Suppose that an unknown random parameter theta with distribution function G is such that given theta, an observable random variable x has conditional probability density f(x / theta) of known form. If a function t = t(x) is used to estimate theta, then the expected squared error with respect to the random variation of both theta and x is: E(t-theta)/sup 2/ = .integral. .integral.(t(x)-theta)/sup 2/ f(x parallel theta)dx dG(theta). For fixed G we can seek to minimize this equation within any desired class of functions t, such as the class of all linear functions A + Bx, or the class of al Borel functions whatsoever.