Minimum mean square error estimation for a class of chaotic systems

We develop the Bayesian maximum a posteriori (MAP) and minimum mean square error (MMSE) estimators for sequences generated by a class of discrete-time chaotic systems. This class (of which the sawtooth map, an archetype of chaotic behavior, is a member) consists of maps of the unit interval whose dynamics are precisely equivalent to the output of a first order, anticausal filter excited by independent, identically distributed Bernoulli noise. This linear formulation reduces the MAP estimation procedure to an M-ary hypothesis testing problem and simplifies the computation of the conditional expectation that represents the MMSE estimate. Estimator performance is evaluated on two systems from the class and compared with the Bayesian linear minimum mean square error estimator.