Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig
11.1. Bayesian Procedures 665 11.1.2.LetX 1 ,X 2 ,...,Xnbe a random sample from a distribution that isb(1,θ). Let the prior of Θ ...
666 Bayesian Statistics 11.2MoreBayesianTerminologyandIdeas SupposeX′=(X 1 ,X 2 ,...,Xn) represents a random sample with likelih ...
11.2. More Bayesian Terminology and Ideas 667 does not exist. However, such priors are used if, when combined with the likelihoo ...
668 Bayesian Statistics Let us change variables to get more familiar results; namely, let t= θ 1 −x s/ √ n andθ 1 =x+ts/ √ n, wi ...
11.2. More Bayesian Terminology and Ideas 669 where Q(θ 1 )= 2 β +n 0 (θ 1 −θ 0 )^2 +[(n−1)s^2 +n(x−θ 1 )^2 ] =(n 0 +n) [( θ 1 − ...
670 Bayesian Statistics has at-distribution with 2α+ndegrees of freedom. Of course, using these degrees of freedom, we can findt ...
11.2. More Bayesian Terminology and Ideas 671 11.2.3.Suppose for the situation of Example 11.2.2,θ 1 has the prior distribution ...
672 Bayesian Statistics 11.2.6.Consider the Bayes model Xi|θ,i=1, 2 ,...,n ∼ iid with distribution Poisson (θ),θ> 0 Θ ∼ h(θ)∝ ...
11.3. Gibbs Sampler 673 N(θ, σ^2 ), whereσ^2 is known. ThenY=Xis a sufficient statistic. Consider the Bayes model Y|θ ∼ N(θ, σ^2 ...
674 Bayesian Statistics Theorem 11.3.1.Suppose we generate random variables by the following algo- rithm: GenerateY∼fY(y), Gene ...
11.3. Gibbs Sampler 675 Example 11.3.1.Suppose the random variableXhas pdf fX(x)= { 2 e−x(1−e−x)0<x<∞ 0elsewhere. (11.3.5) ...
676 Bayesian Statistics pdff(x|yi), whereyiis the observed value ofYi. In advanced texts, it is shown that Yi D → Y∼fY(y) Xi D → ...
11.3. Gibbs Sampler 677 In particular, for largemandn>m, Y=(n−m)−^1 ∑n i=m+1 Yi →P E(Y) (11.3.13) X=(n−m)−^1 ∑n i=m+1 Xi →P E ...
678 Bayesian Statistics 11.3.3.Consider Example 11.3.1. (a)Show thatE(X)=1.5. (b)Obtain the inverse of the cdf ofXand use it to ...
11.4. Modern Bayesian Methods 679 11.4ModernBayesianMethods........................ The prior pdf has an important influence in ...
680 Bayesian Statistics ith step of the algorithm is Θi|x,γi− 1 ∼ g(θ|x,γi− 1 ) Γi|x,θi ∼ g(γ|x,θi). Recall from our discussion ...
11.4. Modern Bayesian Methods 681 which is the pdf of a Γ{a+(1/2),[(θ^2 /2) + (1/b)]−^1 }distribution. Thus the Gibbs sampler fo ...
682 Bayesian Statistics In this last expression, making the change of variabley=1/bwhich has the Jacobian db/dy=−y−^2 ,weobtain ...
11.4. Modern Bayesian Methods 683 Consider, then, the likelihood function m(x|γ)= ∫∞ −∞ f(x|θ)h(θ|γ)dθ. (11.4.13) Using the pdfm ...
684 Bayesian Statistics We can use our solution of this last example to obtain the empirical Bayes estimate for Example 11.4.2 a ...
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