Inference for Trans-dimensional Bayesian Models with Diffusive Nested Sampling

Abstract

Many inference problems involve inferring the number $N$ of components in some region, along with their properties $\{{\mathbf{x}}_i{\}}_{i=1}^N$, from a dataset $\mathcal{D}$. 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.

Only published here for the time being. 17 pages, 10 figures. Software available at https://github.com/eggplantbren/RJObject

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