A nonparametric kernel-based method for realizing Bayes' rule is proposed, based on kernel representations of probabilities in reproducing kernel Hilbert spaces. The prior and conditional probabilities are expressed as empirical kernel mean and covariance operators, respectively, and the kernel mean of the posterior distribution is computed in the form of a weighted sample. The kernel Bayes' rule can be applied to a wide variety of Bayesian inference problems: we demonstrate Bayesian computation without likelihood, and filtering with a nonparametric statespace model. A consistency rate for the posterior estimate is established.

A nonparametric kernel-based method for realizing Bayes' rule is proposed, based on kernel representations of probabilities in reproducing kernel Hilbert spaces. The prior and conditional probabilities are expressed as empirical kernel mean and covariance operators, respectively, and the kernel mean of the posterior distribution is computed in the form of a weighted sample. The kernel Bayes' rule can be applied to a wide variety of Bayesian inference problems: we demonstrate Bayesian computation without likelihood, and filtering with a nonparametric state-space model. A consistency rate for the posterior estimate is established.

We study a nonparametric approach to Bayesian computation via feature means, where the expectation of prior features is updated to yield expected posterior features, based on regression from kernel or neural net features of the observations. All quantities involved in the Bayesian update are learned from observed data, making the method entirely model-free. The resulting algorithm is a novel instance of a kernel Bayes' rule (KBR). Our approach is based on importance weighting, which results in superior numerical stability to the existing approach to KBR, which requires operator inversion. We show the convergence of the estimator using a novel consistency analysis on the importance weighting estimator in the infinity norm. We evaluate our KBR on challenging synthetic benchmarks, including a filtering problem with a state-space model involving high dimensional image observations. The proposed method yields uniformly better empirical performance than the existing KBR, and competitive performance with other competing methods.

A nonparametric kernel-based method for realizing Bayes' rule is proposed, based on kernel representations of probabilities in reproducing kernel Hilbert spaces. The prior and conditional probabilities are expressed as empirical kernel mean and covariance operators, respectively, and the kernel mean of the posterior distribution is computed in the form of a weighted sample. The kernel Bayes' rule can be applied to a wide variety of Bayesian inference problems: we demonstrate Bayesian computation without likelihood, and filtering with a nonparametric state-space model. A consistency rate for the posterior estimate is established. Papers published at the Neural Information Processing Systems Conference.

Nonparametric inference techniques provide promising tools for probabilistic reasoning in high-dimensional nonlinear systems.Most of these techniques embed distributions into reproducing kernel Hilbert spaces (RKHS) and rely on the kernel Bayes' rule (KBR) to manipulate the embeddings. However, the computational demands of the KBR scale poorly with the number of samples and the KBR often suffers from numerical instabilities. In this paper, we present the kernel Kalman rule (KKR) as an alternative to the KBR.The derivation of the KKR is based on recursive least squares, inspired by the derivation of the Kalman innovation update.We apply the KKR to filtering tasks where we use RKHS embeddings to represent the belief state, resulting in the kernel Kalman filter (KKF).We show on a nonlinear state estimation task with high dimensional observations that our approach provides a significantly improved estimation accuracy while the computational demands are significantly decreased.

This paper addresses the problem of filtering with a state-space model. Standard approaches for filtering assume that a probabilistic model for observations (i.e. the observation model) is given explicitly or at least parametrically. We consider a setting where this assumption is not satisfied; we assume that the knowledge of the observation model is only provided by examples of state-observation pairs. This setting is important and appears when state variables are defined as quantities that are very different from the observations. We propose Kernel Monte Carlo Filter, a novel filtering method that is focused on this setting. Our approach is based on the framework of kernel mean embeddings, which enables nonparametric posterior inference using the state-observation examples. The proposed method represents state distributions as weighted samples, propagates these samples by sampling, estimates the state posteriors by Kernel Bayes' Rule, and resamples by Kernel Herding. In particular, the sampling and resampling procedures are novel in being expressed using kernel mean embeddings, so we theoretically analyze their behaviors. We reveal the following properties, which are similar to those of corresponding procedures in particle methods: (1) the performance of sampling can degrade if the effective sample size of a weighted sample is small; (2) resampling improves the sampling performance by increasing the effective sample size. We first demonstrate these theoretical findings by synthetic experiments. Then we show the effectiveness of the proposed filter by artificial and real data experiments, which include vision-based mobile robot localization.

Kernel Bayes' rule has been proposed as a nonparametric kernel-based method to realize Bayesian inference in reproducing kernel Hilbert spaces. However, we demonstrate both theoretically and experimentally that the prediction result by kernel Bayes' rule is in some cases unnatural. We consider that this phenomenon is in part due to the fact that the assumptions in kernel Bayes' rule do not hold in general.

Recent advances of kernel methods have yielded a framework for representing probabilities using a reproducing kernel Hilbert space, called kernel embedding of distributions. In this paper, we propose a Monte Carlo filtering algorithm based on kernel embeddings. The proposed method is applied to state-space models where sampling from the transition model is possible, while the observation model is to be learned from training samples without assuming a parametric model. As a theoretical basis of the proposed method, we prove consistency of the Monte Carlo method combined with kernel embeddings. Experimental results on synthetic models and real vision-based robot localization confirm the effectiveness of the proposed approach.

A nonparametric kernel-based method for realizing Bayes' rule is proposed, based on kernel representations of probabilities in reproducing kernel Hilbert spaces. The prior and conditional probabilities are expressed as empirical kernel mean and covariance operators, respectively, and the kernel mean of the posterior distribution is computed in the form of a weighted sample. The kernel Bayes' rule can be applied to a wide variety of Bayesian inference problems: we demonstrate Bayesian computation without likelihood, and filtering with a nonparametric state-space model. A consistency rate for the posterior estimate is established.