Iterative Optimization in Inverse Problems

188 Iterative Optimization in Inverse Problems

This is false; integrating with respect to x gives 1 on the left side and

1/f

Y

(y|θ) on the right side. Perhaps the equation is not meant to hold for all

x, but just for some x. In fact, if there is a function h such that Y = h(X),

then Equation (13.10) might hold for those x such that h(x)=y.Infact,

this is what happens in the discrete case of probabilities; in that case we

do have

f

Y

(y|θ)=



x∈h

−1

{y}

f

X

(x|θ), (13.11)

where

h

−1

{y} = {x|h(x)=y}.

Consequently,

f

X|Y

(x|y,θ)=f

X

(x|θ)/f

Y

(y|θ), if x ∈ h

−1

{y}, (13.12)

and zero, otherwise. However, this modification of Equation (13.10) fails in

the continuous case of probability ...