Particle ancestor sampling for near - degenerate or intractable state transition models ∗ Fredrik Lindsten , Pete Bunch , Sumeetpal S . Singh , and

We consider Bayesian inference in sequential latent variable models in general, and in nonlinear state space models in particular (i.e., state smoothing). We work with sequential Monte Carlo (SMC) algorithms, which provide a powerful inference framework for addressing this problem. However, for certain challenging and common model classes the state-of-the-art algorithms still struggle. The work is motivated in particular by two such model classes: (i) models where the state transition kernel is (nearly) degenerate, i.e. (nearly) concentrated on a low-dimensional manifold, and (ii) models where point-wise evaluation of the state transition density is intractable. Both types of models arise in many applications of interest, including tracking, epidemiology, and econometrics. The difficulties with these types of models is that they essentially rule out forward-backward-based methods, which are known to be of great practical importance, not least to construct computationally efficient particle Markov chain Monte Carlo (PMCMC) algorithms. To alleviate this, we propose a “particle rejuvenation” technique to enable the use of the forward-backward strategy for (nearly) degenerate models and, by extension, for intractable models. We derive the proposed method specifically within the context of PMCMC, but we emphasise that it is applicable to any forward-backward-based Monte Carlo method. 1 Problem formulation State space models (SSMs) are widely used for modelling time series and dynamical systems. A general, discrete-time SSM can be written as xt |xt−1 ∼ f(xt |xt−1), (1a) yt |xt ∼ g(yt |xt), (1b) ∗Supported by the projects Learning of complex dynamical systems (Contract number: 637-2014-466) and Probabilistic modeling of dynamical systems (Contract number: 621-20135524), both funded by the Swedish Research Council, and the project Bayesian Tracking and Reasoning over Time (Reference: EP/K020153/1), funded by the EPSRC. 1 ar X iv :1 50 5. 06 35 6v 1 [ st at .C O ] 2 3 M ay 2 01 5 where xt ∈ X is the latent state, yt ∈ Y is the observation (both at time t) and f(·) and g(·) are probability density functions (PDFs) encoding the state transition and the observation likelihood, respectively. The initial state is distributed according to x1 ∼ μ(x1). Statistical inference in SSMs typically involves computation of the smoothing distribution, that is, the posterior distribution of a sequence of state variables XT := (x1, . . . , xT ) ∈ X conditionally on a sequence of observations YT := (y1, . . . , yT ) ∈ Y . The smoothing distribution plays a key role both for offline (batch) state inference and for system identification via data augmentation methods, such as expectation maximisation [15] and Gibbs sampling [43]. The motivation for the present work comes from two particularly challenging classes of SSMs: (M1) If the state transition kernel f(·) of the system puts all probability mass on some low-dimensional manifold, we say that the transition is degenerate (for simplicity we use “probability density notation” even in the degenerate case). Degenerate transition kernels arise, e.g., if the dynamical evolution is modeled using additive process noise with a rank-deficient covariance matrix. Models of this type are common in certain application areas, e.g., navigation and tracking; see [26] for several examples. Likewise, if f(·) is concentrated around a lowdimensional manifold (i.e., the transition is highly informative, or the process noise is small) we say that the transition is nearly degenerate. (M2) If the state transition density function f(·) is not available on closed form, the transition is said to be intractable. The typical scenario is that f(·) is a regular (non-degenerate) PDF which it is possible to simulate from, but which nevertheless is intractable. At first, this scenario might seem contrived, but it is in fact quite common in practice. In particular, whenever the dynamical function is defined implicitly by some computer program or black-box simulator, or as a “complicated” nonlinear transformation of a known noise input, it is typically not possible to explicitly write down the corresponding transition PDF; see, e.g., [36, 3] for examples. The main difficulty in performing inference for model classes (M1) and (M2) lies in that the so-called backward kernel of the model (see, e.g., [30]), p(xt |xt+1, y1:t) ∝ f(xt+1 |xt)p(xt | y1:t), (2) will also be (nearly) degenerate or intractable, respectively. This is problematic since many state-of-the-art methods rely on the backward kernel for inference; see, e.g., [24, 31, 47]. In particular, the backward kernel is used to implement the well-known forward-backward smoothing strategy. We will come back to this in the subsequent sections when we discuss the details of these inference methods and how we propose an extension to the methodology geared toward this issue. The main contribution of this paper is constituted by a construction allowing us to replace the simulation from the (problematic) backward kernel, with