7.7 Verify the formulas for Bayesian kriging prediction in Section 7.12. In particular, using the Woodbury formula for the inverse of a matrix

(Ω + FΔFT)

−1 = Ω−1 − Ω−1 FD−1 FTΩ−1

, where D = Δ−1 + FTΩ−1 F

(see Section A.3.6), show that the formula for ????Δ in the posterior mean

(7.69) simplifies to (7.72). You may find it helpful to expand out both formulas and match the terms. In addition, rewrite the equation for D in (7.71)

by multiplying on the left by D−1, on the right by Δ, and rearranging the terms to get Δ = D−1 + D−1FTΩ−1FΔ. Similarly, show that the formula for the posterior variance in the first line of (7.73) simplifies to the second line.

Finally, confirm that as the prior variance matrix Δ gets large, the posterior mean and Bayesian kriging variance converge to the universal kriging predictor and its variance in Section 7.5.

Similarly, show that the posterior mean and variance of ???? converge to the generalized least squares estimate above (7.33) and its variance (FTΩ−1F)

−1.