Expectation maximization algorithms for MAP estimation of jump Markov linear systems

In a jump Markov linear system, the state matrix, observation matrix, and the noise covariance matrices evolve according to the realization of a finite state Markov chain. Given a realization of the observation process, the aim is to estimate the state of the Markov chain assuming known model parameters. Computing conditional mean estimates is infeasible as it involves a cost that grows exponentially with the number of observations. We present three expectation maximization (EM) algorithms for state estimation to compute maximum a posteriori (MAP) state sequence estimates which are also known as Bayesian maximum likelihood state sequence estimates (MLSEs). The first EM algorithm yields the MAP estimate for the entire sequence of the finite state Markov chain. The second EM algorithm yields the MAP estimate of the (continuous) state of the jump linear system. The third EM algorithm computes the joint MAP estimate of the finite and continuous states. The three EM algorithms optimally combine a hidden Markov model (HMM) estimator and a Kalman smoother (KS) in three different ways to compute the desired MAP state sequence estimates. Unlike the conditional mean state estimates, which require computational cost exponential in the data length, the proposed iterative schemes are linear in the data length.