Exploring multimodal distributions with nested sampling
Abstract
In performing a Bayesian analysis, 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. 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. Nested Sampling is a Monte Carlo method targeted at the efficient calculation of the evidence, but also produces posterior inferences as a byproduct and therefore provides means to carry out parameter estimation as well as model selection. The main challenge in implementing Nested Sampling is to sample from a constrained probability distribution. One possible solution to this problem is provided by the Galilean Monte Carlo (GMC) algorithm. We show results of applying Nested Sampling with GMC to some problems which have proven very difficult for standard Markov Chain Monte Carlo (MCMC) and downhill methods, due to the presence of large number of local minima and/or pronounced (curving) degeneracies between the parameters. We also discuss the use of Nested Sampling with GMC in Bayesian object detection problems, which are inherently multimodal and require the evaluation of Bayesian evidence for distinguishing between true and spurious detections.
 Publication:

Bayesian Inference and Maximum Entropy Methods in Science and Engineering: 32nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering
 Pub Date:
 August 2013
 DOI:
 10.1063/1.4819989
 arXiv:
 arXiv:1312.5638
 Bibcode:
 2013AIPC.1553..106F
 Keywords:

 Astrophysics  Instrumentation and Methods for Astrophysics;
 Physics  Data Analysis;
 Statistics and Probability;
 Statistics  Computation
 EPrint:
 Refereed conference proceeding, presented at 32nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering