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Inference to the best explanation of the evidence

1 February 2018 Leave a comment

The p value and Bayesian methods have well known drawbacks when it comes to measuring the strength of the evidence supporting one hypothesis over another. To overcome those drawbacks, this paper proposes an alternative method of quantifying how much support a hypothesis has from evidence consisting of data.

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D. R. Bickel, “The strength of statistical evidence for composite hypotheses: Inference to the best explanation,” Statistica Sinica 22, 1147-1198 (2012). Full article2010 version

The special law of likelihood has many advantages over more commonly used approaches to measuring the strength of statistical evidence. However, it only can measure the support of a hypothesis that corresponds to a single distribution. The proposed general law of likelihood also can measure the extent to which the data support a hypothesis that corresponds to multiple distributions. That is accomplished by formalizing inference to the best explanation.

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False discovery rates are misleadingly low

2 March 2016 Leave a comment

D. R. Bickel, “Correcting false discovery rates for their bias toward false positives,” Working Paper, University of Ottawa, deposited in uO Research at https://goo.gl/GcUjJe (2016). 2016 preprint | Slides: CFDR and RFDR for SSC 2017

12 June 2017: URL updated and slides added

 

Maximum entropy over a set of posteriors

10 August 2015 Leave a comment

D. R. Bickel, “Blending Bayesian and frequentist methods according to the precision of prior information with applications to hypothesis testing,” Statistical Methods & Applications 24, 523-546 (2015). Published article2012 preprint | 2011 preprint | Slides | Simple explanation

SMA

This framework of statistical inference facilitates the development of new methodology to bridge the gap between the frequentist and Bayesian theories. As an example, a simple and practical method for combining p-values with a set of possible posterior probabilities is provided.

In this general approach, Bayesian inference is used when the prior distribution is known, frequentist inference is used when nothing is known about the prior, and both types of inference are blended according to game theory when the prior is known to be a member of some set. (The robust Bayes framework represents knowledge about a prior in terms of a set of possible priors.) If the benchmark posterior that corresponds to frequentist inference lies within the set of Bayesian posteriors derived from the set of priors, then the benchmark posterior is used for inference. Otherwise, the posterior within that set that minimizes the cross entropy to the benchmark posterior is used for inference.

Bayes/non-Bayes blended inference

5 October 2012 Leave a comment

Updated with a new multiple comparison procedure and applications on 30 June 2012 and with slides for a presentation on 5 October 2012:

D. R. Bickel, “Blending Bayesian and frequentist methods according to the precision of prior information with applications to hypothesis testing,” Working Paper, University of Ottawa, deposited in uO Research at http://hdl.handle.net/10393/23124 (2012)2012 preprint | 2011 preprint | Slides

This framework of statistical inference facilitates the development of new methodology to bridge the gap between the frequentist and Bayesian theories. As an example, a simple and practical method for combining p-values with a set of possible posterior probabilities is provided.

In this new approach to statistics, Bayesian inference is used when the prior distribution is known, frequentist inference is used when nothing is known about the prior, and both types of inference are blended according to game theory when the prior is known to be a member of some set. (The robust Bayes framework represents knowledge about a prior in terms of a set of possible priors.) If the benchmark posterior that corresponds to frequentist inference lies within the set of Bayesian posteriors derived from the set of priors, then the benchmark posterior is used for inference. Otherwise, the posterior within that set that is closest to the benchmark posterior is used for inference.

Local FDR estimation software

30 June 2012 1 comment

LFDRenrich is a suite of R functions for the estimation of local false discovery rates by maximum likelihood under a two-component or three-component parametric mixture model of 2X2 tables such as those used in gene enrichment analyses.

LFDRhat is a more general suite of R functions for the estimation of local false discovery rates by maximum likelihood under a two-component or three-component parametric mixture model.

Extending the likelihood paradigm

15 June 2012 Leave a comment

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D. R. Bickel, “The strength of statistical evidence for composite hypotheses: Inference to the best explanation,” Statistica Sinica 22, 1147-1198 (2012). Full article2010 version

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Effect-size estimates from hypothesis probabilities

25 February 2012 Leave a comment

D. R. Bickel, “Empirical Bayes interval estimates that are conditionally equal to unadjusted confidence intervals or to default prior credibility intervals,” Statistical Applications in Genetics and Molecular Biology 11 (3), art. 7 (2012). Full article | 2010 preprint

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The method contributed in this paper adjusts confidence intervals in multiple-comparison problems according to the estimated local false discovery rate. This shrinkage method performs substantially better than standard confidence intervals under the independence of the data across comparisons. A special case of the confidence intervals is the posterior median, which provides an improved method of ranking biological features such as genes, proteins, or genetic variants. The resulting ranks of features lead to better prioritization of which features to investigate further.