Inference of the Transition Matrix in Convolved Hidden Markov Models and the Generalized Baum–Welch Algorithm

A convolved two-level hidden Markov model is defined, with convolved observations as the top layer, continuous responses as the hidden middle layer, and categorical variables following a Markov chain as the hidden bottom layer. The model parameters include the transition probabilities of the Markov chain, response means and variances, and the convolution kernel and error variance. The focus of this study is on categorical deconvolution of the observations into the hidden bottom categorical layer when the transition probability parameters defining the Markov chain transition matrix are unknown. The inversion is cast in a Bayesian setting where the solution is represented by an approximate posterior model. Three algorithms, all of them based on the approximate model, for inference on the unknown parameters are defined and discussed. The inference techniques are compared in two empirical studies and in a seismic case study from a petroleum reservoir offshore Norway. We conclude that an approximate expectation-maximization algorithm, appearing as a generalized Baum–Welch algorithm, is preferable if point parameter estimates and marginal maximum a posterior class predictions are required. If uncertainty quantifications are also required, a more computationally demanding Bayesian Markov chain Monte Carlo algorithm must be used.