Switch-Reset Models : Exact and Approximate Inference

Reset models are constrained switching latent Markov models in which the dynamics either continues according to a standard model, or the latent variable is resampled. We consider exact marginal inference in this class of models and their extension, the switch-reset models. A further convenient class of conjugateexponential reset models is also discussed. For a length T time-series, exact filtering scales with T 2 and smoothing T . We discuss approximate filtering and smoothing routines that scale linearly with T . Applications are given to change-point models and reset linear dynamical systems. 1 LATENT MARKOV MODELS For a time-series of observations y1:T and latent variables x1:T , a latent Markov model defines a joint distribution p(y1:T , x1:T ) = T ∏ t=1 p(yt|xt)p(xt|xt−1) where x0 = ∅. Due to the Markov structure, fig(1a), marginal inference in these well known models can be carried out using the classical p(xt, y1:t) ≡ α(xt), p(yt+1:T |xt) ≡ β(xt), and p(xt|y1:T ) ≡ γ(xt) message passing recursions (Barber, 2011) α(xt+1) = p(yt+1|xt+1) ∫ xt p(xt+1|xt)α(xt) (1.1)