Abstract

MCMC sampling is a methodology that is becoming increasingly important in statistical signal processing. It has been of particular importance to the Bayesian-based approaches to signal processing since it extends significantly the range of problems that they can address. MCMC techniques generate samples from desired distributions by embedding them as limiting distributions of Markov chains. There are many ways of categorizing MCMC methods, but the simplest one is to classify them in one of two groups: the first is used in estimation problems where the unknowns are typically parameters of a model, which is assumed to have generated the observed data; the second is employed in more general scenarios where the unknowns are not only model parameters, but models as well. In this paper, we address the MCMC methods from the second group, which allow for generation of samples from probability distributions defined on unions of disjoint spaces of different dimensions. More specifically, we show why sampling from such distributions is a nontrivial task. It will be demonstrated that these methods genuinely unify the operations of detection and estimation and thereby provide great potential for various important applications. The focus is mainly on the reversible jump MCMC (Green, Biometrika 82 (1995) 711), but other approaches are also discussed. Details of implementation of the reversible jump MCMC are provided for two examples.

Author Keywords: Bayesian model selection; Markov chain Monte Carlo methods; Reversible jump MCMC

Nomenclature

set of real numbers

set of positive real numbers

set with elements (a₁,a₂,…,a_n) where for j=1,…,n

[A]_i,j

ith row, jth column of matrix A

| A |

determinant of matrix A

A^T

matrix A transposed

 

z(z₁,…,z_j−1,z_j,z_j+1,…,z_k)^T,z_−j(z₁,…,z_j−1,z_j+1,…,z_k)^T.

0_n×p

null matrix of dimension n×p

I_n

identity matrix of dimension n×n

indicator function of the set E(1 if zE, 0 otherwise).

δ_x(dz)

delta Dirac measure such that if xA and 0 otherwise

z

highest integer strictly less than z

zπ(z)

z is distributed according to π(z)

z | yπ(z)

the conditional distribution of z given y is π(z)

Probability distribution  

Inverse Gamma  

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Gamma  

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Gaussian  

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Uniform  

Article Outline

Nomenclature

1. Introduction

2. Problem formulation

3. MCMC algorithms for model selection

3.1. The case N=2

3.1.1. Goals

3.1.2. Jumping between Θ₁ and Θ₂: simple case^♦

3.1.3. Jumping between Θ₁ and Θ₂: complicated case^♦

3.1.4. Practical implementation

3.2. Reversible jump MCMC for N2 models

3.2.1. Goals

3.2.2. Practical implementation

3.3. Other approaches^♦

3.3.1. Method of subspace extension and the algorithm of Carlin and Chib

3.3.2. Jump-diffusion sampling

4. Examples

4.1. Analysis of sinusoids in noise

4.1.1. Data models

4.1.2. Overview of the algorithm

4.1.3. The birth and death moves

4.1.4. Merge and split moves?^♦

4.1.5. Example of simulation

4.2. Bernoulli–Gauss deconvolution

4.2.1. Model of the data

4.2.2. Algorithm

5. Conclusions

Acknowledgements

References