Optimal conditional estimation: Average case setting

We consider an estimation problem which appears in the identification of sy stems by means of restricted complexity models: find the optimal approximat ion to an element of a linear normed space (a system) based on noisy inform ation, subject to the restriction that approximations (models) can be selec ted from a prescribed subspace;M of the problem element space. In contrast to the worst-case optimization criterion, which may be pessimistic, in this paper the quality of an identification algorithm is measured by its local average performance. Two types of local average errors are considered: for a given information (measurement) y and for a given unknown element x, the latter in two versions. For a wide spectrum of norms in the measurement spa ce, we define an optimal algorithm and give expressions for its average err ors which show the dependence on information, information errors, unmodelle d dynamics, and norm in the measurement space.