Theoretical Analysis of Resource-Induced Phase Transitions in Estimation Strategies

TL;DR

This result identifies the conditions under which resource limitations alter estimation strategies and analytically reveals the mechanism underlying the emergence of discontinuous, nonmonotonic, and scaling behaviors, which provide a theoretical foundation for understanding how limited resources shape information processing in biological systems.

Abstract

Organisms adapt to volatile environments by integrating sensory information with internal memory, yet their information processing is constrained by resource limitations. Such limitations can fundamentally alter optimal estimation strategies in biological systems. For example, recent experiments suggest that organisms exhibit nonmonotonic phase transitions between memoryless and memory-based estimation strategies depending on sensory reliability. However, an analytical understanding of these resource-induced phase transitions is still missing. This Letter presents an analytical characterization of resource-induced phase transitions in optimal estimation strategies. Our result identifies the conditions under which resource limitations alter estimation strategies and analytically reveals the mechanism underlying the emergence of discontinuous, nonmonotonic, and scaling behaviors. These results provide a theoretical foundation for understanding how limited resources shape information processing in biological systems.

Figures (5)

• Figure 1: (a) Environmental state $x_{t}$. (b) Noisy observation $y_{t}$. (c,e) Internal memory $z_{t}$. (d,f) Optimal estimator $\hat{x}_{t}^{*}$. Thin red and cyan curves are 100 sample trajectories, whereas thick red and magenta curves are their means. While memoryless estimation strategy ($\Pi_{zx},\Pi_{zz}=0$) is optimal at $Q=100$ (c,d), memory-based one ($\Pi_{zx},\Pi_{zz}\neq0$) is optimal at $Q=1000$ (e,f). The rest of the parameters are set to $F=1$ and $M=1$.
• Figure 2: (a,b) $\Pi_{zx}$ and $\Pi_{zz}$ as functions of $Q$ and $E$. Blue, green, and orange dots are the numerical solutions of the observation-based Riccati equation, and correspond to zero, intermediate, and high memory control gains, respectively. The blue and orange dots are optimal in the blue and orange regions, respectively, whereas the green dots never become optimal. (c) Phase boundaries with respect to $E$ and $Q/MF$. Blue, green, and orange curves vary $Q$, $M$, and $F$, respectively. The rest of the parameters are set to $1$.
• Figure 3: Discriminant $\Theta$ [Eq. (\ref{['eq: discriminant 1']})] and optimal memory control gain $\Phi_{zy}^{*}$ [Eq. (\ref{['eq: optimal Pzy 1']})] for $\Phi_{zz}=1$. $D$ is set to 1.
• Figure 4: (a,b,c) $J$ as functions of $\Phi_{zy}$ and $\Phi_{zz}$. Red and magenta curves represent $\partial J/\partial \Phi_{zy}=0$ and $\partial J/\partial \Phi_{zz}=0$, respectively, while cyan lines indicate $\Phi_{zz}=0.26$. (d) $\Phi_{zy}^{*}$ as a function of $Q$. Cyan curves are the intersections between $\partial J/\partial \Phi_{zy}=0$ and $\Phi_{zz}=0.26$, whereas magenta curves correspond to the intersections between $\partial J/\partial \Phi_{zy}=0$ and $\partial J/\partial \Phi_{zz}=0$. The rest of the parameters are set to 1.
• Figure 5: (a) Red, magenta, green, and orange curves indicate $\partial J/\partial \Phi_{zy}=0$, $\partial J/\partial \Phi_{zz}=0$, $\partial J_{Q}/\partial \Phi_{zz}=0$, and $\partial J_{M}/\partial \Phi_{zz}=0$, respectively. The rest of the parameters are set to 1. (b) Magenta, green, and orange curves are the parameter values at which $\partial J/\partial \Phi_{zz}=0$, $\partial J_{Q}/\partial \Phi_{zz}=0$, and $\partial J_{M}/\partial \Phi_{zz}=0$ intersect tangentially with $\partial J/\partial \Phi_{zy}=0$, respectively. The magenta curve ($\Theta_{T}=1$) is obtained numerically, whereas the green ($\Theta_{Q}=1$) and orange ($\Theta_{M}=1$) curves are obtained analytically from Eqs. (\ref{['eq: main-theta-q']}) and (\ref{['eq: main-theta-m']}), respectively. Blue, green, and orange regions correspond to $\Theta_{Q}<1$, $1\leq\Theta_{Q}$ and $\Theta_{M}<1$, and $1\leq\Theta_{M}$, respectively. The control gains $\Pi_{zx}$ and $\Pi_{zz}$ obtained numerically from the observation-based Riccati equation are zero at black dots and take nonzero values at red dots.