noisy importance sampling

A recent short arXival by Fernando Llorente, Luca Martino, Jesse Read, and David Delgado–Gómez in which they analyse settings where (only) a noisy version of the target density is available. Not necessarily in an unbiased fashion although the paper is somewhat unclear as to which integral is targeted in (6), since the integrand is not the original target p(x). The following development is about finding the optimal importance function, which differs from the usual due to the random nature of the approximation, but it does not seem to reconnect with the true target p(x), except when the noisy realisation is unbiased… To me this is a major issue in simulation methodology in that getting away from the unbiasedness constraint opens (rather obviously) a much wider choice of techniques.

what if what???

[Here is a section of the Wikipedia page on Monte Carlo methods which makes little sense to me. What if it was not part of this page?!]

Monte Carlo simulation versus “what if” scenarios

There are ways of using probabilities that are definitely not Monte Carlo simulations – for example, deterministic modeling using single-point estimates. Each uncertain variable within a model is assigned a “best guess” estimate. Scenarios (such as best, worst, or most likely case) for each input variable are chosen and the results recorded.^[55]

By contrast, Monte Carlo simulations sample from a probability distribution for each variable to produce hundreds or thousands of possible outcomes. The results are analyzed to get probabilities of different outcomes occurring.^[56] For example, a comparison of a spreadsheet cost construction model run using traditional “what if” scenarios, and then running the comparison again with Monte Carlo simulation and triangular probability distributions shows that the Monte Carlo analysis has a narrower range than the “what if” analysis. This is because the “what if” analysis gives equal weight to all scenarios (see quantifying uncertainty in corporate finance), while the Monte Carlo method hardly samples in the very low probability regions. The samples in such regions are called “rare events”.

a new rule for adaptive importance sampling

Art Owen and Yi Zhou have arXived a short paper on the combination of importance sampling estimators. Which connects somehow with the talk about multiple estimators I gave at ESM last year in Helsinki. And our earlier AMIS combination. The paper however makes two important assumptions to reach optimal weighting, which is inversely proportional to the variance:

1. the estimators are uncorrelated if dependent;
2. the variance of the k-th estimator is of order a (negative) power of k.

The later is puzzling when considering a series of estimators, in that k appears to act as a sample size (as in AMIS), the power is usually unknown but also there is no reason for the power to be the same for all estimators. The authors propose to use ½ as the default, both because this is the standard Monte Carlo rate and because the loss in variance is then minimal, being 12% larger.

As an aside, Art Owen also wrote an invited discussion “the unreasonable effectiveness of Monte Carlo” of ” Probabilistic Integration: A Role in Statistical Computation?” by François-Xavier Briol, Chris  Oates, Mark Girolami (Warwick), Michael Osborne and Deni Sejdinovic, to appear in Statistical Science, discussion that contains a wealth of smart and enlightening remarks. Like the analogy between pseudo-random number generators [which work unreasonably well!] vs true random numbers and Bayesian numerical integration versus non-random functions. Or the role of advanced bootstrapping when assessing the variability of Monte Carlo estimates (citing a paper of his from 1992). Also pointing out at an intriguing MCMC paper by  Michael Lavine and Jim Hodges to appear in The American Statistician.

Bayesian intelligence in Warwick

This is an announcement for an exciting CRiSM Day in Warwick on 20 March 2019: with speakers

10:00-11:00 Xiao-Li Meng (Harvard): “Artificial Bayesian Monte Carlo Integration: A Practical Resolution to the Bayesian (Normalizing Constant) Paradox”

11:00-12:00 Julien Stoehr (Dauphine): “Gibbs sampling and ABC”

14:00-15:00 Arthur Ulysse Jacot-Guillarmod (École Polytechnique Fedérale de Lausanne): “Neural Tangent Kernel: Convergence and Generalization of Deep Neural Networks”

15:00-16:00 Antonietta Mira (Università della Svizzera italiana e Università degli studi dell’Insubria): “Bayesian identifications of the data intrinsic dimensions”

[whose abstracts are on the workshop webpage] and free attendance. The title for the workshop mentions Bayesian Intelligence: this obviously includes human intelligence and not just AI!

more and more control variates

A few months ago, François Portier and Johan Segers arXived a paper on a question that has always puzzled me, namely how to add control variates to a Monte Carlo estimator and when to stop if needed! The paper is called Monte Carlo integration with a growing number of control variates. It is related to the earlier Oates, Girolami and Chopin (2017) which I remember discussing with Chris when he was in Warwick. The puzzling issue of control variates is [for me] that, while the optimal weight always decreases the variance of the resulting estimate, in practical terms, implementing the method may increase the actual variance. Glynn and Szechtman at MCqMC 2000 identify six different ways of creating the estimate, depending on how the covariance matrix, denoted P(hh’), is estimated. With only one version integrating constant functions and control variates exactly. Which actually happens to be also a solution to a empirical likelihood maximisation under the empirical constraints imposed by the control variates. Another interesting feature is that, when the number m of control variates grows with the number n of simulations the asymptotic variance goes to zero, meaning that the control variate estimator converges at a faster speed.

Creating an infinite sequence of control variates sounds unachievable in a realistic situation. Legendre polynomials are used in the paper, but is there a generic and cheap way to getting these. And … control variate selection, anyone?!