Automatic Bayes factors for testing variances of two independent normal distributions

Research output: Contribution to journalArticleScientificpeer-review

Abstract

Researchers are frequently interested in testing variances of two independent populations. We often would like to know whether the population variances are equal, whether population 1 has smaller variance than population 2, or whether population 1 has larger variance than population 2. In this article we consider the Bayes factor, a Bayesian model selection and hypothesis testing criterion, for this multiple hypothesis test. Application of Bayes factors requires specification of prior distributions for the model parameters. Automatic Bayes factors circumvent the difficult task of prior elicitation by using data-driven mechanisms to specify priors in an automatic fashion. In this article we develop different automatic Bayes factors for testing two variances: first we apply the fractional Bayes factor (FBF) to the testing problem. It is shown that the FBF does not always function as Occam’s razor. Second we develop a new automatic balanced Bayes factor with equal priors for the variances. Third we propose a Bayes factor based on an adjustment of the marginal likelihood in the FBF approach. The latter two methods always function as Occam’s razor. Through theoretical considerations and numerical simulations it is shown that the third approach provides strongest evidence in favor of the true hypothesis.
Original languageEnglish
Pages (from-to)158-170
JournalJournal of Mathematical Psychology
Volume72
DOIs
Publication statusPublished - 2016

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Bayes Factor
Normal Distribution
Normal distribution
Gaussian distribution
Fractional Bayes Factor
Testing
Prior Elicitation
Bayesian Model Selection
Multiple Tests
Marginal Likelihood
Hypothesis Test
Hypothesis Testing
Prior distribution
Specifications
Data-driven
Adjustment
Computer simulation
Specification
Numerical Simulation

Cite this

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title = "Automatic Bayes factors for testing variances of two independent normal distributions",
abstract = "Researchers are frequently interested in testing variances of two independent populations. We often would like to know whether the population variances are equal, whether population 1 has smaller variance than population 2, or whether population 1 has larger variance than population 2. In this article we consider the Bayes factor, a Bayesian model selection and hypothesis testing criterion, for this multiple hypothesis test. Application of Bayes factors requires specification of prior distributions for the model parameters. Automatic Bayes factors circumvent the difficult task of prior elicitation by using data-driven mechanisms to specify priors in an automatic fashion. In this article we develop different automatic Bayes factors for testing two variances: first we apply the fractional Bayes factor (FBF) to the testing problem. It is shown that the FBF does not always function as Occam’s razor. Second we develop a new automatic balanced Bayes factor with equal priors for the variances. Third we propose a Bayes factor based on an adjustment of the marginal likelihood in the FBF approach. The latter two methods always function as Occam’s razor. Through theoretical considerations and numerical simulations it is shown that the third approach provides strongest evidence in favor of the true hypothesis.",
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Automatic Bayes factors for testing variances of two independent normal distributions. / Böing-Messing, F.; Mulder, J.

In: Journal of Mathematical Psychology, Vol. 72, 2016, p. 158-170.

Research output: Contribution to journalArticleScientificpeer-review

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