Testing Precise Hypotheses

Berger, James O.; Delampady, Mohan

doi:10.1214/ss/1177013238

Cited by 616 publications

(431 citation statements)

References 31 publications

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“…The reason for the discrepancy is that the Bayesian hypothesis test punishes H 1 for assigning prior mass to values of d that yield very low likelihoods (i.e., the automatic Ockham's razor discussed previously, see Berger & Delampady, 1987 for a discussion).…”

Section: Unrestricted Analysismentioning

confidence: 99%

Bayesian hypothesis testing for psychologists: A tutorial on the Savage–Dickey method

Wagenmakers

Lodewyckx

Kuriyal

et al. 2010

Cognitive Psychology

652

642

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a b s t r a c tIn the field of cognitive psychology, the p-value hypothesis test has established a stranglehold on statistical reporting. This is unfortunate, as the p-value provides at best a rough estimate of the evidence that the data provide for the presence of an experimental effect. An alternative and arguably more appropriate measure of evidence is conveyed by a Bayesian hypothesis test, which prefers the model with the highest average likelihood. One of the main problems with this Bayesian hypothesis test, however, is that it often requires relatively sophisticated numerical methods for its computation. Here we draw attention to the Savage-Dickey density ratio method, a method that can be used to compute the result of a Bayesian hypothesis test for nested models and under certain plausible restrictions on the parameter priors. Practical examples demonstrate the method's validity, generality, and flexibility.

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Section: Unrestricted Analysismentioning

confidence: 99%

Bayesian hypothesis testing for psychologists: A tutorial on the Savage–Dickey method

Wagenmakers

Lodewyckx

Kuriyal

et al. 2010

Cognitive Psychology

652

642

View full text Add to dashboard Cite

show abstract

“…22 It is noted that in testing a normal mean the sharp null is a good approximation to an interval null as long as the width of the interval is less than about one-half of the standard error of the sample mean. 20,25 The genetic data sets we analyzed here, however, suggested that using a sharp null favors the alternative model (Supplementary Figure S3). This is because the variation of the fatal fraction under the null (estimated from the putative female fetuses) is substantial compared with the sample standard deviations of the autosomal chromosomal dosages from euploid control samples.…”

Section: Discussionmentioning

confidence: 84%

Informative priors on fetal fraction increase power of the noninvasive prenatal screen

Huang

et al. 2018

Genetics in Medicine

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Purpose: Noninvasive prenatal screening (NIPS) sequences a mixture of the maternal and fetal cell-free DNA. Fetal trisomy can be detected by examining chromosomal dosages estimated from sequencing reads. The traditional method uses the Z-test, which compares a subject against a set of euploid controls, where the information of fetal fraction is not fully utilized. Here we present a Bayesian method that leverages informative priors on the fetal fraction.Method: Our Bayesian method combines the Z-test likelihood and informative priors of the fetal fraction, which are learned from the sex chromosomes, to compute Bayes factors. Bayesian framework can account for nongenetic risk factors through the prior odds, and our method can report individual positive/negative predictive values.Results: Our Bayesian method has more power than the Z-test method. We analyzed 3,405 NIPS samples and spotted at least 9 (of 51) possible Z-test false positives.Conclusion: Bayesian NIPS is more powerful than the Z-test method, is able to account for nongenetic risk factors through prior odds, and can report individual positive/negative predictive values.

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“…There are many well-known criticisms to this common procedure, some of which are briefly reviewed below. For further discussion see Jeffreys (1961), Edwards, Lindman and Savage (1963), Rao (1966), Lindley (1972), Good (1983), Berger and Delampady (1987), Berger and Sellke (1987), Matthews (2001), and references therein.…”

Section: Hypothesis Testingmentioning

confidence: 99%

“…One unappealing consequence of this non-regular prior structure, noted by Lindley (1957) and generally known as Lindley's paradox, is that for any fixed value of the pertinent test statistic, the Bayes factor typically increases as √ n with the sample size; hence, with large samples, "evidence" in favor of H 0 may be overwhelming with data sets which are both extremely implausible under H 0 and quite likely under alternative θ values, such as (say) the mleθ. For further discussion of this polemical issue see Bernardo (1980), Shafer (1982), Berger and Delampady (1987), Casella and Berger (1987), Robert (1993), Bernardo (1999), and discussions therein.…”

Section: Bayes Factorsmentioning

confidence: 99%

Bayesian Hypothesis Testing: a Reference Approach

Bernardo¹,

Rueda²

2002

Int Statistical Rev

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SummaryFor any probability model M ≡ {p(x | θ, ω), θ ∈ Θ, ω ∈ Ω} assumed to describe the probabilistic behaviour of data x ∈ X, it is argued that testing whether or not the available data are compatible with the hypothesis H 0 ≡ {θ = θ 0 } is best considered as a formal decision problem on whether to use (a 0 ), or not to use (a 1 ), the simpler probability model (or null model) The BRC criterion provides a general reference Bayesian solution to hypothesis testing which does not assume a probability mass concentrated on M 0 and, hence, it is immune to Lindley's paradox. The theory is illustrated within the context of multivariate normal data, where it is shown to avoid Rao's paradox on the inconsistency between univariate and multivariate frequentist hypothesis testing.

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Testing Precise Hypotheses

Cited by 616 publications

References 31 publications

Bayesian hypothesis testing for psychologists: A tutorial on the Savage–Dickey method

Bayesian hypothesis testing for psychologists: A tutorial on the Savage–Dickey method

Informative priors on fetal fraction increase power of the noninvasive prenatal screen

Bayesian Hypothesis Testing: a Reference Approach

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