Probabilistic inference uses variance V_{p}[x_{q}] = E_{p}[x_{q}^{2}] - (E_{p}[x_{q}])^{2}
of a variable x_{q}, for a fixed probability distribution p(x_{q}).

Lower and upper variances are defined as:

__V__[x_{q}] = _{p} V_{p}[x_{q}]

~~V~~[x_{q}] = _{p} V_{p}[x_{q}]

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__[x_{q}] = _{mu} ( __E__[(x_{q} - mu)^{2}] )
and ~~V~~[x_{q}] = _{mu} ( ~~E~~[(x_{q} - 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 x_{q}).
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(x_{q}) = (x_{q} - mu)^{2}.

© Fabio Cozman[Send Mail?]

Thu Jan 23 15:54:13 EST 1997