Bayes factors for testing inequality constrained hypotheses: Issues with prior specification

Research output: Contribution to journalArticleScientificpeer-review

Abstract

Several issues are discussed when testing inequality constrained hypotheses using a Bayesian approach. First, the complexity (or size) of the inequality constrained parameter spaces can be ignored. This is the case when using the posterior probability that the inequality constraints of a hypothesis hold, Bayes factors based on non-informative improper priors, and partial Bayes factors based on posterior priors. Second, the Bayes factor may not be invariant for linear one-to-one transformations of the data. This can be observed when using balanced priors which are centred on the boundary of the constrained parameter space with a diagonal covariance structure. Third, the information paradox can be observed. When testing inequality constrained hypotheses, the information paradox occurs when the Bayes factor of an inequality constrained hypothesis against its complement converges to a constant as the evidence for the first hypothesis accumulates while keeping the sample size fixed. This paradox occurs when using Zellner's g prior as a result of too much prior shrinkage. Therefore, two new methods are proposed that avoid these issues. First, partial Bayes factors are proposed based on transformed minimal training samples. These training samples result in posterior priors that are centred on the boundary of the constrained parameter space with the same covariance structure as in the sample. Second, a g prior approach is proposed by letting g go to infinity. This is possible because the Jeffreys–Lindley paradox is not an issue when testing inequality constrained hypotheses. A simulation study indicated that the Bayes factor based on this g prior approach converges fastest to the true inequality constrained hypothesis.
Original languageEnglish
Pages (from-to)153-171
JournalBritish Journal of Mathematical and Statistical Psychology
Volume67
Issue number1
DOIs
Publication statusPublished - 2014

Fingerprint

Bayes Factor
Specification
Testing
Paradox
Parameter Space
Covariance Structure
Training Samples
Improper Prior
Converge
Partial
Posterior Probability
Accumulate
Shrinkage
Inequality Constraints
Bayesian Approach
Sample Size
Complement
Infinity
Simulation Study
Invariant

Cite this

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title = "Bayes factors for testing inequality constrained hypotheses: Issues with prior specification",
abstract = "Several issues are discussed when testing inequality constrained hypotheses using a Bayesian approach. First, the complexity (or size) of the inequality constrained parameter spaces can be ignored. This is the case when using the posterior probability that the inequality constraints of a hypothesis hold, Bayes factors based on non-informative improper priors, and partial Bayes factors based on posterior priors. Second, the Bayes factor may not be invariant for linear one-to-one transformations of the data. This can be observed when using balanced priors which are centred on the boundary of the constrained parameter space with a diagonal covariance structure. Third, the information paradox can be observed. When testing inequality constrained hypotheses, the information paradox occurs when the Bayes factor of an inequality constrained hypothesis against its complement converges to a constant as the evidence for the first hypothesis accumulates while keeping the sample size fixed. This paradox occurs when using Zellner's g prior as a result of too much prior shrinkage. Therefore, two new methods are proposed that avoid these issues. First, partial Bayes factors are proposed based on transformed minimal training samples. These training samples result in posterior priors that are centred on the boundary of the constrained parameter space with the same covariance structure as in the sample. Second, a g prior approach is proposed by letting g go to infinity. This is possible because the Jeffreys–Lindley paradox is not an issue when testing inequality constrained hypotheses. A simulation study indicated that the Bayes factor based on this g prior approach converges fastest to the true inequality constrained hypothesis.",
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Bayes factors for testing inequality constrained hypotheses : Issues with prior specification. / Mulder, J.

In: British Journal of Mathematical and Statistical Psychology, Vol. 67, No. 1, 2014, p. 153-171.

Research output: Contribution to journalArticleScientificpeer-review

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AB - Several issues are discussed when testing inequality constrained hypotheses using a Bayesian approach. First, the complexity (or size) of the inequality constrained parameter spaces can be ignored. This is the case when using the posterior probability that the inequality constraints of a hypothesis hold, Bayes factors based on non-informative improper priors, and partial Bayes factors based on posterior priors. Second, the Bayes factor may not be invariant for linear one-to-one transformations of the data. This can be observed when using balanced priors which are centred on the boundary of the constrained parameter space with a diagonal covariance structure. Third, the information paradox can be observed. When testing inequality constrained hypotheses, the information paradox occurs when the Bayes factor of an inequality constrained hypothesis against its complement converges to a constant as the evidence for the first hypothesis accumulates while keeping the sample size fixed. This paradox occurs when using Zellner's g prior as a result of too much prior shrinkage. Therefore, two new methods are proposed that avoid these issues. First, partial Bayes factors are proposed based on transformed minimal training samples. These training samples result in posterior priors that are centred on the boundary of the constrained parameter space with the same covariance structure as in the sample. Second, a g prior approach is proposed by letting g go to infinity. This is possible because the Jeffreys–Lindley paradox is not an issue when testing inequality constrained hypotheses. A simulation study indicated that the Bayes factor based on this g prior approach converges fastest to the true inequality constrained hypothesis.

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