2014/11/14 by Brendon J. Brewer, Brewer, Brendon J.
Mathematics · Physics and Astronomy · #Computation (stat.CO) #Data Analysis #FOS: Computer and information sciences #FOS: Physical sciences #Instrumentation and Methods for Astrophysics (astro-ph.IM) #Statistics and Probability (physics.data-an) #astro-ph.IM #physics.data-an #stat.CO
paper · pdf · doi:10.48550/arxiv.1411.3921
Only published here for the time being. 17 pages, 10 figures. Software available at https://github.com/eggplantbren/RJObject
arxiv created 2015/01/14 · arxiv updated 2015/01/15
Many inference problems involve inferring the number N of components in some region, along with their properties \xi\i=1N, from a dataset D. A common statistical example is finite mixture modelling. In the Bayesian framework, these problems are typically solved using one of the following two methods: i) by executing a Monte Carlo algorithm (such as Nested Sampling) once for each possible value of N, and calculating the marginal likelihood or evidence as a function of N; or ii) by doing a single run that allows the model dimension N to change (such as Markov Chain Monte Carlo with birth/death moves), and obtaining the posterior for N directly. In this paper we present a general approach to this problem that uses trans-dimensional MCMC embedded within a Nested Sampling algorithm, allowing us to explore the posterior distribution and calculate the marginal likelihood (summed over N) even if the problem contains a phase transition or other difficult features such as multimodality. We present two example problems, finding sinusoidal signals in noisy data, and finding and measuring galaxies in a noisy astronomical image. Both of the examples demonstrate phase transitions in the relationship between the likelihood and the cumulative prior mass, highlighting the need for Nested Sampling.