The Nonparametric Kernel Bayes Smoother

Recently, significant progress has been made developing kernel mean expressions for Bayesian inference. An important success in this domain is the nonparametric kernel Bayes’ filter (nKB-filter), which can be used for sequential inference in state space models. We expand upon this work by introducing a smoothing algorithm, the nonparametric kernel Bayes’ smoother (nKB-smoother) which relies on kernel Bayesian inference through the kernel sum rule and kernel Bayes’ rule. We derive the smoothing equations, analyze the computational cost, and show smoothing consistency. We summarize the algorithm, which is simple to implement, requiring only matrix multiplications and the output of the nKB-filter. Finally, we report experimental results that compare the nKB-smoother to previous parametric and nonparametric approaches to Bayesian filtering and smoothing. In the supplementary materials, we show that the combination of the nKB-filter and the nKB-smoother allows marginal kernel mean computation, which gives an alternative to kernel belief propagation.