The Use of Approximate Prior Distributions in a Bayesian Decision Model - Kihlstrom, Richard E. (Econometrica, 1971)
[Consider a Bayesian decision problem in which F is the prior distribution over some parameter space T. If —ψ(d, t) is the product of the loss function and the likelihood function, then the Bayesian solution, d_F, maximizes extlesstex-math extgreater$E_\ F\ (d)= extbackslashint _\ T\ extbackslashpsi (d,t)dF(t)$ extless/tex-math extgreater. Suppose \ F^n\ is a sequence of distribution functions that approach F^0 in the sup-metric topology. Our main theorem gives conditions under which extlesstex-math extgreater$d_\ F extasciicircum\ n\ \ extbackslashrightarrow d_\ F extasciicircum\ 0\ \ $ extless/tex-math extgreater and extlesstex-math extgreater$E_\ F extasciicircum\ 0\ \ (d_\ F extasciicircum\ n\ \ ) extbackslashrightarrow E_\ F extasciicircum\ 0\ \ (d_\ F extasciicircum\ 0\ \ )$ extless/tex-math extgreater.]