Robust bayes analysis in normal linear regression with an improper mixture prior

This paper examines the linear regression model when prior information of the regression parameter β∊R^k and the error variance σ²> 0 is represented by the improper mixture pdf π(β, σ²)α(1-∊)π₀(β,σ²)+∊q(β,σ²), where π₀ is the usual normal inverted gamma base prior, q is a specific noninformative prior and (1-∊), 0 ≤ ∊ ≤1, is the degree of confidence about π₀. This prior is specially useful in multiparameter situations since no new hyperparameters are required in the specification of q. The prior to posterior analysis is shown to produce a mixture pdf which may be multimodal. Robustness is obtained in the following sense: if π₀ is not compatible with the data, its influence in the posterior is discounted, and more weight is automatically placed on the posterior that emerges with the noninformative prior. A numerical example is presented that illustrates the ideas developed in the paper.