Multimodal nested sampling: an efficient and robust alternative to Markov Chain Monte Carlo methods for astronomical data analyses

Abstract

In performing a Bayesian analysis of astronomical data, two difficult problems often emerge. First, in estimating the parameters of some model for the data, the resulting posterior distribution may be multimodal or exhibit pronounced (curving) degeneracies, which can cause problems for traditional Markov Chain Monte Carlo (MCMC) sampling methods. Secondly, in selecting between a set of competing models, calculation of the Bayesian evidence for each model is computationally expensive using existing methods such as thermodynamic integration. The nested sampling method introduced by Skilling, has greatly reduced the computational expense of calculating evidence and also produces posterior inferences as a by‐product. This method has been applied successfully in cosmological applications by Mukherjee, Parkinson & Liddle, but their implementation was efficient only for unimodal distributions without pronounced degeneracies. Shaw, Bridges & Hobson recently introduced a clustered nested sampling method which is significantly more efficient in sampling from multimodal posteriors and also determines the expectation and variance of the final evidence from a single run of the algorithm, hence providing a further increase in efficiency. In this paper, we build on the work of Shaw et al. and present three new methods for sampling and evidence evaluation from distributions that may contain multiple modes and significant degeneracies in very high dimensions; we also present an even more efficient technique for estimating the uncertainty on the evaluated evidence. These methods lead to a further substantial improvement in sampling efficiency and robustness, and are applied to two toy problems to demonstrate the accuracy and economy of the evidence calculation and parameter estimation. Finally, we discuss the use of these methods in performing Bayesian object detection in astronomical data sets, and show that they significantly outperform existing MCMC techniques. An implementation of our methods will be publicly released shortly.

An Improved Distance and Mass Estimate for SGR A* From a Multistar Orbit Analysis

“…Each individual fit then results in a set of samples of the posterior distribution of the star's six orbital parameters. These individual fits are performed using the Bayesian multimodal nested sampling algorithm MULTINEST (Feroz & Hobson 2008;Feroz et al 2009), which was implemented in our code by Leo Meyer. This algorithm samples the posterior distribution more efficiently than traditional Markov chain Monte Carlo sampling schemes, especially when the distribution is multimodal.…”

“…parameters (in addition to S0-2ʼs sixKeplerian orbital parameters). This 13-dimensional orbital fit was done using the Bayesian multimodal nested sampling algorithm called MULTIN-EST (see Feroz & Hobson 2008;Feroz et al 2009). We also use the results of this orbital fit of S0-2 alone to compare with the results of fitting S0-38 alone and S0-2 and S0-38 simultaneously (Section 3.4).…”

Detecting Eccentric Supermassive Black Hole Binaries With Pulsar Timing Arrays: Resolvable Source Strategies

“…The binary parameter space is searched using a python wrapper (Buchner et al 2014) to the nested sampling package MULTINEST (Feroz & Hobson 2008;Feroz et al 2009Feroz et al , 2013, and we have cross-checked our results with a sampler utilizing advanced Markov chain Monte Carlo techniques. 9 The product of these analyses are samples from the posterior probability distribution of the signal parameters space, allowing us to quantify the measurement precision of parameters based on the Bayesian credible regions, and also permitting model selection via computation of competing models' Bayesian evidence.…”

Reconstructing the sky location of gravitational-wave detected compact binary systems: Methodology for testing and comparison

“…Using two classes of stochastic sampling techniques, Markov-Chain Monte Carlo [20][21][22] and nested sampling [23][24][25], LALINFERENCE coherently analyzes the data from all the interferometers in the network and generates the multidimensional PDF on the full set of parameters needed to describe a binary system before marginalizing over all parameters other than the sky location (a binary in circular orbit is described by 9 to 15 parameters, depending on whether spins of the binary components are included in the model). (ii) A much faster low-latency technique, that we will call TIMING++ [8], uses data products from the search stage of the analysis, and can construct sky maps on (sub)minute time scales by using primarily time-delay information between different detector sites.…”