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Archive for the ‘imprecise probability’ Category

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.

How to combine statistical methods

29 August 2012 1 comment

D. R. Bickel, “Game-theoretic probability combination with applications to resolving conflicts between statistical methods,” International Journal of Approximate Reasoning 53, 880-891 (2012). Full article | 2011 preprint | Slides | Simple explanation

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This paper proposes both a novel solution to the problem of combining probability distributions and a framework for using the new method to combine the results of differing statistical methods that may legitimately be used to analyze the same data set. While the paper emphasizes theoretical development, it is motivated by the need to combine two conflicting estimators of the probability of differential gene expression.

Confidence-based decision theory

1 May 2012 Leave a comment

D. R. Bickel, “Coherent frequentism: A decision theory based on confidence sets,” Communications in Statistics – Theory and Methods 41, 1478-1496 (2012). Full article (open access) | 2009 version | Simple explanation (link added 27 June 2018)

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To combine the self-consistency of Bayesian statistics with the objectivity of frequentist statistics, this paper formulates a framework of inference for developing novel statistical methods. The framework is based on a confidence posterior, a parameter probability distribution that does not require any prior distribution. While the Bayesian posterior is defined in terms of a conditional distribution given the observed data, the confidence posterior is instead defined such that the probability that the parameter value lies in any fixed subset of parameter space, given the observed data, is equal to the coverage rate of the corresponding confidence interval. Inferences based on the confidence posterior are reliable in the sense that the certainty level of a composite hypothesis is a weakly consistent estimate of the 0-1 indicator of hypothesis truth. At the same time, the confidence posterior is as non-contradictory as the Bayesian posterior since both satisfy the same coherence axioms. Using the theory of coherent upper and lower probabilities, the confidence posterior is generalized for situations in which no approximate or exact confidence set is available. Examples of hypothesis testing and estimation illustrate the range of applications of the proposed framework.

Additional summaries appear in the abstract and in Section 1.3 of the paper.

How to use priors with caution

13 April 2012 Leave a comment

D. R. Bickel, “Controlling the degree of caution in statistical inference with the Bayesian and frequentist approaches as opposite extremes,” Electronic Journal of Statistics 6, 686-709 (2012). Full text (open access) | 2011 preprint

Electronic Journal of Statistics

This paper reports a novel probability-interval framework for combining strengths of frequentist and Bayesian methods on the basis of game-theoretic first principles. It enables data analysis on the basis of the posterior distribution that is a blend between a set of plausible Bayesian posterior distributions and a parameter distribution that represents an alternative method of data analysis. This paper’s framework of statistical inference is intended to facilitate the development of new methods to bridge the gap between the frequentist and Bayesian approaches. Four concrete examples illustrate how such intermediate methods can leverage strengths of the two extreme approaches.

Combining inferences from different methods

28 November 2011 Leave a comment

D. R. Bickel, “Resolving conflicts between statistical methods by probability combination: Application to empirical Bayes analyses of genomic data,” Technical Report, Ottawa Institute of Systems Biology, arXiv:1111.6174 (2011). Full preprint

This paper proposes a solution to the problem of combining the results of differing statistical methods that may legitimately be used to analyze the same data set. The motivating application is the combination of two estimators of the probability of differential gene expression: one uses an empirical null distribution, and the other uses the theoretical null distribution. Since there is usually not any reliable way to predict which null distribution will perform better for a given data set and since the choice between them often has a large impact on the conclusions, the proposed hedging strategy addresses a pressing need in statistical genomics. Many other applications are also mentioned in the abstract and described in the introduction.