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Abstract:
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We consider the classic linear regression model where the error variance is assumed to be unknown. In the Bayesian setting, traditional approaches to this problem often involve the use of "conjugate" priors where the prior on the regression coefficients depends on the error variance. Surprisingly, this approach is problematic for maximum a posteriori (MAP) estimation. Here, the resulting estimates for the error variance cannot adapt to the underlying sparsity of the coefficients, resulting in severe underestimates. We present examples demonstrating this phenomenon and recommend instead independent priors on the coefficients and variance. Interestingly, this recommendation is connected to recent results for penalized likelihood estimation with unknown error variance.
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