Kernel Bayesian Filtering

• Kernel Bayesian filtering is a nonparametric method that uses RKHS embeddings where probability distributions are represented as kernel means, enabling algebraic Bayesian updates.
• It facilitates closed-form posterior, prediction, and expectation computations without density estimation or Monte Carlo integration, handling both linear and nonlinear systems.
• The framework extends classical filtering by incorporating operator theory, regularization, and scalable computational strategies with proven consistency and convergence guarantees.

Kernel Bayesian filtering is a nonparametric framework for sequential Bayesian inference in dynamical systems that leverages the power of reproducing kernel Hilbert space (RKHS) embeddings of probability distributions. By representing all relevant distributions—prior, transition, likelihood, and posterior—via their canonical kernel means and covariance operators, the filtering problem is reformulated into algebraic operations in an RKHS. This approach subsumes conventional and many recent nonparametric state-space methods, enabling closed-form Bayesian updates, prediction, and expectation computation without explicit density estimation, Monte Carlo integration, or parametric assumptions. The methodology comprises foundational operator theory, consistent empirical estimation, and computational strategies that span Gaussian and non-Gaussian, linear and nonlinear, low- and high-dimensional settings.

1. Foundations: Kernel Mean Embeddings and Covariance Operators

The core abstraction is the kernel mean embedding of a probability law $P$ on a measurable space $X$ into an RKHS $\mathcal H_X$ with kernel $k_X$, defined by

$$\mu_P := \mathbb E_{X\sim P}[\,k_X(\cdot, X)\,] \in \mathcal H_X$$

for all bounded, positive-definite $k_X$. The reproducing property ensures that for all $f\in \mathcal H_X$, $\langle f, \mu_P\rangle = \mathbb E_P[f(X)]$. Characteristic kernels yield injective mean embeddings, so protein family distributions can be uniquely encoded by their kernel means.

Given a joint distribution $P_{XY}$ on $(X, Y)$ with associated kernels $k_X$, $k_Y$ and RKHSs $\mathcal H_X$, $\mathcal H_Y$, the (uncentered) covariance operators are

$$C_{XX}: \mathcal H_X \to \mathcal H_X,\,\, \langle f, C_{XX}g\rangle = \mathbb E[f(X)g(X)]$$

$$C_{YX}: \mathcal H_X \to \mathcal H_Y,\,\, \langle h, C_{YX}f\rangle = \mathbb E[f(X)h(Y)]$$

These operators form the backbone for representing conditional expectations and for generalizing Bayes' rule to the RKHS setting (Fukumizu et al., 2010).

2. Kernel Bayes' Rule: Operator and Empirical Formulations

The kernel Bayes' rule (KBR) provides an RKHS-level analog of classical Bayes' rule, targeting the conditional mean embedding

$$\mu_{X|y} = \mathbb E[\,k_X(\cdot, X)\mid Y = y\,] \in \mathcal H_X$$

without requiring explicit density estimation. At the population level, the update is expressed as

$$\mu_{X|y} = C_{XY} (C_{YY} + \lambda I)^{-1} k_Y(\cdot, y)$$

Alternative Tikhonov regularizations, such as the "squared-operator" scheme,

$$\mu_{X|y} = \widehat C_{ZW} (\widehat C_{WW}^2 + \delta I)^{-1} \widehat C_{WW} k_Y(\cdot, y)$$

are used to ensure numerical stability in finite samples.

Given $n$ observed pairs $(X_i, Y_i)$ and a prior embedding as a weighted sum over $U_j$ with weights $\gamma_j$, empirical KBR constructs Gram matrices $G_X$, $G_Y$ and prior mean vector $m_\pi$. The coefficients for the posterior mean are computed as

$$\alpha = [(1/n)G_X + n^{-1/3} I]^{-1} m_\pi\,,\quad \Lambda = \mathrm{diag}(\alpha)$$

followed by

$$R_{X|Y} = \Lambda G_Y [( \Lambda G_Y )^2 + \delta I ]^{-1} \Lambda,\qquad \rho = R_{X|Y} k_Y(y)$$

resulting in

$$\widehat\mu_{X|y} = \sum_{i=1}^n \rho_i k_X(\cdot, X_i)$$

This facilitates all posterior computations as linear algebra on kernel matrices (Fukumizu et al., 2010).

3. Recursive Kernel Filtering: State-Space Model and Update Steps

For a Markovian dynamical system $X_{t+1}\sim q(X_{t+1}| X_t)$, $Y_t \sim p(Y_t| X_t)$, kernel Bayesian filtering recursively tracks the mean embedding of the posterior at each time $t$, $\mu_{t|t} := \mu_{X_t|y_{1:t}} \in \mathcal H_X$, represented as a kernel expansion over training samples.

