Archive for Monte Carlo error


Posted in Books, Statistics with tags , , , , , , , , , , , , , on June 25, 2018 by xi'an

A recent arXival on a new version of ABC based on kernel estimators (but one could argue that all ABC versions are based on kernel estimators, one way or another.) In this ABC-CDE version, Izbicki,  Lee and Pospisilz [from CMU, hence the picture!] argue that past attempts failed to exploit the full advantages of kernel methods, including the 2016 ABCDE method (from Edinburgh) briefly covered on this blog. (As an aside, CDE stands for conditional density estimation.) They also criticise these attempts at selecting summary statistics and hence failing in sufficiency, which seems a non-issue to me, as already discussed numerous times on the ‘Og. One point of particular interest in the long list of drawbacks found in the paper is the inability to compare several estimates of the posterior density, since this is not directly ingrained in the Bayesian construct. Unless one moves to higher ground by calling for Bayesian non-parametrics within the ABC algorithm, a perspective which I am not aware has been pursued so far…

The selling points of ABC-CDE are that (a) the true focus is on estimating a conditional density at the observable x⁰ rather than everywhere. Hence, rejecting simulations from the reference table if the pseudo-observations are too far from x⁰ (which implies using a relevant distance and/or choosing adequate summary statistics). And then creating a conditional density estimator from this subsample (which makes me wonder at a double use of the data).

The specific density estimation approach adopted for this is called FlexCode and relates to an earlier if recent paper from Izbicki and Lee I did not read. As in many other density estimation approaches, they use an orthonormal basis (including wavelets) in low dimension to estimate the marginal of the posterior for one or a few components of the parameter θ. And noticing that the posterior marginal is a weighted average of the terms in the basis, where the weights are the posterior expectations of the functions themselves. All fine! The next step is to compare [posterior] estimators through an integrated squared error loss that does not integrate the prior or posterior and does not tell much about the quality of the approximation for Bayesian inference in my opinion. It is furthermore approximated by  a doubly integrated [over parameter and pseudo-observation] squared error loss, using the ABC(ε) sample from the prior predictive. And the approximation error only depends on the regularity of the error, that is the difference between posterior and approximated posterior. Which strikes me as odd, since the Monte Carlo error should take over but does not appear at all. I am thus unclear as to whether or not the convergence results are that relevant. (A difficulty with this paper is the strong dependence on the earlier one as it keeps referencing one version or another of FlexCode. Without reading the original one, I spotted a mention made of the use of random forests for selecting summary statistics of interest, without detailing the difference with our own ABC random forest papers (for both model selection and estimation). For instance, the remark that “nuisance statistics do not affect the performance of FlexCode-RF much” reproduces what we observed with ABC-RF.

The long experiment section always relates to the most standard rejection ABC algorithm, without accounting for the many alternatives produced in the literature (like Li and Fearnhead, 2018. that uses Beaumont et al’s 2002 scheme, along with importance sampling improvements, or ours). In the case of real cosmological data, used twice, I am uncertain of the comparison as I presume the truth is unknown. Furthermore, from having worked on similar data a dozen years ago, it is unclear why ABC is necessary in such context (although I remember us running a test about ABC in the Paris astrophysics institute once).

seeking the error in nested sampling

Posted in pictures, Statistics, Travel with tags , , , , , , on April 13, 2017 by xi'an

A newly arXived paper on the error in nested sampling, written by Higson and co-authors, and read in Berlin, looks at the difficult task of evaluating the sampling error of nested sampling. The conclusion is essentially negative in that the authors recommend multiple runs of the method to assess the magnitude of the variability of the output by bootstrap, i.e. to call for the most empirical approach…

The core of this difficulty lies in the half-plug-in, half-quadrature, half-Monte Carlo (!) feature of the nested sampling algorithm, in that (i) the truncation of the unit interval is based on a expectation of the mass of each shell (i.e., the zone between two consecutive isoclines of the likelihood, (ii) the evidence estimator is a quadrature formula, and (iii) the level of the likelihood at the truncation is replaced with a simulated value that is not even unbiased (and correlated with the previous value in the case of an MCMC implementation). As discussed in our paper with Nicolas, the error in the evidence approximation is of the same order as other Monte Carlo methods in that it gets down like the square root of the number of terms at each iteration. Contrary to earlier intuitions that focussed on the error due to the quadrature.

