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Uncertainty propagation for empirical Bayes interval estimates: A fiducial approach

1 December 2017 Comments off

D. R. Bickel, “Confidence distributions applied to propagating uncertainty to inference based on estimating the local false discovery rate: A fiducial continuum from confidence sets to empirical Bayes set estimates as the number of comparisons increases,” Communications in Statistics – Theory and Methods 46, 10788-10799 (2017). Published article | Free access (limited time)2014 preprint

Publication Cover

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.

Empirical Bayes single-comparison procedure

1 July 2016 Comments off

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 version2011 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.

Inference after checking the prior & sampling model

1 September 2015 Comments off

D. R. Bickel, “Inference after checking multiple Bayesian models for data conflict and applications to mitigating the influence of rejected priors,” International Journal of Approximate Reasoning 66, 53–72 (2015). Simple explanation | Published version2014 preprint | Slides

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The proposed procedure combines Bayesian model checking with robust Bayes acts to guide inference whether or not the model is found to be inadequate:

  1. The first stage of the procedure checks each model within a large class of models to determine which models are in conflict with the data and which are adequate for purposes of data analysis.
  2. The second stage of the procedure applies distribution combination or decision rules developed for imprecise probability.

This proposed procedure is illustrated by the application of a class of hierarchical models to a simple data set.

The link Simple explanation was added on 6 June 2017.

Maximum entropy over a set of posteriors

10 August 2015 Comments off

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

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.

Erratum: “Simple estimators of false discovery rates given as few as one or two p-values without strong parametric assumptions”

31 March 2015 Comments off

Coherent fiducial distributions

20 August 2013 Comments off

D. R. Bickel and M. Padilla, “A prior-free framework of coherent inference and its derivation of simple shrinkage estimators,” Journal of Statistical Planning and Inference 145, 204–221 (2014). 2012 version

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Small dimensional empirical Bayes inference

9 May 2013 Comments off

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

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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:

  1. proven asymptotic conservatism;
  2. simplicity of calculation without the tuning of smoothing parameters;
  3. no strong parametric assumptions;
  4. 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.