JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 6, 1.52-154 (1962) On the Stochastic Approximation Method and Optimal Filtering Theory Yu CHI Ho* Harvard University, Cambridge, Massachusetts and The RAND Corporation, Santa Monica, California Submitted by Richard Bellman In this short paper we wish to establish some connections among the maxi- mum likelihood estimate, the optimal filtering, and the stochastic approxima- tion solutions to the following well-known problem: Consider the vector- matrix equations Ax + vk = bk K = 1, 2, a** (1) where A is a given Y x n matrix x is an unknown n-vector and ok is a random r-vector with E(Q) = 0 E(v&) = 16(K - j) b, is a r-vector of observation One wishes to determine an estimate x: for the unknown parameters x which is optimal in some sense. First, we shall make a maximum likelihood estimate for X. It is well- known [l, 21 that a recursive method for calculating the estimate in the case of gaussian noise is where %+, = 4, + P,+d’@,+, - Af,) Li$ = 0 P& = P;’ + A’A P, = given n x n positive definite matrix (2) (3) * Any views expressed in this paper are,those of the author. They should not be interpreted as reflecting the views of The RAND Corporation or the official opinion or policy of any of its governmental or private research sponsors. Papers are repro- duced by The RAND Corporation as a courtesy to members of its staff. 152