Abstract
Bayesian inference involves two main computational challenges. First, in estimating the parameters of some model for the data, the posterior distribution may well be highly multi-modal: a regime in which the convergence to stationarity of traditional Markov Chain Monte Carlo (MCMC) techniques becomes incredibly slow. Second, in selecting between a set of competing models the necessary estimation of the Bayesian evidence for each is, by definition, a (possibly high-dimensional) integration over the entire parameter space; again this can be a daunting computational task, although new Monte Carlo (MC) integration algorithms offer solutions of ever increasing efficiency. Nested sampling (NS) is one such contemporary MC strategy targeted at calculation of the Bayesian evidence, but which also enables posterior inference as a by-product, thereby allowing simultaneous parameter estimation and model selection. The widely-used MultiNest algorithm presents a particularly efficient implementation of the NS technique for multi-modal posteriors. In this paper we discuss importance nested sampling (INS), an alternative summation of the MultiNest draws, which can calculate the Bayesian evidence at up to an order of magnitude higher accuracy than `vanilla' NS with no change in the way MultiNest explores the parameter space. This is accomplished by treating as a (pseudo-)importance sample the totality of points collected by MultiNest, including those previously discarded under the constrained likelihood sampling of the NS algorithm. We apply this technique to several challenging test problems and compare the accuracy of Bayesian evidences obtained with INS against those from vanilla NS.
Highlights
The last two decades in astrophysics and cosmology have seen the arrival of vast amounts of high quality data
With the availability of vast amounts of high quality data, statistical inference is increasingly playing an important role in cosmology and astroparticle physics
Markov Chain Monte Carlo (MCMC) techniques and more recently algorithms based on nested sampling have been employed successfully in a variety of different areas
Summary
Bayesian inference involves two main computational challenges. First, in estimating the parameters of some model for the data, the posterior distribution may well be highly multi-modal: a regime in which the convergence to stationarity of traditional Markov Chain Monte Carlo (MCMC) techniques becomes incredibly slow. In this paper we discuss importance nested sampling (INS), an alternative summation of the MULTINEST draws, which can calculate the Bayesian evidence at up to an order of magnitude higher accuracy than ‘vanilla’ NS with no change in the way MULTINEST explores the parameter space. This is accomplished by treating as a (pseudo)importance sample the totality of points collected by MULTINEST, including those previously discarded under the constrained likelihood sampling of the NS algorithm.
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