4.14 The purpose of this exercise is to confirm that Eqs. (4.74)–(4.76) imply

(4.81). Let xT be a possible value of the random field and define yT by yt = 0 and y∖t = x∖t. Writing pt(xt|x∖t) = p(xt|x∖t), show that p(xt|x∖t)

p(yt|y∖t) = p(xt|x∖t)pT∖{t}(x∖t)

p(yt|y∖t)pT∖{t}(y∖t)

= p(xT)

p(yT)

= exp [

∑

Λ⊂T

{GΛ(xΛ) − GΛ(yΛ)}]

.

If t ∉ Λ, note that xΛ = yΛ, so the corresponding term disappears from the sum. If t ∈ Λ, show that GΛ(yΛ) = 0. Hence, confirm that the formula for p(xt|x∖t)∕p(yt|y∖t) reduces to the right-hand side of (4.81).