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# Calculation of variance

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:

V[xq] = minp Vp[xq]

V[xq] = maxp Vp[xq]

Calculation of bounds for variances in a Quasi-Bayesian network is a great challenge because the expression for Vp[xq] is quadratic on the probability values.

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] = minmu ( E[(xq - mu)2] ) and V[xq] = minmu ( 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.

© Fabio Cozman[Send Mail?]

Thu Jan 23 15:54:13 EST 1997