Pattern Recognition and Machine Learning

512 10. APPROXIMATE INFERENCE

The factor approximations will therefore take the form of exponential-quadratic

functions of the form

̃fn(θ)=snN(θ|mn,vnI) (10.213)

wheren=1,...,N, and we set ̃f 0 (θ)equal to the priorp(θ). Note that the use of
N(θ|·,·)does not imply that the right-hand side is a well-defined Gaussian density
(in fact, as we shall see, the variance parametervncan be negative) but is simply a
convenient shorthand notation. The approximations ̃fn(θ), forn=1,...,N, can
be initialized to unity, corresponding tosn=(2πvn)D/^2 ,vn→∞andmn= 0 ,
whereDis the dimensionality ofxand hence ofθ. The initialq(θ), defined by
(10.191), is therefore equal to the prior.
We then iteratively refine the factors by taking one factorfn(θ)at a time and
applying (10.205), (10.206), and (10.207). Note that we do not need to revise the
Exercise 10.37 termf 0 (θ)because an EP update will leave this term unchanged. Here we state the
results and leave the reader to fill in the details.
First we remove the current estimate ̃fn(θ)fromq(θ)by division using (10.205)
Exercise 10.38 to giveq\n(θ), which has mean and inverse variance given by

m\n = m+v\nvn−^1 (m−mn) (10.214)

(v\n)−^1 = v−^1 −vn−^1. (10.215)

Next we evaluate the normalization constantZnusing (10.206) to give

Zn=(1−w)N(xn|m\n,(v\n+1)I)+wN(xn| 0 ,aI). (10.216)

Similarly, we compute the mean and variance ofqnew(θ)by finding the mean and
Exercise 10.39 variance ofq\n(θ)fn(θ)to give

m = m\n+ρn

v\n

v\n+1

(xn−m\n) (10.217)

v = v\n−ρn

(v\n)^2

v\n+1

+ρn(1−ρn)

(v\n)^2 ‖xn−m\n‖^2

D(v\n+1)^2

(10.218)

where the quantity

ρn=1−

w

Zn

N(xn| 0 ,aI) (10.219)

has a simple interpretation as the probability of the pointxnnot being clutter. Then

we use (10.207) to compute the refined factor ̃fn(θ)whose parameters are given by

vn−^1 =(vnew)−^1 −(v\n)−^1 (10.220)

mn = m\n+(vn+v\n)(v\n)−^1 (mnew−m\n) (10.221)

sn =

Zn

(2πvn)D/^2 N(mn|m\n,(vn+v\n)I)

. (10.222)

This refinement process is repeated until a suitable termination criterion is satisfied,

for instance that the maximum change in parameter values resulting from a complete