## Model fusion & multiple testing in the likelihood paradigm

D. R. Bickel, “Model fusion and multiple testing in the likelihood paradigm: Shrinkage and evidence supporting a point null hypothesis,” Working Paper, University of Ottawa, deposited in uO Research at http://hdl.handle.net/10393/31897 (2014). 2014 preprint | Supplement (link added 10 February 2015)

Errata for Theorem 4:

- The weights of evidence should not be conditional.
- Some of the equal signs should be “is a member of” signs.

## Fiducial error propagation for empirical Bayes set estimates

D. R. Bickel, “A fiducial continuum from confidence sets to empirical Bayes set estimates as the number of comparisons increases,” Working Paper, University of Ottawa, deposited in uO Research at http://hdl.handle.net/10393/31898 (2014). 2014 preprint

Two problems confronting the eclectic approach to statistics result from its lack of a unifying theoretical foundation. First, there is typically no continuity between a p-value reported as a level of evidence for a hypothesis in the absence of the information needed to estimate a relevant prior on one hand and an estimated posterior probability of a hypothesis reported in the presence of such information on the other hand. Second, the empirical Bayes methods recommended do not propagate the uncertainty due to estimating the prior.

The latter problem is addressed by applying a coherent form of fiducial inference to hierarchical models, yielding empirical Bayes set estimates that reflect uncertainty in estimating the prior. Plugging in the maximum likelihood estimator, while not propagating that uncertainty, provides continuity from single comparisons to large numbers of comparisons.

## Causality, Probability, and Time (by Kleinberg)—a review

Kleinberg, Samantha

Causality, probability, and time. Cambridge University Press, Cambridge, 2013. viii+259 pp. ISBN: 978-1-107-02648-3

60A99 (03A05 03B48 62A01 62P99 68T27 91G80 92C20)

Arguably an equally valuable contribution of the book is its integration of relevant work in philosophy, computer science, and statistics. While the first two disciplines have benefited from the productive interactions exemplified in [J. Pearl, Probabilistic reasoning in intelligent systems: networks of plausible inference, Morgan Kaufmann Ser. Represent. Reason., Morgan Kaufmann, San Mateo, CA, 1988; MR0965765 (90g:68003)] and [J. Williamson, Bayesian nets and causality, Oxford Univ. Press, Oxford, 2005; MR2120947 (2005k:68198)], the statistics community has developed its own theory of causal inference in relative isolation. Rather than following S. L. Morgan and C. Winship [Counterfactuals and causal inference: methods and principles for social research, Cambridge Univ. Press, New York, 2007] and others in bringing that theory into conversation with that of Pearl [op. cit.], the author creatively employs recent developments in statistical inference to identify causes.

For the specific situation in which many putative causes are tested but only a few are true causes, she explains how to estimate the local rate of discovering false causes. In this context, the local false discovery rate (LFDR) corresponding to a putative cause is a posterior probability that it is not a true cause. This is an example of an empirical Bayes method in that the prior distribution is estimated from the data rather than assigned.

Building on [P. Suppes, A probabilistic theory of causality, North-Holland, Amsterdam, 1970; MR0465774 (57 #5663)], the book emphasizes the importance for prediction not only of whether something is a cause but also of the strength of a cause. A cause is ε–significant if its causal strength, defined in terms of changing the probability of its effect, is at least ε, where ε is some nonnegative number. Otherwise, it is ε-insignificant.

The author poses an important problem and comes close to solving it, i.e., the problem of inferring whether a cause is ε-significant. The solution attempted in Section 4.2 confuses causal significance (ε-significance) with statistical significance (LFDR estimate below some small positive number α). This is by no means a fatal criticism of the approach since it can be remedied in principle by defining a false discovery as a discovery of an ε-insignificant cause. This tests the null hypothesis that the cause is ε-insignificant for a specified value of ε rather than the book’s null hypothesis, which in effect asserts that the cause is limε→0ε-insignificant, i.e., ε-insignificant for all ε>0. In the case of a specified value of ε, a cause should be considered ε-significant if the estimated LFDR is less than α, provided that the LFDR is defined in terms of the null hypothesis of ε-insignificance. The need to fill in the technical details and to answer more general questions arising from this distinction between causal significance and statistical significance opens up exciting opportunities for further research guided by insights from the literature on seeking substantive significance as well as statistical significance [see, e.g., M. A. van de Wiel and K. I. Kim, Biometrics 63 (2007), no. 3, 806–815; MR2395718].

Reviewed by David R. Bickel

This review first appeared at Causality, Probability, and Time (Mathematical Reviews) and is used with permission from the American Mathematical Society.

## Small dimensional empirical Bayes inference

D. R. Bickel, “Simple estimators of false discovery rates given as few as one or two p-values without strong parametric assumptions,” *Statistical Applications in Genetics and Molecular Biology* **12**, 529–543 (2013). 2011 version | erratum

To address multiple comparison problems in small-to-high-dimensional biology, this paper introduces estimators of the local false discovery rate (LFDR), reports their main properties, and illustrates their use with proteomics data. The new estimators have the following advantages:

- proven asymptotic conservatism;
- simplicity of calculation without the tuning of smoothing parameters;
- no strong parametric assumptions;
- applicability to very small numbers of hypotheses as well as to very large numbers of hypotheses.

The link to the erratum was added 31 March 2015.

## Estimates of the local FDR

Z. Yang, Z. Li, and D. R. Bickel, “Empirical Bayes estimation of posterior probabilities of enrichment: A comparative study of five estimators of the local false discovery rate,” *BMC Bioinformatics* **14**, art. 87 (2013). published version | 2011 version | 2010 version

This paper adapts novel empirical Bayes methods for the problem of detecting enrichment in the form of differential representation of genes associated with a biological category with respect to a list of genes identified as differentially expressed. Read more…

## Optimal strength of evidence

D. R. Bickel, “Minimax-optimal strength of statistical evidence for a composite alternative hypothesis,” *International Statistical Review* **81**, 188-206 (2013). 2011 version | Simple explanation (added 2 July 2017)

This publication generalizes the likelihood measure of evidential support for a hypothesis with the help of tools originally developed by information theorists for minimizing the number of letters in a message. The approach is illustrated with an application to proteomics data.

## Local FDR estimation for low-dimensional data

M. Padilla and D. R. Bickel, “Estimators of the local false discovery rate designed for small numbers of tests,” *Statistical Applications in Genetics and Molecular Biology* **11** (5), art. 4 (2012). Full article | 2010 & 2012 preprints

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