Fully Bayes factors with a generalized g-prior

FULLY BAYES FACTORS WITH A GENERALIZED g-PRIOR

where α is an unknown intercept parameter, 1n is an n× 1 vector each component of which is one, XF = (x1, . . . ,xp) is an n×p design matrix, βF is a p×1 vector of unknown regression coefficients, In is an n × n identity matrix and σ 2 is an unknown positive scalar. (The subscript F denotes the full model.) We assume that the columns of XF have been standardized so that for 1 ≤ i ≤ p, x′i1n = 0...

No Need for Bayes Factors: A Fully Bayesian Evidence Synthesis

Empirical Bayes vs. Fully Bayes Variable Selection

For the problem of variable selection for the normal linear model, fixed penalty selection criteria such as AIC, Cp, BIC and RIC correspond to the posterior modes of a hierarchical Bayes model for various fixed hyperparameter settings. Adaptive selection criteria obtained by empirical Bayes estimation of the hyperparameters have been shown by George and Foster [2000. Calibration and Empirical B...

Optimal Hypothesis Testing: From Semi to Fully Bayes Factors.

We propose and examine statistical test-strategies that are somewhat between the maximum likelihood ratio and Bayes factor methods that are well addressed in the literature. The paper shows an optimality of the proposed tests of hypothesis. We demonstrate that our approach can be easily applied to practical studies, because execution of the tests does not require deriving of asymptotical analyt...

The Uniform Prior Bayes Estimate

For the Gaussian sequence model Xi = μi + σZi, i = 1, 2, · · · , n, where Zi are iid standard normal, and σ is considered known, we give three equivalent exact formulas for the posterior mean of the mean vector μ = (μ1, · · · , μn) when μ has a uniform prior distribution in an n-dimensional solid ball of a general radius c. The exact formulas have interesting and even curious connections to non...