Approximate forward-backward algorithm for a switching linear Gaussian model

Motivated by the application of seismic inversion in the petroleum industry we consider a hidden Markov model with two hidden layers. 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 assumed Gaussian with mean values that are linear functions of the second hidden layer. This model class, which we call switching linear Gaussian models, has clear similarities with what is known as switching linear dynamical systems and conditionally Gaussian state space models. The important difference is that we allow the observations to depend on both past and future values of the hidden Gaussian process. The forward-backward algorithms is not directly feasible for switching linear Gaussian models as the recursions then involve a mixture of Gaussian densities where the number of terms grows exponentially with the length of the Markov chain. We propose an approximate forward-backward algorithm by dropping the less important terms and thereby obtain a computationally feasible algorithm that generates samples from an approximation to the conditional distribution of the unobserved layers given the data. We also use this approximation as a proposal distribution in a Metropolis–Hastings setting and obtain high acceptance rates and good mixing properties. We demonstrate the effectiveness and quality of the approximation in simulation examples.