General Theory of Optimality

Abstract

In this chapter the only requirements made of the loss function L(x^T, z^T, u^T−1) are that it be measurable, non-negative, and possibly infinite. The principal idea exploited here is that of the minimal conditional loss functional $\hat F_t \left( {z^t ,\left\{ u \right\}} \right)$. For {u} fixed $\hat F_t$, is a measurable function of z^t, and for z^t fixed it depends on the truncated law {u^t−1} — hence the term “functional.” In the usual treatment of the dynamic programming problem (Blackwell [2], Hinderer [8], etc) the minimal conditional loss function depends only on the value of the truncated law u^t−1(z^t−1) at the observation point z and not on the entire law {u^t−l}. While the complexity of the functional $\hat F_t \left( {z^t ,\left\{ u \right\}} \right)$ over the function $\hat F_t \left( {z^t ,u^{t - 1} } \right)$ may be a disadvantage, it appears to be outweighed by the “natural” properties of $\hat F_t$,. Section 4.4, in which a comparison of the two criteria is made is not required for the development of the argument and is included primarily to show the relation of the present approach to that of the literature of dynamic programming.