Archive for Monte Carlo Statistical Methods

Monte Carlo with determinantal processes [reply from the authors]

Posted in Books, Statistics with tags , , , , , , , , , , , , , , on September 22, 2016 by xi'an

[Rémi Bardenet and Adrien Hardy have written a reply to my comments of today on their paper, which is more readable as a post than as comments, so here it is. I appreciate the intention, as well as the perfect editing of the reply, suited for a direct posting!]

Thanks for your comments, Xian. As a foreword, a few people we met also had the intuition that DPPs would be relevant for Monte Carlo, but no result so far was backing this claim. As it turns out, we had to work hard to prove a CLT for importance-reweighted DPPs, using some deep recent results on orthogonal polynomials. We are currently working on turning this probabilistic result into practical algorithms. For instance, efficient sampling of DPPs is indeed an important open question, to which most of your comments refer. Although this question is out of the scope of our paper, note however that our results do not depend on how you sample. Efficient sampling of DPPs, along with other natural computational questions, is actually the crux of an ANR grant we just got, so hopefully in a few years we can write a more detailed answer on this blog! We now answer some of your other points.

“one has to examine the conditions for the result to operate, from the support being within the unit hypercube,”
Any compactly supported measure would do, using dilations, for instance. Note that we don’t assume the support is the whole hypercube.

“to the existence of N orthogonal polynomials wrt the dominating measure, not discussed here”
As explained in Section 2.1.2, it is enough that the reference measure charges some open set of the hypercube, which is for instance the case if it has a density with respect to the Lebesgue measure.

“to the lack of relation between the point process and the integrand,”
Actually, our method depends heavily on the target measure μ. Unlike vanilla QMC, the repulsiveness between the quadrature nodes is tailored to the integration problem.

“changing N requires a new simulation of the entire vector unless I missed the point.”
You’re absolutely right. This is a well-known open issue in probability, see the discussion on Terence Tao’s blog.

“This requires figuring out the upper bounds on the acceptance ratios, a “problem-dependent” request that may prove impossible to implement”
We agree that in general this isn’t trivial. However, good bounds are available for all Jacobi polynomials, see Section 3.

“Even without this stumbling block, generating the N-sized sample for dimension d=N (why d=N, I wonder?)”
This is a misunderstanding: we do not say that d=N in any sense. We only say that sampling from a DPP using the algorithm of [Hough et al] requires the same number of operations as orthonormalizing N vectors of dimension N, hence the cubic cost.

1. “how does it relate to quasi-Monte Carlo?”
So far, the connection to QMC is only intuitive: both rely on well-spaced nodes, but using different mathematical tools.

2. “the marginals of the N-th order determinantal process are far from uniform (see Fig. 1), and seemingly concentrated on the boundaries”
This phenomenon is due to orthogonal polynomials. We are investigating more general constructions that give more flexibility.

3. “Is the variance of the resulting estimator (2.11) always finite?”
Yes. For instance, this follows from the inequality below (5.56) since ƒ(x)/K(x,x) is Lipschitz.

4. and 5. We are investigating concentration inequalities to answer these points.

6. “probabilistic numerics produce an epistemic assessment of uncertainty, contrary to the current proposal.”
A partial answer may be our Remark 2.12. You can interpret DPPs as putting a Gaussian process prior over ƒ and sequentially sampling from the posterior variance of the GP.

Bayesian model selection without evidence

Posted in Books, Statistics, University life with tags , , , , , , , on September 20, 2016 by xi'an

“The new method circumvents the challenges associated with accurate evidence calculations by computing posterior odds ratios using Bayesian parameter estimation”

One paper leading to another, I had a look at Hee et al. 2015 paper on Bayes factor estimation. The “novelty” stands in introducing the model index as an extra parameter in a single model encompassing all models under comparison, the “new” parameterisation being in (θ,n) rather than in θ. With the distinction that the parameter θ is now made of the union of all parameters across all models. Which reminds us very much of Carlin and Chib (1995) approach to the problem. (Peter Green in his Biometrika (1995) paper on reversible jump MCMC uses instead a direct sum of parameter spaces.) The authors indeed suggest simulating jointly (θ,n) in an MCMC or nested sampling scheme. Rather than being updated by arbitrary transforms as in Carlin and Chib (1995) the useless parameters from the other models are kept constant… The goal being to estimate P(n|D) the marginal posterior on the model index, aka the posterior probability of model n.

Now, I am quite not certain keeping the other parameter constants is a valid move: given a uniform prior on n and an equally uniform proposal, the acceptance probability simplifies into the regular Metropolis-Hastings ratio for model n. Hence the move is valid within model n. If not, I presume the previous pair (θ⁰,n⁰) is repeated. Wait!, actually, this is slightly more elaborate: if a new value of n, m, is proposed, then the acceptance ratio involves the posteriors for both n⁰ and m, possibly only the likelihoods when the proposal is the prior. So the move will directly depend on the likelihood ratio in this simplified case, which indicates the scheme could be correct after all. Except that this neglects the measure theoretic subtleties that led to reversible jump symmetry and hence makes me wonder. In other words, it follows exactly the same pattern as reversible jump without the constraints of the latter… Free lunch,  anyone?!

