not the “downward” move is truly downwards and the “upward” move is truly upwards obviously depends on the generating distribution: I find it rather surprising that the authors consider the *same* random walk density in both cases as I would have imagined relying on a more dispersed distribution for the downward move in order to reach more easily other modes. For instance, the downward move could have been based on an *anti*-Langevin proposal, relying on the gradient to proceed further down…

This special choice of a single proposal however simplifies the acceptance ratio (and keeps the overall proposal symmetric). The final acceptance ratio still requires a ratio of intractable normalising constants that the authors bypass by Møller et al. (2006) auxiliary variable trick. While the authors mention the alternative pseudo-marginal approach of Andrieu and Roberts (2009), they do not try to implement it, although this would be straightforward here: since the normalising constants are the probabilities of accepting a downward and an upward move, respectively. Those can easily be evaluated at a cost similar to the use of the auxiliary variables. That is,

– generate a few moves from the current value and record the proportion *p* of accepted downward moves;

– generate a few moves from the final proposed value and record the proportion *q* of accepted downward moves;

and replace the ratio of intractable normalising constants with *p/q*. It is not even clear that one needs those extra moves since the algorithm requires an acceptance in the downward and upward moves, hence generate Geometric variates associated with those probabilities p and q, variates that can be used for estimating them. From a theoretical perspective, I also wonder if forcing the downward and upward moves truly leads to an improved convergence speed. Considering the case when the random walk is poorly calibrated for either the downward or upward move, the number of failed attempts before an acceptance may get beyond the reasonable.

As XXL and David pointed out to me, the unusual aspect of the approach is that here the proposal density is intractable, rather than the target density itself. This makes using Andrieu and Roberts (2009) seemingly less straightforward. However, as I was reminded this afternoon at the statistics and probability seminar in Bristol, the argument for the pseudo-marginal based on an unbiased estimator is that w Q(w|x) has a marginal in x equal to π(x) when the expectation of w is π(x). In thecurrent problem, the proposal in x can extended into a proposal in (x,w), w P(w|x) whose marginal is the proposal on x.

If we complement the target π(x) with the conditional P(w|x), the acceptance probability would then involve

{π(x’) P(w’|x’) / π(x) P(w|x)} / {w’ P(w’|x’) / w P(w|x)} = {π(x’) / π(x)} {w/w’}

so it seems the pseudo-marginal (or auxiliary variable) argument also extends to the proposal. Here is a short experiment that shows no discrepancy between target and histogram:

nozero=1e-300 #love-hate move move<-function(x){ bacwa=1;prop1=prop2=rnorm(1,x,2) while (runif(1)>{pi(x)+nozero}/{pi(prop1)+nozero}){ prop1=rnorm(1,x,2);bacwa=bacwa+1} while (runif(1)>{pi(prop2)+nozero}/{pi(prop1)+nozero}) prop2=rnorm(1,prop1,2) y=x if (runif(1)<pi(prop2)*bacwa/pi(x)/fowa){ y=prop2;assign("fowa",bacwa)} return(y)} #arbitrary bimodal target pi<-function(x){.25*dnorm(x)+.75*dnorm(x,mean=5)} #running the chain T=1e5 x=5*rnorm(1);luv8=rep(x,T) fowa=1;prop1=rnorm(1,x,2) #initial estimate while (runif(1)>{pi(x)+nozero}/{pi(prop1)+nozero}){ fowa=fowa+1;prop1=rnorm(1,x,2)} for (t in 2:T) luv8[t]=move(luv8[t-1])