Statistics Seminars: Bayesian Inference with Sets of Conjugate Priors: Parameter Set Shapes and Model Behaviour.
6 March 2013 14:10 in CM221
Baysian inference is one of the main approaches to statistical inference. It requires to express (subjective) knowledge on the parameters of interest not incorporated in the data by a so-called prior distribution. Via Bayes' rule, prior information and data are combined to the posterior distribution, on which all inferences are based. The adequate choice of priors has always been an intensive matter of debate in the Bayesian literature, and it becomes especially relevant when data is scarce or do not contain much information about the parameters of interest.
An example for the former case is common-cause failure modelling, where data must be complemented with expert assessments in order to obtain sensible inferences. Usual modelling of expert information is through conjugate priors, whose parameters (the so-called hyperparameters) are easy to interpret and elicit. However, posterior inferences are very sensitive to the choice of hyperparameters. Therefore, sets of hyperparameters are considered, leading to sets of conjugate priors, where ambiguity in the prior information is reflected, e.g., by interval-valued expectations. In the context of these imprecise/interval probability models, the choice of hyperparameter set shape is discussed, as it influences posterior inferences. Here, one important aspect is prior-data conflict, where information from oulier-free data suggests hyperparameter values which are very surprising from the viewpoint of prior information. As it may not be clear whether the prior specifications or the integrity of the data collecting method should be questioned, such a conflict should be reflected in the posterior, and lead to very cautious inferences. Contrary to prior information modeling with single conjugate priors, where prior-data conflict is mostly averaged out, certain choices of hyperparameter set shapes exhibit the desired behaviour, leading to cautious inferences if, and only if, caution is needed.
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