G-prior

Slide 64: Bayesian Core: A Practical Approach to Computational Bayesian Statistics Regression and variable selection Zellner's noninformative G-prior Computational details Both terms involve in fi nite summations on c The denominator in both cases is the normalising constant of the posterior ∞ f (y | X, c) c − 1 c = 1 256 / 785

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Slide 60: Bayesian Core: A Practical Approach to Computational Bayesian Statistics Regression and variable selection Zellner's noninformative G-prior Zellner's noninformative G-prior Di ff erence with informative G-prior setup is that we now consider c as unknown (relief!)

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Slide 29: Bayesian Core: A Practical Approach to Computational Bayesian Statistics Regression and variable selection Zellner's informative G-prior Experimenter dilemma Problem of the choice of M or of c if M = Ik+1 / c Example (Processionary caterpillar) ˜ No precise prior information about β, M, a and b.

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Slide 52: Bayesian Core: A Practical Approach to Computational Bayesian Statistics Regression and variable selection Zellner's informative G-prior T HPD A 1 − α HPD interval on βi is thus given by − 1 − 1 τi − κ (i, i) Fn (1 − α / 2), τi + κ (i, i) Fn (1 − α / 2).

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Slide 31: Bayesian Core: A Practical Approach to Computational Bayesian Statistics Regression and variable selection Zellner's informative G-prior Zellner's informative G-prior Constraint Allow the experimenter to introduce information about the location parameter of the regression while bypassing the most di ffi cult aspects of the prior speci fi cation, namely the derivation of the prior correlation structure.

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Slide 50: Bayesian Core: A Practical Approach to Computational Bayesian Statistics Regression and variable selection Zellner's informative G-prior Credible regions Highest posterior density (HPD) regions on subvectors of the parameter β derived from the marginal posterior distribution of β.

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Slide 59: Bayesian Core: A Practical Approach to Computational Bayesian Statistics Regression and variable selection Zellner's noninformative G-prior Zellner's noninformative G-prior Di ff erence with informative G-prior setup is that we now consider c as unknown (relief!) 251 / 785

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Slide 105: Bayesian Core: A Practical Approach to Computational Bayesian Statistics Regression and variable selection Variable selection Prior de fi nitions For the full model, Zellner's G-prior: (i) ˜ β | σ 2, X ∼ Nk+1 (β, cσ 2 (X T X) − 1) and σ 2 ∼ π (σ 2 | X) = σ

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Slide 46: Bayesian Core: A Practical Approach to Computational Bayesian Statistics Regression and variable selection Zellner's informative G-prior Gaussian predictive Conditional on σ 2, the future vector of observations has a Gaussian distribution with ˜ ˜ ˜

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Slide 48: Bayesian Core: A Practical Approach to Computational Bayesian Statistics Regression and variable selection Zellner's informative G-prior Predictor A predictor under squared error loss is the posterior predictive mean ˜ ˆ ˜ β + cβ, X c+1

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