On the Complexity of Computing the Minimum Mean Square Error of Causal Prediction

This paper gives a complete characterization of the complexity of computing the minimum mean square pre-diction error for wide-sense stationary stochastic processes. It shows that if the spectral density of the stationary process is a strictly positive, computable continuous function then the minimum mean square error (MMSE) is always a computable number. It is also shown that the computation of the MMSE is a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\neq P_{1}$</tex> complete problem on the set of strictly positive, polynomial-time computable, continuous spectral densities. This means that if, as widely assumed, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$FP_{1}\neq\neq P_{1}$</tex>, then there exist strictly positive, polynomial-time computable continuous spectral densities for which the computation of the MMSE is not polynomial-time computable. So under the widely accepted assumptions of complexity theory, the computation of the MMSE is generally much harder than <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$NP_{1}$</tex> complete problems.