The purpose of this appendix is to demonstrate that the function:
is a convex function of the variational parameters 
.
We note first that affine transformations do not change convexity
properties. Thus convexity in 
implies convexity in the variational parameters 
. It remains
to show that
is a convex function of the vector 
;
here we have indicated the discrete values in the range of the random
variable X by 
and denoted the probability measure on
such values by 
. Taking the gradient of f with respect to 
gives:
where 
defines a probability distribution. The convexity is
revealed by a positive semi-definite Hessian 
, whose
components in this case are
To see that is positive semi-definite, consider
where 
is the variance of a discrete random variable Z
which takes the values 
with probability 
.