Mutual Kolmogorov-Sinai entropy approach to nonlinear estimation

For a general nonlinear estimation problem, the authors develop an upper bound on the correlation coefficient in terms of the mutual Komogorov-Sinai entropy. This upper bound may be reached by means of a nonlinear transformation such that, after transformation, the processes are jointly Gaussian. Furthermore, to minimize the minimum mean-square estimation (MMSE) error, an approach is used based on the calculus of variations, to find the vector nonlinear functions whose elements turn out to be the eigenfunctions of two vector integral operators that can be concurrently solved from two vector integral equations. The relationship between the minimum mean-square estimation error and the mutual Kolmogorov-Sinai entropy is discussed. It is shown that the mutual Kolmogorov-Sinai entropy rate being equal to 0.5 is an important threshold in MMSE.<>