[Paper Review] Kalman filter in quantum language

Shiro Ishikawa, Kohshi Kikuchi|arXiv (Cornell University)|Apr 10, 2014

Quantum Mechanics and Applications9 references3 citations

TL;DR

This paper proposes that the Kalman filter can be more intuitively understood through quantum language (quantum measurement theory), reformulating its recursive Bayesian estimation as a sequential measurement process in a $W^*$-algebra framework. By modeling state and observation dynamics using observables and conditional probabilities within a statistical measurement theory (SMT) formalism, the authors derive a Bayes-Kalman operator that unifies prediction, filtering, and smoothing under a single quantum-linguistic framework, demonstrating that the Kalman filter's logic becomes clearer in this formalism than in classical statistics.

ABSTRACT

Recently, we proposed measurement theory ( or. quantum language) as a linguistic turn of quantum mechanics (with the Copenhagen interpretation). This theory has a great power of scientific descriptions. In fact, we have continued asserting that even statistics can be described in terms of measurement theory. Thus, we believe that quantum language is future statistics (i.e., statistics will develop into quantum language). However, now we think that our arguments were too abstract and philosophical, that is, we should have presented concrete examples much more. Thus, in this paper, we show that the calculation of Kalman filter is more understandable in terms of quantum language than in terms of usual statistics. For this, we devote ourselves to statistical measurement theory, in which the Bertrand paradox is discussed.

Motivation & Objective

• To demonstrate that the Kalman filter's recursive estimation process is more naturally expressed in quantum language than in classical statistical frameworks.
• To bridge the gap between Bayesian statistics and quantum measurement theory by applying statistical measurement theory (SMT) to dynamic systems.
• To reformulate the Kalman filter as a sequential measurement process involving observables, states, and conditional probabilities in a $W^*$-algebraic structure.
• To show that the conceptual clarity of the Kalman filter improves when viewed through the lens of quantum language, especially in handling prediction, filtering, and smoothing.

Proposed method

• Formalizes the Kalman filter within classical statistical measurement theory (SMT) using a $W^*$-algebraic framework, where states are represented by probability densities and observables by projection-valued measures.
• Models the system state and measurement dynamics using Gaussian-like observables with parameters $a_t, b_t, r_t$ and $c_t, d_t, q_t$, respectively, to represent state transitions and observation processes.
• Derives the Bayes-Kalman operator as a conditional update rule that combines prior state density $\rho_s(\omega_s)$ with likelihood $\widetilde{f}_{x_s}(\omega_s)$ to yield a posterior density via normalization.
• Uses iterative propagation of densities through time, expressing the posterior at time $s$ as a product of the prior and the likelihood, normalized via $L^\infty$-norm, enabling recursive computation.
• Introduces a transformation of variables via $\widetilde{u}_t, \widetilde{v}_t$ and $u_{t-1}, v_{t-1}$ to simplify the convolution of Gaussians into a single Gaussian form, preserving the structure of the Kalman update.
• Applies the framework to three cases: prediction ($s > n$), filtering ($s = n$), and smoothing ($s < n$), unified under a single operator $B_{\widehat{\mathsf{O}}_{t_0}}^s$.

Experimental results

Research questions

• RQ1 Can the Kalman filter's recursive Bayesian update be more naturally interpreted using quantum language rather than classical statistical formulations?
• RQ2 How does the $W^*$-algebraic formulation of statistical measurement theory clarify the structure of Kalman filtering, prediction, and smoothing?
• RQ3 What is the role of the Bayes-Kalman operator in unifying prediction, filtering, and smoothing within a single measurement-theoretic framework?
• RQ4 How do the transformations $\widetilde{u}_t, \widetilde{v}_t$ and $u_{t-1}, v_{t-1}$ simplify the product of Gaussian densities in the Kalman update?

Key findings

• The Kalman filter's recursive update is reformulated as a sequential measurement process in a $W^*$-algebraic framework, where the posterior state is obtained via a normalized product of prior density and likelihood, expressed as $[B_{\widehat{\mathsf{O}}_{t_0}}^s(\times_{t\in T}\{x_t\})]\rho_0(\omega_s) = \frac{\widetilde{f}_{x_s}(\omega_s) \cdot \rho_s(\omega_s)}{\int \widetilde{f}_{x_s}(\omega_s) \cdot \rho_s(\omega_s) d\omega_s}$.
• The likelihood function $\widetilde{f}_{x_s}(\omega_s)$ is shown to be approximately Gaussian, with effective parameters $\widetilde{u}_{s}, \widetilde{v}_{s}$ derived from recursive transformations of the original system and observation parameters.
• The transformation $u_{t-1} = -\frac{a_t \widetilde{u}_t}{\sqrt{1 + r_t^2 \widetilde{u}_t^2}}$, $v_{t-1} = \frac{b_t \widetilde{u}_t - \widetilde{v}_t}{\sqrt{1 + r_t^2 \widetilde{u}_t^2}}$ enables the merging of state transition and observation noise into a single effective Gaussian likelihood.
• The final posterior density $\widetilde{f}_{x_s}(\omega_s)$ is derived as a product of two Gaussians: one from the prior state and one from the likelihood, resulting in a single Gaussian with updated precision and mean, consistent with the standard Kalman update.
• The framework unifies prediction, filtering, and smoothing under a single operator $B_{\widehat{\mathsf{O}}_{t_0}}^s$, with the case $s = n$ corresponding to filtering and $s < n$ to smoothing.
• The authors conclude that the quantum language formulation provides greater conceptual clarity for the Kalman filter than classical statistics, suggesting that future statistics will evolve into quantum language.