But the situation is much less understood when the resulting sample is used for estimation of quantities related with the posterior distribution. With no clear approach to assess and even less correct the resulting error, since it is not solely a Monte Carlo error. As noted by the authors, the quadrature approximation to the univariate integral replaces the unknown prior weight of a shell with its Beta order statistic expectation and the average of the likelihood over the shell with a single (uniform???) realisation. Or the mean value of a transform of the parameter with a single (biased) realisation. Since most posterior expectations can be represented as integrals over likelihood levels of the average value over an iso-likelihood contour. The approach advocated in the paper involved multiple threads of an “unwoven nested sampling run”, which means launching n nested sampling runs with one living term from the n currents living points in the current nested sample. (Those threads may then later be recombined into a single nested sample.) This is the starting point to a nested flavour of bootstrapping, where threads are sampled with replacement, from which confidence intervals and error estimates can be constructed. (The original notion appears in Skilling’s 2006 paper, but I missed it.)

The above graphic is an attempt within the paper at representing the (marginal) posterior of a transform f(θ). That I do not fully understand… The notations are rather horrendous as X is not the data but the prior probability for the likelihood to be above a given bound which is actually the corresponding quantile. (There is no symbol for data and £ is used for the likelihood function as well as realisations of the likelihood function…) A vertical slice on the central panel gives the posterior distribution of f(θ) given the event that the likelihood is in the corresponding upper tail. Or given the corresponding shell (?).

control functionals for Monte Carlo integration

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , on June 28, 2016 by xi'an

img_2451A paper on control variates by Chris Oates, Mark Girolami (Warwick) and Nicolas Chopin (CREST) appeared in a recent issue of Series B. I had read and discussed the paper with them previously and the following is a set of comments I wrote at some stage, to be taken with enough gains of salt since Chris, Mark and Nicolas answered them either orally or in the paper. Note also that I already discussed an earlier version, with comments that are not necessarily coherent with the following ones! [Thanks to the busy softshop this week, I resorted to publish some older drafts, so mileage can vary in the coming days.]

First, it took me quite a while to get over the paper, mostly because I have never worked with reproducible kernel Hilbert spaces (RKHS) before. I looked at some proofs in the appendix and at the whole paper but could not spot anything amiss. It is obviously a major step to uncover a manageable method with a rate that is lower than √n. When I set my PhD student Anne Philippe on the approach via Riemann sums, we were quickly hindered by the dimension issue and could not find a way out. In the first versions of the nested sampling approach, John Skilling had also thought he could get higher convergence rates before realising the Monte Carlo error had not disappeared and hence was keeping the rate at the same √n speed.

The core proof in the paper leading to the 7/12 convergence rate relies on a mathematical result of Sun and Wu (2009) that a certain rate of regularisation of the function of interest leads to an average variance of order 1/6. I have no reason to mistrust the result (and anyway did not check the original paper), but I am still puzzled by the fact that it almost immediately leads to the control variate estimator having a smaller order variance (or at least variability). On average or in probability. (I am also uncertain on the possibility to interpret the boxplot figures as establishing super-√n speed.)

Another thing I cannot truly grasp is how the control functional estimator of (7) can be both a mere linear recombination of individual unbiased estimators of the target expectation and an improvement in the variance rate. I acknowledge that the coefficients of the matrices are functions of the sample simulated from the target density but still…

Another source of inner puzzlement is the choice of the kernel in the paper, which seems too simple to be able to cover all problems despite being used in every illustration there. I see the kernel as centred at zero, which means a central location must be know, decreasing to zero away from this centre, so possibly missing aspects of the integrand that are too far away, and isotonic in the reference norm, which also seems to preclude some settings where the integrand is not that compatible with the geometry.

I am equally nonplussed by the existence of a deterministic bound on the error, although it is not completely deterministic, depending on the values of the reproducible kernel at the points of the sample. Does it imply anything restrictive on the function to be integrated?

A side remark about the use of intractable in the paper is that, given the development of a whole new branch of computational statistics handling likelihoods that cannot be computed at all, intractable should possibly be reserved for such higher complexity models.