Leitura do Dia - Bayesian heavy-tailed models and con‡flict resolution: a review

Bayesian heavy-tailed models and con‡flict resolution: a review

Anthony O’Hagan and Luis Pericchi

April 5, 2011

Abstract
We review a substantial literature, spanning 50 years, concerning the resolution of confl‡icts using Bayesian heavy-tailed models. Con‡icts arise when different sources of information about the model parameters (e.g. prior information, or the information in individual observations) sug-
gest quite different plausible regions for those parameters. Traditional Bayesian models based on normal distributions or other conjugate structures typically resolve con‡icts by centring the posterior at some compromise position, but this is not a realistic resolution when it means that
the posterior is then in con‡ict with the different information sources.
Bayesian modelling with heavy-tailed distributions has been shown to produce more reasonable con‡ict resolution, typically by favouring one source of information over the other. The less favoured source is ultimately wholly or partially rejected as the con‡ict becomes increasingly
extreme.
The literature reviewed here provides formal proofs of con‡ict resolution by asymptotic rejection of some information sources. Results are given for a variety of models, from the simplest case of a single observation relating to a single location parameter up to models with many
location parameters, location and scale parameters, or other kinds of parameters. However, these results do not begin to address models of the kind of complexity that are routinely used in practical Bayesian modelling. In addition to reviewing the available theory, we also identify clearly the gaps in the literature that need to be …filled in order for modellers to be able to develop applications with appropriate ‘built-in robustness’.

Keywords: rejection of information, partial rejection of information, built-in robustness, heavy-tailed modelling, theory of conffl‡ict resolution,
outliers