Kernel embedding of maps for sequential Bayesian inference: The variational mapping particle filter
This work addresses the challenge of high-dimensional Bayesian inference for researchers in fields like atmospheric dynamics and epidemiology, though it appears incremental as it builds on existing particle filter methods with a new mapping approach.
The authors tackled the problem of efficient sampling in high-dimensional state spaces with limited particles by introducing a novel sequential Monte Carlo filter that uses kernel-embedded mappings to minimize Kullback-Leibler divergence, achieving results where the number of effective particles remains close to the total number without requiring resampling in experiments on chaotic and epidemic models.
In this work, a novel sequential Monte Carlo filter is introduced which aims at efficient sampling of high-dimensional state spaces with a limited number of particles. Particles are pushed forward from the prior to the posterior density using a sequence of mappings that minimizes the Kullback-Leibler divergence between the posterior and the sequence of intermediate densities. The sequence of mappings represents a gradient flow. A key ingredient of the mappings is that they are embedded in a reproducing kernel Hilbert space, which allows for a practical and efficient algorithm. The embedding provides a direct means to calculate the gradient of the Kullback-Leibler divergence leading to quick convergence using well-known gradient-based stochastic optimization algorithms. Evaluation of the method is conducted in the chaotic Lorenz-63 system, the Lorenz-96 system, which is a coarse prototype of atmospheric dynamics, and an epidemic model that describes cholera dynamics. No resampling is required in the mapping particle filter even for long recursive sequences. The number of effective particles remains close to the total number of particles in all the experiments.