Connections between sequential Bayesian inference and evolutionary dynamics

TL;DR

This work establishes a rigorous bridge between continuous-time sequential Bayesian inference and evolutionary dynamics by linking the Kushner-Stratonovich filtering PDE to the Crow-Kimura/replicator-mutator framework. It introduces a time-varying, piecewise-smooth observation construction that yields a generalized Zakai equation in the limit, and it reveals a gradient-flow interpretation of the replicator dynamics under the Fisher-Rao metric. In the linear-Gaussian regime, the study connects nonlocal fitness with covariance inflation mechanisms in ensemble Kalman-Bucy filters, and it analyzes misspecified-model filtering to identify optimal (r,s) parameter settings that minimize time-asymptotic MSE while preserving realistic uncertainty through covariance calculations. The results generalize classical filtering (Zakai/Kushner-Stratonovich) to a nonlocal, mutation-inclusive dynamics with practical implications for designing robust sampling and filtering algorithms under model misspecification. Collectively, the paper advances both theoretical understanding and algorithmic possibilities at the intersection of evolutionary biology and Bayesian inference.

Abstract

It has long been posited that there is a connection between the dynamical equations describing evolutionary processes in biology and sequential Bayesian learning methods. This manuscript describes new research in which this precise connection is rigorously established in the continuous time setting. Here we focus on a partial differential equation known as the Kushner-Stratonovich equation describing the evolution of the posterior density in time. Of particular importance is a piecewise smooth approximation of the observation path from which the discrete time filtering equations, which are shown to converge to a Stratonovich interpretation of the Kushner-Stratonovich equation. This smooth formulation will then be used to draw precise connections between nonlinear stochastic filtering and replicator-mutator dynamics. Additionally, gradient flow formulations will be investigated as well as a form of replicator-mutator dynamics which is shown to be beneficial for the misspecified model filtering problem. It is hoped this work will spur further research into exchanges between sequential learning and evolutionary biology and to inspire new algorithms in filtering and sampling.

Figures (5)

• Figure 3.1: Snapshot in time for the above filtering problem with $H=2, m_0=0, P_0 = 0.3, x_0^\ast = 5, \Xi = 1$ and the smooth observations are constructed with $\delta_d = 500 \Delta t$. Clearly the Ito interpretation (pink line) is not the correct limit for the Crow-Kimura with smooth approximation. The correct Stratonovich interpretation (black line) aligns closely with the Crow-Kimura with smooth obs (cyan line). Note that the Stratonovich interpretation \ref{['eq:zakaistratlingauss']} coincides with the familar Zakai equation from filtering \ref{['eq:zakai']}.
• Figure 4.1: Plot of asymptotic MSE $E_\infty$ for various s vs r values for system 2. The optimal (in terms of asymptotic mse) values are indicated by the red line, calculated using \ref{['eq:optsr']}. The colourbar shows corresponding values of the asymptotic MSE $E_\infty$. The dashed red line on the left plot shows the theoretical expression for $s^l$ as given in lemma \ref{['lem:optimalrs']}
• Figure 4.2: Plot of asymptotic MSE $E_\infty$ for various s vs r values for system 2. The optimal (in terms of asymptotic mse) values are indicated by the red line, calculated using \ref{['eq:optsr']}. The colourbar shows corresponding values of the asymptotic MSE $E_\infty$.
• Figure 4.3: Demonstration of more realistic/representative covariances that can be obtained with the non-local replicator mutator, i.e. with $s \neq 0$. Left plot indicates system 1, right plot indicates system 2. The red solid line indicates $C_\infty$ vs $r$ using \ref{['eq:asympCscalar']}. The blue horizontal dashed line indicates the analytic $E_\infty^{\text{opt}}$, i.e. the minimum asymptotic MSE obtained by using $r^{\text{opt}}, s^{\text{opt}}$ in \ref{['eq:asympmsescalar']}, the pink horizontal dashed line indicates $C_\infty$ for $r=1$ (i.e. $\hat{C}_\infty$ as defined in \ref{['eq:covperfect']} , the covariance in the perfect model case). Finally, the black dashed line indicates $C_\infty$ for $(s=0, r^\text{opt}_0)$, i.e. the optimal $r$ for multiplicative covariance inflation.
• Figure 4.4: Plot of optimal $(r,s)$ values for system 1 (left plot) and system 2 (right plot). The blue line indicates the $(r,s)$ pairs minimising mse only, obtained from \ref{['eq:optsr']} and the cyan line indicates the $(r,s)$ pairs such that $C_\infty = E_\infty$, obtained from \ref{['eq:rminuscond']}. The point of intersection of the two lines indicates the $(r,s)$ pair that achieves both. The red square indicates the optimal $r$ value corresponding to the multiplicative covariance inflation case $(s=0)$. The blue squares indicate the possible $r$ values corresponding to the optimal $s$ in terms of $E_\infty$. In system 1, multiplicative covariance inflation leads to overconfident estimates ($C_\infty$ too small), whereas in system 2, it is not so far off from the optimal choice of $s = -0.0135, r = 0.99$.

Theorems & Definitions (25)

• Theorem
• Lemma 2.1
• Lemma 3.1
• Example 3.2
• Theorem 3.1
• Lemma 4.1
• Lemma 4.4
• Lemma 4.5
• Lemma 4.6
• Lemma 4.7