Summary: We compute new state and mode estimation algorithms for discrete-time Gauss--Markov models whose parameter sets switch according to a known Markov law. An important feature of our algorithms is that they are based upon the exact filter dynamics computed in [\textit{R. J. Elliott}, \textit{F. Dufour} and \textit{D. Sworder}, IEEE Trans. Autom. Control 41, 1807--1810 (1996; Zbl 0873.93077)]. The fundamental and well-known obstacle in estimation of jump Markov systems is managing the geometrically growing history of candidate hypotheses. In our scheme, we address this issue by proposing an extension of an idea due to Viterbi. Our scheme maintains a fixed number of candidate paths in a history, each identified by an optimal subset of estimated mode probabilities. We compute finite-dimensional suboptimal filters and smoothers, which estimate the hidden state process and the mode probability. Our smoothers are based upon a duality between forward and backward dynamics. Further, our smoothing algorithms are general and can be configured into the standard forms of fixed point, fixed lag, and fixed interval smoothers. A computer simulation is included to demonstrate performance.