The two main recursions are:

• Prediction (Time-Update):

$$\mu_{t+1|t} = \widehat C_{X_{+1}, X}( \widehat C_{XX} + \lambda I )^{-1} \mu_{t|t}$$

In coefficient form:

$$\alpha^{(t+1|t)} = [ G_X + T\lambda I ]^{-1} G_{XX+1} \alpha^{(t)}$$

• Correction (Measurement-Update):

$\alpha^{(t+1)} = R_{X|Y}^{(t+1)} k_Y(y_{t+1})$

with $R_{X|Y}^{(t+1)}$ constructed analogously to the batch update but using the predicted coefficients as input weights.

This structure parallels the classical Kalman filter while operating entirely in the data-defined RKHS (Fukumizu et al., 2010).

4. Consistency, Rates, and Theoretical Guarantees

Kernel Bayesian filtering enjoys consistency guarantees under sufficient smoothness conditions. If the prior $\pi/p_X$ lies in the range of $C_{XX}^\beta$, and expected conditional expectations belong to the range of $C_{WW}^\nu$, and regularization is appropriately decayed with sample size, then for any fixed $y$,

$$| \langle f, \widehat\mu_{X|y} \rangle - \mathbb E[ f(X)\mid Y=y ] | = O_p(n^{-\rho})$$

for some $\rho > 0$. The precise rate is controlled by the regularity parameters $\beta, \nu$, and the rate of prior mean estimation. Stronger average-case (in $L^2(Q_Y)$) convergence bounds are also established (Fukumizu et al., 2010).

5. Practical Considerations: Kernel Choices, Regularization, and Efficient Computation

• Kernel Selection: Gaussian RBF kernels are universal and characteristic. Bandwidths are typically set using the median heuristic or cross-validation.
• Regularization: Both the main operator inversions ($\lambda$) and the squared-operator in the measurement update ($\delta$) require tuning to balance bias and numerical stability. Cross-validation on marginal consistency or predictive performance is standard.
• Computation: Naive Gram matrix inversions are $O(n^3)$ per step. Common accelerations include incomplete Cholesky or Nyström low-rank approximations, reducing cost to $O(n r^2)$. Storage is dominated by the leading $r$ eigenpairs or factors.
• Storage: Only the $n$-dimensional expansion coefficients and the associated kernel evaluations need to be retained at each step.

These considerations enable kernel Bayesian filtering to scale, with the caveat that $n$ is limited by available computational resources (Fukumizu et al., 2010).

6. Illustrative Applications and Performance

• Likelihood-free Bayesian Computation: KBR has been demonstrated for posterior inference where only simulations are possible, outperforming likelihood-free methods such as rejection ABC in terms of mean-squared error versus computational cost.
• Nonparametric State-Space Filtering: In nonlinear oscillator models and higher-dimensional SO$(3)$ camera rotation tracking ($1200$-dimensional RGB), kernel Bayesian filtering achieves convergence in MSE to ground truth and outperforms extended and unscented Kalman filters once sufficient training data are available. Notably, in highly nonlinear or non-Gaussian cases where parametric methods mis-specify the model, kernel methods retain robustness (Fukumizu et al., 2010).

The essential insight is that all discrete distributions are represented in RKHS via canonical kernel means, and Bayesian updates are linear-algebraic manipulations in feature space, bypassing the need for explicit density estimation or Monte Carlo integration.

7. Extensions, Variants, and Ongoing Developments

The original KBR framework has catalyzed several subsequent advances:

• Posterior Regularization: Embedding regularization at the posterior distribution level improves stability and convergence in challenging nonlinear state-space models. Thresholding strategies and direct regression formulations have been proposed with established consistency and sum-to-one guarantees (Song et al., 2016).
• Importance Weighting: Reformulating kernel Bayes' rule as a two-stage importance weighting estimator results in positive definite operator estimates and improved numerical stability. This approach shows uniform empirical performance gains, especially in high-dimensional filtering with complex observations (Xu et al., 2022).
• Adaptive Kalman-Like Filtering in RKHS: The adaptive kernel Kalman filter (AKKF) combines Kalman-style covariance updates with kernel mean embeddings of both particles and empirical distributions. Substantially fewer particles (tens instead of thousands) suffice compared to particle or unscented Kalman filters, bypassing particle degeneracy (Sun et al., 2022).
• Operator-theoretic Perspectives: Lifting nonlinear dynamics or measurements into RKHS enables direct application of Kalman filtering or minimimum variance estimation to the Koopman operator, further broadening the class of nonlinear, nonparametric models tractable by kernel Bayesian filtering (Li et al., 2024, Li et al., 2019).

These developments continue to broaden the reach, scalability, and robustness of kernel-based Bayesian filtering in both theory and application.