522 10. APPROXIMATE INFERENCE
whereZjis the normalization constant defined by (10.197). By applying this result
recursively, and initializing withp 0 (D)=1, derive the result
p(D)
∏
j
Zj. (10.243)
10.37 ( ) www Consider the expectation propagation algorithm from Section 10.7, and
suppose that one of the factorsf 0 (θ)in the definition (10.188) has the same expo-
nential family functional form as the approximating distributionq(θ). Show that if
the factor ̃f 0 (θ)is initialized to bef 0 (θ), then an EP update to refine ̃f 0 (θ)leaves
̃f 0 (θ)unchanged. This situation typically arises when one of the factors is the prior
p(θ), and so we see that the prior factor can be incorporated once exactly and does
not need to be refined.
10.38 ( ) In this exercise and the next, we shall verify the results (10.214)–(10.224)
for the expectation propagation algorithm applied to the clutter problem. Begin by
using the division formula (10.205) to derive the expressions (10.214) and (10.215)
by completing the square inside the exponential to identify the mean and variance.
Also, show that the normalization constantZn, defined by (10.206), is given for the
clutter problem by (10.216). This can be done by making use of the general result
(2.115).
10.39 ( ) Show that the mean and variance ofqnew(θ)for EP applied to the clutter
problem are given by (10.217) and (10.218). To do this, first prove the following
results for the expectations ofθandθθTunderqnew(θ)
E[θ]=m\n+v\n∇m\nlnZn (10.244)
E[θTθ]=2(v\n)^2 ∇v\nlnZn+2E[θ]Tm\n−‖m\n‖^2 (10.245)
and then make use of the result (10.216) forZn. Next, prove the results (10.220)–
(10.222) by using (10.207) and completing the square in the exponential. Finally,
use (10.208) to derive the result (10.223).
