## Inference to the best explanation of the evidence

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.

D. R. Bickel, “The strength of statistical evidence for composite hypotheses: Inference to the best explanation,” *Statistica Sinica* **22**, 1147-1198 (2012). Full article | 2010 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.

## Estimates of the local false discovery rate based on prior information: Application to GWAS

A. Karimnezhad and D. R. Bickel, “Incorporating prior knowledge about genetic variants into the analysis of genetic association data: An empirical Bayes approach,” Working Paper, University of Ottawa, deposited in uO Research at http://hdl.handle.net/10393/34889 (2016). 2016 preprint

## Empirical Bayes single-comparison procedure

D. R. Bickel, “Small-scale inference: Empirical Bayes and confidence methods for as few as a single comparison,” *International Statistical Review ***82**, 457-476 (2014). Published version | 2011 preprint | Simple explanation (link added 21 June 2017)

Parametric empirical Bayes methods of estimating the local false discovery rate by maximum likelihood apply not only to the large-scale settings for which they were developed, but, with a simple modification, also to small numbers of comparisons. In fact, data for a single comparison are sufficient under broad conditions, as seen from applications to measurements of the abundance levels of 20 proteins and from simulation studies with confidence-based inference as the competitor.

## Meaningful constraints and meaningless priors

Constraints versus priors.

*SIAM/ASA J. Uncertain. Quantif.*3 (2015), no. 1, 586–598.

62A01 (62C10 62C20 62G15)

In this lucid expository paper, Stark advances several arguments for using frequentist methods instead of Bayesian methods in statistical inference and decision problems. The main examples involve restricted-parameter problems, those of inferring the value of a parameter of interest that is constrained to lie in an unusually restrictive set. When the parameter is restricted, frequentist methods can lead to solutions markedly different from those of Bayesian methods. For even when the prior distribution is a default intended to be weakly informative, it actually carries substantial information.

Stark calls routine Bayesian practice into question since priors are not selected according to the analyst’s beliefs but rather for reasons that have no apparent support from the Dutch book argument, the featured rationale for Bayesianism as a rational norm (pp. 589–590; [see D. V. Lindley, *Understanding uncertainty*, revised edition, Wiley Ser. Probab. Stat., Wiley, Hoboken, NJ, 2014; MR3236718]). Uses of the prior beyond the scope of the paper include those encoding (1) empirical Bayes estimates of parameter variability [e.g., B. Efron, *Large-scale inference*, Inst. Math. Stat. Monogr., 1, Cambridge Univ. Press, Cambridge, 2010; MR2724758 (2012a:62006)], (2) the beliefs of subject-matter experts [e.g., A. O’Hagan et al., *Uncertain judgements: eliciting experts’ probabilities*, Wiley, West Sussex, 2006, doi:10.1002/0470033312], or (3) the beliefs of archetypical agents of wide scientific interest [e.g., D. J. Spiegelhalter, K. R. Abrams and J. P. Myles, *Bayesian approaches to clinical trials and health-care evaluation*, Wiley, West Sussex, 2004 (Section 5.5), doi:10.1002/0470092602].

Stark finds Bayesianism to lack not only normative force but also descriptive power. He stresses that he does not know anyone who updates personal beliefs according to Bayes’s theorem in everyday life (pp. 588, 590).

In the conclusions section, Stark asks, “Which is the more interesting question: what would happen if Nature generated a new value of the parameter and the data happened to remain the same, or what would happen for the same value of the parameter if the measurement were repeated?” For the Bayesian who sees parameter distributions more in terms of beliefs than random events, the missing question is, “What should one believe about the value of a parameter given what happened and the information encoded in the prior and other model specifications?” That question would interest Stark only to the extent that the prior encodes meaningful information (p. 589).

Reviewed by David R. Bickel

This review first appeared at “Constraints versus priors” (Mathematical Reviews) and is used with permission from the American Mathematical Society.

## Small-scale empirical Bayes & fiducial estimators

M. Padilla and D. R. Bickel, “Empirical Bayes and fiducial effect-size estimation for small numbers of tests,” Working Paper, University of Ottawa, deposited in uO Research at http://hdl.handle.net/10393/32151 (2015). 2015 preprint