# Subjective Bayesian statistics: agreement between prior and data

1 SELECT - Model selection in statistical learning
LMO - Laboratoire de Mathématiques d'Orsay, Inria Saclay - Ile de France
Abstract : When Bayesian inference is required to estimate the parameter of a decision-making model, the prior modelling must be specified with great care, especially when subjective knowledge is used. Indeed, Bayesian inference is jeopardized when there is a too large discrepancy between prior knowledge and observed data. From a theoretical point of view, prior and likelihood constitute the complete Bayesian model and it seems relevant to study the agreement of the data to this model. In the fact, the detection of a possible conflict between the prior and the observed likelihood remains a significant preliminary to subjective Bayesian inference, particularly in industrial reliability, and seems not to have been much studied before. In the present report, we propose to use a criterion $\calCoh(\pi;X_n)$ measuring the agreement of the available prior $\pi$ and the data $X_n$ information with a ratio of Kullback-Leibler distances between the proposed prior and benchmark weakly or noninformative prior distributions. If $\calCoh(\pi;X_n)\leq1$ then both informations are declared in agreement. To use it we have to define a class of minimaly informative proper prior distributions from data. Therefore we propose a general approach using minimal training samples and posterior priors, overcoming some difficulties to model ignorance. This approach is applied to the exponential and Weibull models, the most used distributions in lifetime problems. Finally we discuss the use of the $\calCoh$ criterion as a tool of calibration for subjective prior modelling.
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Reports
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https://hal.inria.fr/inria-00071367
Contributor : Rapport de Recherche Inria <>
Submitted on : Tuesday, May 23, 2006 - 4:54:24 PM
Last modification on : Wednesday, October 14, 2020 - 4:00:28 AM
Long-term archiving on: : Sunday, April 4, 2010 - 10:07:20 PM

### Identifiers

• HAL Id : inria-00071367, version 1

### Citation

Nicolas Bousquet. Subjective Bayesian statistics: agreement between prior and data. [Research Report] RR-5900, INRIA. 2006. ⟨inria-00071367⟩

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