A review of sequential Monte Carlo methods for real-time disease modeling

We consider the problem of high-dimensional filtering of state-space models (SSMs) at discrete times. This problem is particularly challenging as analytical solutions are typically not available and many numerical approximation methods can have a cost that scales exponentially with the dimension of the hidden state. Inspired by lag-approximation methods for the smoothing problem, we introduce a lagged approximation of the smoothing distribution that is necessarily biased. For certain classes of SSMs, particularly those that forget the initial condition exponentially fast in time, the bias of our approximation is shown to be uniformly controlled in the dimension and exponentially small in time. We develop a sequential Monte Carlo (SMC) method to recursively estimate expectations with respect to our biased filtering distributions. Moreover, we prove for a class of class of SSMs that can contain dependencies amongst coordinates that as the dimension $d\rightarrow\infty$ the cost to achiev …

Latent variable time-series models are among the most heavily used tools from machine learning and applied statistics. These models have the advantage of learning latent structure both from noisy observations and from the temporal ordering in the data, where it is assumed that meaningful correlation structure exists across time. A few highly-structured models, such as the linear dynamical system with linear-Gaussian observations, have closed-form inference procedures (e.g. the Kalman Filter), but this case is an exception to the general rule that exact posterior inference in more complex generative models is intractable. Consequently, much work in time-series modeling focuses on approximate inference procedures for one particular class of models. Here, we extend recent developments in stochastic variational inference to develop a `black-box' approximate inference technique for latent variable models with latent dynamical structure. We propose a structured Gaussian variational approxima …

A new approach to inference in state space models is proposed, based on approximate Bayesian computation (ABC). ABC avoids evaluation of the likelihood function by matching observed summary statistics with statistics computed from data simulated from the true process; exact inference being feasible only if the statistics are sufficient. With finite sample sufficiency unattainable in the state space setting, we seek asymptotic sufficiency via the maximum likelihood estimator (MLE) of the parameters of an auxiliary model. We prove that this auxiliary model-based approach achieves Bayesian consistency, and that - in a precise limiting sense - the proximity to (asymptotic) sufficiency yielded by the MLE is replicated by the score. In multiple parameter settings a separate treatment of scalar parameters, based on integrated likelihood techniques, is advocated as a way of avoiding the curse of dimensionality. Some attention is given to a structure in which the state variable is driven by a c …

Filtering and smoothing algorithms for linear discrete- time state-space models with skewed and heavy-tailed measurement noise are presented. The algorithms use a variational Bayes approximation of the posterior distribution of models that have normal prior and skew-$t$-distributed measurement noise. The proposed filter and smoother are compared with conventional low- complexity alternatives in a simulated pseudorange positioning scenario. In the simulations the proposed methods achieve better accuracy than the alternative methods, the computational complexity of the filter being roughly 5 to 10 times that of the Kalman filter.

This paper presents a new filter for state–space models based on Bellman’s dynamic-programming principle, allowing for nonlinearity, non-Gaussianity and degeneracy in the observation and/or state-transition equations. The resulting Bellman filter is a direct generalisation of the (iterated and extended) Kalman filter, enabling scalability to higher dimensions while remaining computationally inexpensive. It can also be extended to enable smoothing. Under suitable conditions, the Bellman-filtered states are stable over time and contractive towards a region around the true state at every time step. Static (hyper)parameters are estimated by maximising a filter-implied pseudo log-likelihood decomposition. In univariate simulation studies, the Bellman filter performs on par with state-of-the-art simulation-based techniques at a fraction of the computational cost. In two empirical applications, involving up to 150 spatial dimensions or highly degenerate/nonlinear state dynamics, the Bellman f …