The Bayesian Choice: From Decision-Theoretic Foundations to by Christian P. Robert

By Christian P. Robert

This is often an advent to Bayesian records and choice idea, together with complex subject matters comparable to Monte Carlo equipment. This re-creation includes numerous revised chapters and a brand new bankruptcy on version selection.

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Additional info for The Bayesian Choice: From Decision-Theoretic Foundations to Computational Implementation, 2nd Edition

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3. π(σ 2 |x) ∝ Obviously, these limiting arguments are ad-hoc expedients which are not always justified, in particular because the resulting estimator may depend on the choice of the converging sequence.

4 Prior and posterior distributions Let us assume at this point that, in addition to the sample distribution, f (x|θ), a prior distribution on θ, π(θ), is available, that is, that we deal with a complete Bayesian model. Chapter 3 considers the preliminary problem of deriving this distribution from the prior information. Given these two distributions, we can construct several distributions, namely: (a) the joint distribution of (θ, x), ϕ(θ, x) = f (x|θ)π(θ) ; (b) the marginal distribution of x, m(x) = ϕ(θ, x) dθ = f (x|θ)π(θ) dθ ; (c) the posterior distribution of θ, obtained by Bayes’s formula, π(θ|x) = = f (x|θ)π(θ) f (x|θ)π(θ) dθ f (x|θ)π(θ) ; m(x) (d) the predictive distribution of y, when y ∼ g(y|θ, x), obtained by g(y|x) = g(y|θ, x)π(θ|x)dθ .

Given a probability distribution π on θ, the Bayesian inferential scope is much larger than the classical perspective. For instance, not only the mean, mode, or median of π(θ|x) can be computed, but also evaluations of the performances of these estimators (through their variance and higherorder moments) are available. Moreover, the knowledge of the posterior distribution also allows for the derivation of confidence regions through highest posterior density (HPD) regions, that is, regions of the form {θ; π(θ|x) ≥ k}, in both unidimensional and multidimensional cases.

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