Evaluation of marginal likelihoods via the density of states

Michael Habeck
Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics, PMLR 22:486-494, 2012.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v22-habeck12, title = {Evaluation of marginal likelihoods via the density of states}, author = {Habeck, Michael}, booktitle = {Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics}, pages = {486--494}, year = {2012}, editor = {Lawrence, Neil D. and Girolami, Mark}, volume = {22}, series = {Proceedings of Machine Learning Research}, address = {La Palma, Canary Islands}, month = {21--23 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v22/habeck12/habeck12.pdf}, url = {https://proceedings.mlr.press/v22/habeck12.html}, abstract = {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.} }
Endnote
%0 Conference Paper %T Evaluation of marginal likelihoods via the density of states %A Michael Habeck %B Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2012 %E Neil D. Lawrence %E Mark Girolami %F pmlr-v22-habeck12 %I PMLR %P 486--494 %U https://proceedings.mlr.press/v22/habeck12.html %V 22 %X 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.
RIS
TY - CPAPER TI - Evaluation of marginal likelihoods via the density of states AU - Michael Habeck BT - Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics DA - 2012/03/21 ED - Neil D. Lawrence ED - Mark Girolami ID - pmlr-v22-habeck12 PB - PMLR DP - Proceedings of Machine Learning Research VL - 22 SP - 486 EP - 494 L1 - http://proceedings.mlr.press/v22/habeck12/habeck12.pdf UR - https://proceedings.mlr.press/v22/habeck12.html AB - 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. ER -
APA
Habeck, M.. (2012). Evaluation of marginal likelihoods via the density of states. Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 22:486-494 Available from https://proceedings.mlr.press/v22/habeck12.html.

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