By Christian P. Robert

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Extra info for The Bayesian Choice

Example text

5). When such improper prior distributions are derived by automatic methods from the density f (x|θ) (see Chapter 3), they seem more open to criticism, but let us point out the following points. (1) These automatic approaches are usually the only way to derive prior distributions in noninformative settings, that is, in cases where the only available (or retained) information is the knowledge of the sample distribution, f (x|θ). This generalization of the usual Bayesian paradigm thus makes possible a further extension of the scope of Bayesian techniques.

Show that, when a spherically symmetric distribution has a density, it is a function of xt x only. b. Show that, if the density of x is ϕ(xt x), the density of r = ||x|| is proportional to r p−1 ϕ(r 2 ), and give the proportionality coeﬃcient. c. Show that, if x = (x1 , x2 ) with x1 ∈ IRq and x2 ∈ IRp−q , and ||x||2 = ||x1 ||2 + ||x2 ||2 , the density of (r1 , r2 ) = (||x1 ||, ||x2 ||) is proportional to r1q−1 r2p−q−1 ϕ r12 + r22 . d. Deduce that ||x1 ||2 ||x1 ||2 + ||x2 ||2 is distributed according to a beta distribution Be(q/2, (p − q)/2).

Xn−1 ) f (xn |θ)π(θ|x1 , . . , xn−1 )dθ f (xn |θ)f (xn−1 |θ)π(θ|x1 , . . , xn−2 ) f (xn |θ)f (xn−1 |θ)π(θ|x1 , . . , xn−2 )dθ ... f (xn |θ)f (xn−1 |θ) . . f (x1 |θ)π(θ) . f (xn |θ)f (xn−1 |θ) . . f (x1 |θ)π(θ)dθ It may happen that the observations do not modify the distribution of some parameters. This is obviously the case when the distribution of x does not depend on these parameters, as in some nonindentiﬁable settings. 2 Consider one observation x from a normal N θ1 + θ2 ,1 2 24 Introduction 1 distribution, with a prior π on (θ1 , θ2 ) such that π(θ1 , θ2 ) = π1 (θ1 + θ2 )π2 (θ1 − θ2 ).