ABSTRACT
Department of Microbiology and Immunology, Rega Institute, KU Leuven, Leuven, Belgium
Philippe Lemey
Department of Microbiology and Immunology, Rega Institute, KU Leuven, Leuven, Belgium
CONTENTS
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.2 Prior and posterior-based estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2.1 Integrating the likelihood against the model prior . . . . . . 62 4.2.2 The arithmetic mean estimator . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.2.3 The harmonic mean estimator(s) . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2.4 Akaike and Bayesian Information Criterion through
Markov chain Monte Carlo: AICM and BICM . . . . . . . . . . 64 4.2.5 Generalized harmonic mean estimator (GHME) and
inflated density ratio (IDR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2.5.1 GHME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2.5.2 Inflated density ratio (IDR) . . . . . . . . . . . . . . . . . . 66
4.3 Path sampling approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.3.1 Path sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3.1.1 Marginal likelihood estimation . . . . . . . . . . . . . . . 69 4.3.1.2 Direct Bayes factor estimation . . . . . . . . . . . . . . . 69
4.3.2 Stepping-stone sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.3.2.1 Marginal likelihood estimation . . . . . . . . . . . . . . . 71 4.3.2.2 Direct Bayes factor estimation . . . . . . . . . . . . . . . 72
4.3.3 Generalized stepping-stone sampling . . . . . . . . . . . . . . . . . . . . 74 4.4 Simulation study: uncorrelated relaxed clocks . . . . . . . . . . . . . . . . . . . 76
4.4.1 Relaxed molecular clock models . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.4.2 Comparing model selection to model averaging . . . . . . . . . 78 4.4.3 Simulation settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.5 Practical example on demographic models . . . . . . . . . . . . . . . . . . . . . . 85
Algorithms, and
4.5.1 HIV-1 group M dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.5.2 Demographic/coalescent models . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.5.3 Proper priors and the Bayesian skyride model . . . . . . . . . . 87 4.5.4 Computational requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
In this chapter, we discuss recent advances in the field of Bayesian model testing and focus on methods that aim at either estimating (log) marginal likelihoods or at directly estimating (log) Bayes factors. We start by introducing several of the most popular (log) marginal likelihood estimators, which are attractive from a computational perspective. Because these estimators have recently been shown to perform poorly, we discuss computationally more demanding, but also more accurate path sampling approaches that can be used to either estimate (log) marginal likelihoods for different models, but also to directly estimate (log) Bayes factors between two competing models. For a specific class of evolutionary models, i.e., the relaxed molecular clock models, we also discuss how such methods compare to specific Bayesian model averaging approaches that allow constructing a classifier to approximate (log) Bayes factors between the models in the candidate model set. To demonstrate their practical use, we apply the presented techniques in a simulation study on relaxed molecular clocks and in a demographic model selection study that focuses on an HIV-1 dataset.
