Nested Hypothesis Testing: The Bayesian Reference Criterion

It is argued that hypothesis testing problems are best considered as decision problems concerning the choice of a useful probability model. Decision theory, information measures and reference analysis, are combined to propose a non-subjective Bayesian approach to nested hypothesis testing, the Bayesian Reference Criterion (BRC). The results are compared both with frequentist based procedures, and with the use of Bayes factors. The theory is illustrated with stylized examples, where alternative approaches may easily be compared.

[1]  José M Bernardo and Adrian F M Smith,et al.  BAYESIAN THEORY , 2008 .

[2]  J. Bernardo,et al.  An introduction to Bayesian reference analysis: inference on the ratio of multinomial parameters , 1998 .

[3]  C. Robert,et al.  Model choice in generalised linear models: A Bayesian approach via Kullback-Leibler projections , 1998 .

[4]  A. O’Hagan,et al.  Properties of intrinsic and fractional Bayes factors , 1997 .

[5]  Huaiyu Zhu.  On Information and Sufficiency , 1997 .

[6]  C. Robert,et al.  Noninformative Bayesian testing and neutral Bayes factors , 1996 .

[7]  G. Datta.  On priors providing frequentist validity of Bayesian inference for multiple parametric functions , 1996 .

[8]  J. Berger,et al.  The Intrinsic Bayes Factor for Model Selection and Prediction , 1996 .

[9]  Keying Ye,et al.  Frequentist validity of posterior quantiles for a two-parameter exponential family , 1996 .

[10]  Malay Ghosh,et al.  Some remarks on noninformative priors , 1995 .

[11]  Jayanta K. Ghosh,et al.  Noninformative priors for maximal invariant parameter in group models , 1995 .

[12]  M. Ghosh,et al.  Hierarchical Bayes estimators of the error variance in one-way ANOVA models , 1995 .

[13]  A. O'Hagan,et al.  Fractional Bayes factors for model comparison , 1995 .

[14]  D. Gorenstein,et al.  Outline of proof , 1994 .

[15]  J. Q. Smith,et al.  1. Bayesian Statistics 4 , 1993 .

[16]  Giovanni Parmigiani,et al.  Bayesian Model Criticism , 1993 .

[17]  Raúl Rueda,et al.  A Bayesian alternative to parametric hypothesis testing , 1992 .

[18]  M. Mouchart,et al.  Bayesian testing and testing Bayesians , 1992 .

[19]  J. Berger,et al.  Interpreting the stars in precise hypothesis testing , 1991 .

[20]  M. Aitkin.  Posterior Bayes Factors , 1991 .

[21]  David R. Cox,et al.  Role of Models in Statistical Analysis , 1990 .

[22]  E. Lehmann.  Model Specification: The Views of Fisher and Neyman, and Later Developments , 1990 .

[23]  W. Jefferys.  Bayesian Analysis of Random Event Generator Data , 1990 .

[24]  E. Bas,et al.  Testing a Point Null Hypothesis: Asymptotic Robust Bayesian Analysis with Respect to the Priors Given on a Subsigma Field , 1989 .

[25]  James O. Berger,et al.  Estimating a Product of Means: Bayesian Analysis with Reference Priors , 1989 .

[26]  J. Berger,et al.  Testing Precise Hypotheses , 1987 .

[27]  Piero Veronese,et al.  COHERENT DISTRIBUTIONS AND LINDLEY'S PARADOX (*) , 1987 .

[28]  Analisis bayesiano de los contrastes de hipotesis parametricos , 1985 .

[29]  E. Jaynes,et al.  E. T. Jaynes: Papers on Probability, Statistics and Statistical Physics , 1983 .

[30]  B. Reiser,et al.  An Exponential Subfamily which Admits UMPU Tests Based on a Single Test Statistic , 1982 .

[31]  D. Spiegelhalter,et al.  Bayes Factors for Linear and Log‐Linear Models with Vague Prior Information , 1982 .

[32]  Contraste de modelos probabilisticos desde una perspectiva bayesiana , 1982 .

[33]  G. Shafer.  Lindley's Paradox , 1982 .

[34]  M. Bayarri.  Inferencia bayesiana sobre el coeficiente de correlacion de una poblacion normal bivariante , 1981 .

[35]  J. Bernardo.  A Bayesian analysis of classical hypothesis testing , 1980 .

[36]  D. Spiegelhalter,et al.  Bayes Factors and Choice Criteria for Linear Models , 1980 .

[37]  J. Bernardo.  Expected Information as Expected Utility , 1979 .

[38]  J. Bernardo.  Reference Posterior Distributions for Bayesian Inference , 1979 .

[39]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[40]  D. Lindley,et al.  Inference for a Bernoulli Process (a Bayesian View) , 1976 .

[41]  H. Akaike.  A new look at the statistical model identification , 1974 .

[42]  H. Akaike,et al.  Information Theory and an Extension of the Maximum Likelihood Principle , 1973 .

[43]  K. Gaver,et al.  Posterior probabilities of alternative linear models , 1971 .

[44]  Cedric A. B. Smith,et al.  Personal Probability and Statistical Analysis , 1965 .

[45]  G. C. Tiao,et al.  A Further Look at Robustness via Bayes's Theorem , 1962 .

[46]  Howard Raiffa,et al.  Applied Statistical Decision Theory. , 1961 .

[47]  M. Bartlett.  A comment on D. V. Lindley's statistical paradox , 1957 .

[48]  D. Lindley.  A STATISTICAL PARADOX , 1957 .

[49]  L. Stein,et al.  Probability and the Weighing of Evidence , 1950 .