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### Coupling techniques for nonlinear ensemble filtering

We consider filtering in high-dimensional non-Gaussian state-space models with intractable transition kernels, nonlinear and possibly chaotic dynamics, and sparse observations in space and time. We propose a novel filtering methodology that harnesses transportation of measures, convex optimization, and ideas from probabilistic graphical models to yield robust ensemble approximations of the filtering distribution in high dimensions. Our approach can be understood as the natural generalization of the ensemble Kalman filter (EnKF) to nonlinear updates, using stochastic or deterministic couplings. The use of nonlinear updates can reduce the intrinsic bias of the EnKF at a marginal increase in computational cost. We avoid any form of importance sampling and introduce non-Gaussian localization approaches for dimension scalability. Our framework achieves state-of-the-art tracking performance on challenging configurations of the Lorenz-96 model in the chaotic regime.

### Online Natural Gradient as a Kalman Filter

We establish a full relationship between Kalman filtering and Amari's natural gradient in statistical learning. Namely, using an online natural gradient descent on data log-likelihood to evaluate the parameter of a probabilistic model from a series of observations, is exactly equivalent to using an extended Kalman filter to estimate the parameter (assumed to have constant dynamics). In the i.i.d. case, this relation is a consequence of the "information filter" phrasing of the extended Kalman filter. In the recurrent (state space, non-i.i.d.) case, we prove that the joint Kalman filter over states and parameters is a natural gradient on top of real-time recurrent learning (RTRL), a classical algorithm to train recurrent models. This exact algebraic correspondence provides relevant settings for natural gradient hyperparameters such as learning rates or initialization and regularization of the Fisher information matrix.

### The Kernel Kalman Rule — Efficient Nonparametric Inference with Recursive Least Squares

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.

### Generalized Bayesian Filtering via Sequential Monte Carlo

We introduce a framework for inference in general state-space hidden Markov models (HMMs) under likelihood misspecification. In particular, we leverage the loss-theoretic perspective of generalized Bayesian inference (GBI) to define generalized filtering recursions in HMMs, that can tackle the problem of inference under model misspecification. In doing so, we arrive at principled procedures for robust inference against observation contamination through the $\beta$-divergence. Operationalizing the proposed framework is made possible via sequential Monte Carlo methods (SMC). The standard particle methods, and their associated convergence results, are readily generalized to the new setting. We demonstrate our approach to object tracking and Gaussian process regression problems, and observe improved performance over standard filtering algorithms.

### Kernel Bayes' Rule

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.