Fig 10. Posteriors of switching state-space models.

(a) True posterior distribution. This is the resultant graph encoding the conditional independence relation after conditioning on the observed data {y_t} up to time T. The new edges between the hidden states make exact inference intractable. (b) Approximate posterior distribution. Compared to the true posterior, a structured approximation decouples the hidden Gaussian state-space models from each other and from the switching state. On this approximate distribution, efficient closed-form inference can be performed. The marginal distributions of the Gaussian hidden states {x_t} and the discrete-valued switching state {s_t} are now inter-dependent through variational summary statistics $g_t^{\left(m\right)}$ and $h_t^{\left(m\right)}$.