Probabilistic inference uses variance Vp[xq] = Ep[xq2] - (Ep[xq])2 of a variable xq, for a fixed probability distribution p(xq).
Lower and upper variances are defined as:
To produce a convergent algorithm for calculation of lower and upper
variances, we can use Walley's variance envelope theorem
[Walley1991, Theorem G2,], which demonstrates that
V[xq] = E[(xq - mu)2] ) .
The calculation of lower and upper variances becomes a unidimensional
optimization problem, which can be solved by discretizing mu
(note that mu must be larger than zero and smaller than the square
of the largest value of xq).
The computational burden of this procedure is very intense
since for each value of mu it is necessary to obtain the bounds
for expected value of u(xq) = (xq - mu)2.
Thu Jan 23 15:54:13 EST 1997