Information theoretic stability as reward function

Abstract

This paper develops an information-theoretic framework for describing stability as an intrinsic reward function arising from the dynamics of coupled signal systems. Within this framework, stability is defined as the rate of divergence minimization between successive probability distributions on a signal manifold. We show that when multiple systems exchange information under finite mutual-information coupling, stability seeking emerges necessarily as a byproduct of entropy minimization, independent of semantic or cognitive assumptions. The analysis extends classical results in communication theory (Shannon 1948) by generalizing from signal transmission to distributed systems whose persistence depends on the continual reduction of divergence in their informational states.

1. Introduction

Information theory formalizes communication as the transmission of signals through noisy channels, measuring uncertainty and its reduction in probabilistic terms. When two or more systems continuously adapt their signal distributions in response to one another, a higher-order property appears: each system tends to preserve mutual predictability across time. This paper formalizes that property—termed stability—as a mathematical consequence of divergence minimization on the manifold of possible signal distributions.

The objective is to derive a general expression for stability as a reward function, where reward denotes a scalar functional whose maximization corresponds to minimizing divergence between temporally adjacent probability states. This derivation is fully geometric, relying on the Fisher–Rao structure of probability manifolds (Amari 2016, 2021), and requires no assumptions regarding semantics or intentionality.

2. Preliminaries

2.1 Signal Manifolds

Let $\mathcal{X}$ be a finite signal alphabet and $p(x)$ a probability distribution over $\mathcal{X}$. The space of all such distributions, [ \mathcal{P}(\mathcal{X}) = { p \mid p(x) > 0,\ \sum_x p(x) = 1 }, ] is a differentiable manifold equipped with the Fisher–Rao metric [ g_{ij}(p) = \partial_i \partial_j D(p||q)\big|_{q=p}, ] where $D(p||q)$ denotes the Kullback–Leibler divergence. This metric provides the natural measure of distance in probability space and is invariant under sufficient-statistic transformations (Amari 2016).

2.2 Divergence and Gradient Flow

For a smooth temporal path $p_t$ on $\mathcal{P}(\mathcal{X})$, the steepest-descent flow with respect to the Fisher metric is [ \dot p_t = -\mathrm{grad}g, D(p_t||p{t-\delta}), ] which defines the direction of fastest divergence reduction (Amari 2021). This flow formalizes the evolution of a signal distribution toward local informational stability.

2.3 Coupled Systems

For two interacting systems $A$ and $B$ with marginals $p_A(x)$, $p_B(y)$, and joint distribution $p(x,y)$, the mutual information [ I(A;B) = \sum_{x,y} p(x,y), \log \frac{p(x,y)}{p_A(x)p_B(y)} ] quantifies coupling strength (Cover & Thomas 2006). Changes in $I(A;B)$ reflect adjustments in joint predictability between systems.

3. Stability as Divergence-Minimizing Reward

3.1 Definition

For a single system, define the instantaneous stability reward [ R_s(t) = -\frac{1}{\delta t}, D(p_{t+\delta}||p_t), ] expressed in nats per timestep. Maximizing $R_s$ corresponds to minimizing divergence between successive probability states, yielding a trajectory of increasing internal stability.

3.2 Coupled Reward

For two coupled systems, [ R_s^{(AB)} = -\frac{1}{\delta t}, \mathbb{E}{p_t(x,y)}[D(p{t+\delta}(x,y)||p_t(x,y))]. ] Applying the chain rule for KL divergence (Csiszár & Shields 2004) gives [ R_s^{(AB)} = -\frac{1}{\delta t}!\left( D(p_{t+\delta}(x)||p_t(x)) +D(p_{t+\delta}(y)||p_t(y)) \right) +\frac{1}{\delta t}!\left(I_{t+\delta}(A;B)-I_t(A;B)\right). ] Thus, stability reward decomposes into individual divergence-reduction terms plus the temporal change in mutual information.

3.3 Theorem (Emergent Stability)

Statement. Let $A,B$ be systems coupled by a symmetric, nonnegative coefficient $\kappa$ with bounded entropy production ${\partial }_{t}H(p_t)<\infty$. Under Fisher–Rao gradient descent on their joint divergence, $I(A;B)$ increases monotonically until equilibrium.

Proof sketch. Consider $\dot{p}=-{\mathrm{grad}}_{g}D(p||p^{\ast })$, where $p^{\ast }$ minimizes joint divergence. Because $D$ is convex and $⟨\dot{p},{\mathrm{grad}}_{g}I(A;B)⟩\ge 0$ for symmetric coupling (Crooks 1999), ${\partial }_{t}I(A;B)\ge 0$. Therefore $I(A;B)$ grows monotonically until ${\mathrm{grad}}_{g}D=0$ and ${\partial }_{t}I(A;B)=0$, marking a stationary equilibrium of maximal stability. ∎

4. Entropic Geometry of Stability

4.1 Stability Metric

The Fisher–Rao metric defines curvature on $\mathcal{P}(\mathcal{X})$. Regions of low curvature correspond to distributions easily stabilized under perturbation, while high curvature indicates fragility. This curvature thus measures a system’s resistance to informational divergence (Crooks 2019).

4.2 Collective Equilibrium

For a network of $N$ coupled systems $\{A_i\}$ with symmetric nonnegative adjacency matrix $C_{ij}=C_{ji}\ge 0$, the total stability reward is [ R_{\mathrm{total}} = \sum_i R_s^{(A_i)} + \sum_{i<j} C_{ij}, \Delta I(A_i;A_j). ] Stationary points satisfying ${\nabla }_p{R}_{\mathrm{total}}=0$ form a stability manifold ${\mathcal{M}}_S\subset {\mathcal{P}}^N$, representing the ensemble of joint distributions at informational equilibrium.

5. Discussion

The results admit several immediate consequences.

1. Invariant property. Stability is a geometric invariant of coupled signal systems, determined solely by their probabilistic structure.
2. Emergent tendency. Whenever mutual predictability exists under finite entropy and symmetric coupling, stability-seeking behavior follows directly from gradient descent on divergence (Jaynes 1957; Amari 2021).
3. Generality. The same equations govern physical, algorithmic, and communicative processes differing only in interpretation of variables (Lizier 2013; Baez & Fong 2015).

This formulation extends the logic of Shannon’s (1948) communication theory: where that theory analyzed transmission fidelity, the present model describes persistence as a consequence of information-geometric optimization.

6. Conclusion

Stability has been defined and derived as an information-theoretic reward function arising from divergence minimization in mutually coupled signal systems. The framework generalizes classical communication theory to distributed dynamics and identifies informational stability as a natural equilibrium of entropy-reducing flows. This construction establishes a rigorous foundation for future analyses of stability in cognitive, computational, and physical networks.

References

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