# THE FORMAL DEFINITION OF REFERENCE PRIORS

@article{Berger2009THEFD, title={THE FORMAL DEFINITION OF REFERENCE PRIORS}, author={James O. Berger and Jos{\'e} M. Bernardo and Dongchu Sun}, journal={Annals of Statistics}, year={2009}, volume={37}, pages={905-938} }

Reference analysis produces objective Bayesian inference, in the sense that inferential statements depend only on the assumed model and the available data, and the prior distribution used to make an inference is least informative in a certain information-theoretic sense. Reference priors have been rigorously defined in specific contexts and heuristically defined in general, but a rigorous general definition has been lacking. We produce a rigorous general definition here and then show how an…

## 340 Citations

Objective priors in the empirical Bayes framework

- Mathematics, Computer Science
- 2016

A non-parametric, transformation invariant estimator for the prior distribution is introduced in terms of the missing information similar to the reference prior, which implies a natural interpretation as a trade-off between choosing the least informative prior and incorporating the information provided by the data, a symbiosis between the objective and empirical Bayes methodologies.

Integrated Objective Bayesian Estimation and Hypothesis Testing

- Computer Science
- 2011

The combined use of the intrinsic discrepancy, an invariant information-based loss function, and appropriately defined reference priors provides an integrated objective Bayesian solution to both estimation and hypothesis testing problems.

Bayesian Methodology in Statistics

- Mathematics
- 2009

Bayesian methods provide a complete paradigm for statistical inference under uncertainty. These may be derived from an axiomatic system and provide a coherent methodology which makes it possible to…

Integrated Objective Bayesian Estimation and Hypothesis Testing

- Computer Science
- 2010

The combined use of the intrinsic discrepancy, an invariant information-based loss function, and appropriately defined reference priors provides an integrated objective Bayesian solution to both estimation and hypothesis testing problems.

Learning Approximately Objective Priors

- Computer Science, MathematicsUAI
- 2017

Techniques for learning reference prior approximations are proposed that select a parametric family and optimize a black-box lower bound on the reference prior objective to find the member of the family that serves as a good approximation.

Non-informative priors in GUM Supplement 1

- Mathematics
- 2011

Abstract Supplement 1 to the ‘Guide to the Expression of Uncertainty in Measurement’ (GUM S1) proposes a Monte Carlo method for the propagation of the probability density functions (PDFs) assigned to…

Objective Priors: An Introduction for Frequentists

- Mathematics
- 2011

Bayesian methods are increasingly applied in these days in the theory and practice of statistics. Any Bayesian inference depends on a likelihood and a prior. Ideally one would like to elicit a prior…

Objective Bayes models for compatibility assessment and bias estimation

- Mathematics
- 2017

Abstract The paper derives a class of non-informative, probability matching priors and of default, data-dependent priors for the difference in two normal means when the variance(s) are unknown. These…

Overall Objective Priors

- Computer Science, Mathematics
- 2015

This paper considers three methods for selecting a single objective prior and study, in a variety of problems including the multinomial problem, whether or not the resulting prior is a reasonable overall prior.

Posterior propriety in Bayesian extreme value analyses using reference priors

- Mathematics
- 2015

The Generalized Pareto (GP) and Generalized extreme value (GEV) distributions play an important role in extreme value analyses, as models for threshold excesses and block maxima respectively. For…

## References

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This chapter describes reference analysis, a method to produce Bayesian inferential statements which only depend on the assumed model and the available data. Statistical information theory is used to…

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- Mathematics
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SUMMARY In this paper, reference priors are derived for three cases where partial information is available. If a subjective conditional prior is given, two reasonable methods are proposed for finding…

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For certain mixture models, improper priors are undesirable because they yield improper posteriors. However, proper priors may be undesirable because they require subjective input. We propose the use…

Estimating a Product of Means: Bayesian Analysis with Reference Priors

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Abstract Suppose that we observe X ∼ N(α, 1) and, independently, Y ∼ N(β, 1), and are concerned with inference (mainly estimation and confidence statements) about the product of means θ = αβ. This…

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Various approaches to the development of a noninformative prior for the AR(1) model are considered and compared. Particular attention is given to the reference prior approach, which seems to work…

Objective Bayesian Analysis of Spatially Correlated Data

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Spatially varying phenomena are often modeled using Gaussian random fields, specified by their mean function and covariance function. The spatial correlation structure of these models is commonly…

Reference priors in multiparameter nonregular cases

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SummaryThe reference prior in the sense of Bernardo is derived in some multiparameter nonregular cases. The family of densities we consider have discontinuities as some points which depend on one…

Jeffreys' prior is asymptotically least favorable under entropy risk

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We provide a rigorous proof that Jeffreys’ prior asymptotically maximizes Shannon’s mutual information between a sample of size n and the parameter. This was conjectured by Bernard0 (1979) and,…

Ordered group reference priors with application to the multinomial problem

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SUMMARY Noninformative priors are developed, using the reference prior approach, for multiparameter problems in which there may be parameters of interest and nuisance parameters. For a given grouping…

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- Mathematics
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Bayesian statistical practice makes extensive use of versions of ob- jective Bayesian analysis. We discuss why this is so, and address some of the criticisms that have been raised concerning…