importance sampling and necessary sample size

Posted in Books, Statistics with tags , , , , , on September 7, 2016 by xi'an

Daniel Sanz-Alonso arXived a note yesterday where he analyses importance sampling from the point of view of empirical distributions. With the difficulty that unnormalised importance sampling estimators are not associated with an empirical distribution since the sum of the weights is not one. For several f-divergences, he obtains upper bounds on those divergences between the empirical cdf and a uniform version, D(w,u), which translate into lower bounds on the importance sample size. I however do not see why this divergence between a weighted sampled and the uniformly weighted version is relevant for the divergence between the target and the proposal, nor how the resulting Monte Carlo estimator is impacted by this bound. A side remark [in the paper] is that those results apply to infinite variance Monte Carlo estimators, as in the recent paper of Chatterjee and Diaconis I discussed earlier, which also discussed the necessary sample size.

conditional sampling

Posted in R, Statistics with tags , , , , on September 5, 2016 by xi'an

An interesting question about stratified sampling came up on X validated last week, namely how to optimise a Monte Carlo estimate based on two subsequent simulations, one, X, from a marginal and one or several Y from the corresponding conditional given X, when the costs of producing those two simulations significantly differ. When looking at the variance of the Monte Carlo estimate, this variance can be minimised in the number of simulations from the marginal under a computing budget. However, when the costs of producing x, y given or (x,y) are about the same, it does not pay to replicate simulations from y given x or x given y, because this increases the randomness of the estimator and induces positive correlation between some terms in the sum. Assuming that the conditional variances are always computable, we could envision an extension to the question where for each new value of a simulated x (or y), a variable number of conditionally simulated y (or x) could be produced. Or even when those conditional variances are unknown but can be replaced with empirical versions.

The illustration comes from a bivariate normal model with correlation, for a rather arbitrary function , but the pattern remains the same, namely that iid simulations of the pair (X,Y invariably leads to a smaller variance of the estimator compared with simulation with a 1:10 (10 Y’s for one X) or 10:1 ratio between x’s and y’s. Depending on the function and the relative variances, the 1:10 or 10:1 schemes may have a similar variability.

zigma=c(9,1,-.9*sqrt(1*9))

geney=function(x,n=1){
  return(rnorm(n,mean=zigma[3]*x/zigma[1],sd=sqrt(zigma[2]-
    zigma[3]^2/zigma[1])))}
genex=function(y,n=1){
  return(rnorm(n,mean=zigma[3]*y/zigma[2],sd=sqrt(zigma[1]-
     zigma[3]*zigma[3]/zigma[2])))}
targ=function(x,y){ log(x^2*y^4)+x^2*cos(x^2)/y*sin(y^2)}

T=1e4;N=1e3
vales=matrix(0,N,3)
for (i in 1:N){
   xx=rnorm(T,sd=sqrt(zigma[1]))
   vales[i,1]=mean(targ(xx,geney(xx,n=T)))
   xx=rep(rnorm(T/10,sd=sqrt(zigma[1])),10)
   vales[i,2]=mean(targ(xx,geney(xx,n=T)))
   yy=rep(rnorm(T/10,sd=sqrt(zigma[2])),10)
   vales[i,3]=mean(targ(enex(yy,n=T),yy))}

MCqMC 2016 [#4]

Posted in Mountains, pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , , , on August 21, 2016 by xi'an

In his plenary talk this morning, Arnaud Doucet discussed the application of pseudo-marginal techniques to the latent variable models he has been investigating for many years. And its limiting behaviour towards efficiency, with the idea of introducing correlation in the estimation of the likelihood ratio. Reducing complexity from O(T²) to O(T√T). With the very surprising conclusion that the correlation must go to 1 at a precise rate to get this reduction, since perfect correlation would induce a bias. A massive piece of work, indeed!

The next session of the morning was another instance of conflicting talks and I hoped from one room to the next to listen to Hani Doss’s empirical Bayes estimation with intractable constants (where maybe SAME could be of interest), Youssef Marzouk’s transport maps for MCMC, which sounds like an attractive idea provided the construction of the map remains manageable, and Paul Russel’s adaptive importance sampling that somehow sounded connected with our population Monte Carlo approach. (With the additional step of considering transform maps.)

An interesting item of information I got from the final announcements at MCqMC 2016 just before heading to Monash, Melbourne, is that MCqMC 2018 will take place in the city of Rennes, Brittany, on July 2-6. Not only it is a nice location on its own, but it is most conveniently located in space and time to attend ISBA 2018 in Edinburgh the week after! Just moving from one Celtic city to another Celtic city. Along with other planned satellite workshops, this occurrence should make ISBA 2018 more attractive [if need be!] for participants from oversea.

MCqMC 2016 [#2]

Posted in pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , , , , on August 17, 2016 by xi'an

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…)

MCqMC 2016 [#1]

Posted in Statistics, Travel, University life with tags , , , , , , , , on August 16, 2016 by xi'an

mcqmc1This week, I attend the MCqMC 2016 conference in Stanford, which is quite an exciting gathering of researchers involved in various aspects of Monte Carlo methods. As Art Owen put it in his welcoming talk, the whole Carlo family is there! (Not to mention how pleasant the Stanford Campus currently is, after the scorching heat we met the past week in Northern California inlands.) My talk is on folded Markov chains, which is a proposal Randal and I have been working on for quite a while, with Gareth joining us more recently. The basic idea was inspired from a discussion I had about a blog post, so long ago that I cannot even trace it! Namely, when defining an inside set A and an outside set, such that the outside set can be projected onto the inside set, one can fold both the target and the proposal, essentially looking at a collection of values for each step of the Markov chain. In other words, the problem can be reduced to A at essentially no cost and with the benefits of a compact support A and of a possibly uniformly ergodic Markov chain. We are still working on the paper, but the idea is both cool and straightforward, so we decided to talk about it at Nordstat 2016 and now MCqMC 2016.