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Conference Paper

Evaluation of marginal likelihoods via the density of states


Habeck,  M
Dept. Empirical Inference, Max Planck Institute for Intelligent Systems, Max Planck Society;
Max Planck Institute for Developmental Biology, Max Planck Society;
Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Habeck, M. (2012). Evaluation of marginal likelihoods via the density of states. In M. Lawrence, & N. Girolami (Eds.), Artificial Intelligence and Statistics, 21-23 April 2012, La Palma, Canary Islands (pp. 486-494). Madison, WI, USA: International Machine Learning Society.

Cite as: https://hdl.handle.net/21.11116/0000-0001-8F45-1
Bayesian model comparison involves the evaluation of the marginal likelihood, the expectation of the likelihood under the prior distribution. Typically, this high-dimensional integral over all model parameters is approximated using Markov chain Monte Carlo methods. Thermodynamic integration is a popular method to estimate the marginal likelihood by using samples from annealed posteriors. Here we show that there exists a robust and flexible alternative. The new method estimates the density of states, which counts the number of states associated with a particular value of the likelihood. If the density of states is known, computation of the marginal likelihood reduces to a one- dimensional integral. We outline a maximum likelihood procedure to estimate the density of states from annealed posterior samples. We apply our method to various likelihoods and show that it is superior to thermodynamic integration in that it is more flexible with regard to the annealing schedule and the family of bridging distributions. Finally, we discuss the relation of our method with Skilling's nested sampling.