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.

B’day party!

transport Monte Carlo

Read this recent arXival by Leo Duan (from UF in Gainesville) on transport approaches to approximate Bayesian computation, in connection with normalising flows. The author points out a “lack of flexibility in a large class of normalizing flows”  to bring forward his own proposal.

“…we assume the reference (a multivariate uniform distribution) can be written as a mixture of many one-to-one transforms from the posterior”

The transportation problem is turned into defining a joint distribution on (β,θ) such that θ is marginally distributed from the posterior and β is one of an infinite collection of transforms of θ. Which sounds quite different from normalizing flows, to be sure. Reverting the order, if one manages to simulate β from its marginal the resulting θ is one of the transforms. Chosen to be a location-scale modification of β, s⊗β+m. The weights when going from θ to β are logistic transforms with Dirichlet distributed scales. All with parameters to be optimised by minimising the Kullback-Leibler distance between the reference measure on β and its inverse mixture approximation, and resorting to gradient descent. (This may sound a wee bit overwhelming as an approximation strategy and I actually had to make a large cup of strong macha to get over it, but this may be due to the heat wave occurring at the same time!) Drawing θ from this approximation is custom-made straightforward and an MCMC correction can even be added, resulting in an independent Metropolis-Hastings version since the acceptance ratio remains computable. Although this may defeat the whole purpose of the exercise by stalling the chain if the approximation is poor (hence suggesting this last step being used instead as a control.)

The paper also contains a theoretical section that studies the approximation error, going to zero as the number of terms in the mixture, K, goes to infinity. Including a Monte Carlo error in log(n)/n (and incidentally quoting a result from my former HoD at Paris 6, Paul Deheuvels). Numerical experiments show domination or equivalence with some other solutions, e.g. being much faster than HMC, the remaining $1000 question being of course the on-line evaluation of the quality of the approximation.

JB³ [Junior Bayes beyond the borders]

Bocconi and j-ISBA are launcing a webinar series for and by junior Bayesian researchers. The first talk is on 25 June, 25 at 3pm UTC/GMT (5pm CET) with Francois-Xavier Briol, one of the laureates of the 2020 Savage Thesis Prize (and a former graduate of OxWaSP, the Oxford-Warwick doctoral training program), on Stein’s method for Bayesian computation, with as a discussant Nicolas Chopin.

As pointed out on their webpage,

Due to the importance of the above endeavor, JB³ will continue after the health emergency as an annual series. It will include various refinements aimed at increasing the involvement of the whole junior Bayesian community and facilitating a broader participation to the online seminars all over the world via various online solutions.

Thanks to all my friends at Bocconi for running this experiment!

Computing Bayes: Bayesian Computation from 1763 to the 21st Century

Last night, Gael Martin, David Frazier (from Monash U) and myself arXived a survey on the history of Bayesian computations. This project started when Gael presented a historical overview of Bayesian computation, then entitled ‘Computing Bayes: Bayesian Computation from 1763 to 2017!’, at ‘Bayes on the Beach’ (Queensland, November, 2017). She then decided to build a survey from the material she had gathered, with her usual dedication and stamina. Asking David and I to join forces and bring additional perspectives on this history. While this is a short and hence necessary incomplete history (of not everything!), it hopefully brings some different threads together in an original enough fashion (as I think there is little overlap with recent surveys I wrote). We welcome comments about aspects we missed, skipped or misrepresented, most obviously!