Calculation of variance

 

Probabilistic inference uses variance V_p[x_q] = E_p[x_q²] - (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] = min_p V_p[x_q]

V[x_q] = max_p V_p[x_q]

Calculation of bounds for variances in a Quasi-Bayesian network is a great challenge because the expression for V_p[x_q] 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[x_q] = min_mu ( E[(x_q - mu)²] ) and V[x_q] = min_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.