Natural nested sampling

“The nested sampling algorithm solves otherwise challenging, high-dimensional integrals by evolving a collection of live points through parameter space. The algorithm was immediately adopted in cosmology because it partially overcomes three of the major difficulties in Markov chain Monte Carlo, the algorithm traditionally used for Bayesian computation. Nested sampling simultaneously returns results for model comparison and parameter inference; successfully solves multimodal problems; and is naturally self-tuning, allowing its immediate application to new challenges.”

I came across a review on nested sampling in Nature Reviews Methods Primers of May 2022, with a large number of contributing authors, some of whom I knew from earlier papers in astrostatistics. As illustrated by the above quote from the introduction, the tone is definitely optimistic about the capacities of the method, reproducing the original argument that the evidence is the posterior expectation of the likelihood L(θ) under the prior. Which representation, while valid, is not translating into a dimension-free methodology since parameters θ still need be simulated.

“Nested sampling lies in a class of algorithms that form a path of bridging distributions and evolves samples along that path. Nested sampling stands out because the path is automatic and smooth — compression along log X by, on average, 1/𝑛at each iteration — and because along the path is compressed through constrained priors, rather than from the prior to the posterior. This was a motivation for nested sampling as it avoids phase transitions — abrupt changes in the bridging distributions — that cause problems for other methods, including path samplers, such as annealing.”

The elephant in the room is eventually processed, namely the simulation from the prior constrained to the likelihood level sets that in my experience (with, e.g., mixture posteriors) proves most time consuming. This stems from the fact that these level sets are notoriously difficult to evaluate from a given sample: all points stand within the set but they hardly provide any indication of the boundaries of saif set… Region sampling requires to construct a region that bounds the likelihood level set, which requires some knowledge of the likelihood variations to have a chance to remain efficient, incl. in cosmological applications, while regular MCMC steps require an increasing number of steps as the constraint gets tighter and tighter. For otherwise it essentially amounts to duplicating a live particle.

on control variates

A few months ago, I had to write a thesis evaluation of Rémi Leluc’s PhD, which contained several novel Monte Carlo proposals on control variates and importance techniques. For instance, Leluc et al. (Statistics and Computing, 2021) revisits the concept of control variables by adding a perspective of control variable selection using LASSO. This prior selection is relevant since control variables are not necessarily informative about the objective function being integrated and my experience is that the more variables the less reliable the improvement. The remarkable feature of the results is in obtaining explicit and non-asymptotic bounds.

The author obtains a concentration inequality on the error resulting from the use of control variables, under strict assumptions on the variables. The associated numerical experiment illustrates the difficulties of practically implementing these principles due to the number of parameters to calibrate. I found the example of a capture-recapture experiment on ducks (European Dipper) particularly interesting, not only because we had used it in our book but also because it highlights the dependence of estimates on the dominant measure.

Based on a NeurIPS 2022 poster presentation Chapter 3 is devoted to the use of control variables in sequential Monte Carlo, where a sequence of importance functions is constructed based on previous iterations to improve the approximation of the target distribution. Under relatively strong assumptions of importance functions dominating the target distribution (which could generally be achieved by using an increasing fraction of the data in a partial posterior distribution), of sub-Gaussian tails of an intractable distribution’s residual, a concentration inequality is established for the adaptive control variable estimator.

This chapter uses a different family of control variables, based on a Stein operator introduced in Mira et al. (2016). In the case where the target is a mixture in IR^d, one of our benchmarks in Cappé et al. (2008), remarkable gains are obtained for relatively high dimensions. While the computational demands of these improvements are not mentioned, the comparison with an MCMC approach (NUTS) based on the same number of particles demonstrates a clear improvement in Bayesian estimation.

Chapter 4 corresponds to a very recent arXival and presents a very original approach to control variate correction by reproducing the interest rate law through an approximation using the closest neighbor (leave-one-out) method. It requires neither control function nor necessarily additional simulation, except for the evaluation of the integral, which is rather remarkable, forming a kind of parallel with the bootstrap. (Any other approximation of the distribution would also be acceptable if available at the same computational cost.) The thesis aims to establish the convergence of the method when integration is performed by a Voronoi tessellation, which leads to an optimal rate of order n^-1-2/d for quadratic error (under conditions of integrand regularity). In the alternative where the integral must be evaluated by Monte Carlo, this optimality disappears, unless a massive amount of simulations are used. Numerical illustrations cover SDEs and a Bayesian hierarchical modeling already used in Oates et al. (2017), with massive gain in both cases.

back to a correction of the harmonic mean estimator

In a 2009 JCGS paper, Peter Lenk proposed a bias correction of the harmonic mean estimator, which is somewhat surprising given that the estimator usually has no variance and hence that its consistency is purely formal, since no speed of convergence can be taken for granted. In particular, the conjugate Normal model serving as a motivation leads to an infinite variance. The author is however blaming the poor behaviour of the harmonic mean estimator on the overly concentrated support of the posterior distribution, despite having no reservation about the original identity (with standard notations)

but suggesting the corrected

although this is only true when A is within the support of the posterior. (In which case it connects with our own 2009 correction.) Opting for a set A corresponding to a “simulation support” of the posterior with a very vague meaning, if somewhat connected with the nested sampling starting set.

library, whatzat???

MCM in Paris, 2023

The next MCM conference takes place in (downtown) Paris next 26-30 June. Deadlines are 31 December for mini-symposia/invited sessions and 28 February for contributed talks/posters. I appreciate very much the effort in lowering the registration fees to 80€ for students and 170€ for others, whilst including lunches into the deal! (The view of Paris in the above logo is actually taking from Paris Jussieu campus.)