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Computing the Bayes Factor from a Markov chain Monte Carlo Simulation of the Posterior Distribution

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dc.contributor.authorWeinberg, Martin D.
dc.contributor.departmentUniversity of Massachusetts Amherst
dc.date2023-09-22T21:59:20.000
dc.date.accessioned2024-04-26T08:26:06Z
dc.date.available2024-04-26T08:26:06Z
dc.date.issued2010-01-01
dc.descriptionThis is a pre-published version which is collected from arXiv link.
dc.description.abstractComputation of the marginal likelihood from a simulated posterior distribution is central to Bayesian model selection but is computationally difficult. I argue that the marginal likelihood can be reliably computed from a posterior sample by careful attention to the numerics of the probability integral. Posing the expression for the marginal likelihood as a Lebesgue integral, we may convert the harmonic mean approximation from a sample statistic to a quadrature rule. As a quadrature, the harmonic mean approximation suffers from enormous truncation error as consequence . In addition, I demonstrate that the integral expression for the harmonic-mean approximation converges slowly at best for high-dimensional problems with uninformative prior distributions. These observations lead to two computationally-modest families of quadrature algorithms that use the full generality sample posterior but without the instability. The first algorithm automatically eliminates the part of the sample that contributes large truncation error. The second algorithm uses the posterior sample to assign probability to a partition of the sample space and performs the marginal likelihood integral directly. This eliminates convergence issues. The first algorithm is analogous to standard quadrature but can only be applied for convergent problems. The second is a hybrid of cubature: it uses the posterior to discover and tessellate the subset of that sample space was explored and uses quantiles to compute a representive field value. Neither algorithm makes strong assumptions about the shape of the posterior distribution and neither is sensitive outliers. [abridged]
dc.description.pages737-770
dc.identifier.doihttps://doi.org/10.1214/12-BA725
dc.identifier.urihttps://hdl.handle.net/20.500.14394/2765
dc.relation.urlhttps://scholarworks.umass.edu/cgi/viewcontent.cgi?article=1107&context=astro_faculty_pubs&unstamped=1
dc.source.issue3
dc.source.issue7
dc.source.statuspublished
dc.subjectBayesian computation
dc.subjectmarginal likelihood
dc.subjectalgorithm
dc.subjectBayes factors
dc.subjectmodel selection
dc.subjectAstrophysics and Astronomy
dc.titleComputing the Bayes Factor from a Markov chain Monte Carlo Simulation of the Posterior Distribution
dc.typearticle
dc.typearticle
digcom.contributor.authorisAuthorOfPublication|email:weinberg@astro.umass.edu|institution:University of Massachusetts Amherst|Weinberg, Martin D.
digcom.identifierastro_faculty_pubs/114
digcom.identifier.contextkey1610517
digcom.identifier.submissionpathastro_faculty_pubs/114
dspace.entity.typePublication
relation.isAuthorOfPublication6387ef0a-b08d-4ee2-b19d-775a09801631
relation.isAuthorOfPublication.latestForDiscovery6387ef0a-b08d-4ee2-b19d-775a09801631
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