7.12 Suppose x: p 1, and L(x|) = N(,1I), for 1 known. Show that if L(...

7.12 Suppose x: p × 1, and L(x|θ) = N(θ,σ1¹⁵I), for σ1¹⁵ known. Show that if L(θ Φ) = N(Φe,σ2¹⁵I), where Φ is a scalar, e is a p-vector of ones, and σ2¹⁵ is known, and if the density of Φ is proportional to a constant, the posterior density of θ given x is Normal, E(θ x) = Gx, where G denotes an intraclass covariance matrix, and in particular, for θ = (θi), i = 1,..., p,

where hi = σi—2, i = 1, 2, and ≡ (xi/p). Note that the mean of θ1 is a weighted average of the traditional maximum likelihood estimator, x1, and thecomponent mean, , and the weights are the precisions of the sampling and prior distributions. The effect of this averaging is to pull the estimator away from x1 toward the component mean, (See also the Lindley discussion of Stein, 1962.)