Approximate forward–backward algorithm for a switching linear Gaussian model

A hidden Markov model with two hidden layers is considered. The bottom layer is a Markov chain and given this the variables in the second hidden layer are assumed conditionally independent and Gaussian distributed. The observation process is Gaussian with mean values that are linear functions of the second hidden layer. The forward–backward algorithm is not directly feasible for this model as the recursions result in a mixture of Gaussian densities where the number of terms grows exponentially with the length of the Markov chain. By dropping the less important Gaussian terms an approximate forward–backward algorithm is defined. Thereby one gets a computationally feasible algorithm that generates samples from an approximation to the conditional distribution of the unobserved layers given the data. The approximate algorithm is also used as a proposal distribution in a Metropolis–Hastings setting, and this gives high acceptance rates and good convergence and mixing properties. The model considered is related to what is known as switching linear dynamical systems. The proposed algorithm can in principle also be used for these models and the potential use of the algorithm is therefore large. In simulation examples the algorithm is used for the problem of seismic inversion. The simulations demonstrate the effectiveness and quality of the proposed approximate algorithm.