We can apply the method of section 5.3.2 approximate the
asymptotic mean of MCMC-UMS for doubly-bounded data.
In other words, we propose to model
the distribution of MCMC-UMS generated
data by (6.6).
We then maximize the entropy of this distribution
over the mean under the constraint that the mean
is on the manifold. More precisely,
we propose to maximize the entropy of (6.6)
subject to (5.12).
Unfortunately, the entropy is written in terms of
. But, it can be shown
that for a given , there is a unique
This is true in one or more dimensions. Therefore,
are alternative parameterizations for
the multivariate truncated exponential distribution.
Solving for asymptotic mean of MCMC-UMS
In the same manner as in Sections 4.4 and 5,
we use as the free variable
under the constraint (5.12).
So, to maximize (6.6), we need
the derivatives of (6.6)
with respect to the elements of .
Using the derivative chain-rule, we can write the first
derivative of (6.6) with respect to
where, from (6.6)
From (6.3) ,
And, from (5.10),
After (a lot of) cancellations, we get
resulting in the condition for maximization of entropy: