Kernel embedded nonlinear observational mappings in the variational mapping particle filter

arXiv.org Machine Learning

Recently, some works have suggested methods to combine variational probabilistic inference with Monte Carlo sampling. One promising approach is via local optimal transport. In this approach, a gradient steepest descent method based on local optimal transport principles is formulated to transform deterministically point samples from an intermediate density to a posterior density. The local mappings that transform the intermediate densities are embedded in a reproducing kernel Hilbert space (RKHS). This variational mapping method requires the evaluation of the log-posterior density gradient and therefore the adjoint of the observational operator. In this work, we evaluate nonlinear observational mappings in the variational mapping method using two approximations that avoid the adjoint, an ensemble based approximation in which the gradient is approximated by the particle covariances in the state and observational spaces the so-called ensemble space and an RKHS approximation in which the observational mapping is embedded in an RKHS and the gradient is derived there. The approximations are evaluated for highly nonlinear observational operators and in a low-dimensional chaotic dynamical system. The RKHS approximation is shown to be highly successful and superior to the ensemble approximation.


High-dimensional Filtering using Nested Sequential Monte Carlo

arXiv.org Machine Learning

Sequential Monte Carlo (SMC) methods comprise one of the most successful approaches to approximate Bayesian filtering. However, SMC without good proposal distributions struggle in high dimensions. We propose nested sequential Monte Carlo (NSMC), a methodology that generalises the SMC framework by requiring only approximate, properly weighted, samples from the SMC proposal distribution, while still resulting in a correct SMC algorithm. This way we can exactly approximate the locally optimal proposal, and extend the class of models for which we can perform efficient inference using SMC. We show improved accuracy over other state-of-the-art methods on several spatio-temporal state space models.


Toward Practical N2 Monte Carlo: the Marginal Particle Filter

arXiv.org Machine Learning

Sequential Monte Carlo techniques are useful for state estimation in non-linear, non-Gaussian dynamic models. These methods allow us to approximate the joint posterior distribution using sequential importance sampling. In this framework, the dimension of the target distribution grows with each time step, thus it is necessary to introduce some resampling steps to ensure that the estimates provided by the algorithm have a reasonable variance. In many applications, we are only interested in the marginal filtering distribution which is defined on a space of fixed dimension. We present a Sequential Monte Carlo algorithm called the Marginal Particle Filter which operates directly on the marginal distribution, hence avoiding having to perform importance sampling on a space of growing dimension. Using this idea, we also derive an improved version of the auxiliary particle filter. We show theoretic and empirical results which demonstrate a reduction in variance over conventional particle filtering, and present techniques for reducing the cost of the marginal particle filter with N particles from O(N2) to O(N logN).


Sequential Monte Carlo Methods for System Identification

arXiv.org Machine Learning

One of the key challenges in identifying nonlinear and possibly non-Gaussian state space models (SSMs) is the intractability of estimating the system state. Sequential Monte Carlo (SMC) methods, such as the particle filter (introduced more than two decades ago), provide numerical solutions to the nonlinear state estimation problems arising in SSMs. When combined with additional identification techniques, these algorithms provide solid solutions to the nonlinear system identification problem. We describe two general strategies for creating such combinations and discuss why SMC is a natural tool for implementing these strategies.