codex
72 lines
4.8 KiB
TeX
72 lines
4.8 KiB
TeX
\documentclass[preprint,review,12pt]{elsarticle}
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\usepackage[utf8]{inputenc}
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\usepackage{amsmath,amssymb,amsfonts,amsthm}
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\usepackage{graphicx}
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\usepackage{hyperref}
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\newtheorem{theorem}{Theorem}
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\newtheorem{lemma}{Lemma}
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\newtheorem{definition}{Definition}
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\journal{Journal of Theoretical Biology}
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\begin{document}
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\begin{frontmatter}
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\title{Information Bottlenecks and Bounded Rational Decision Making: A Mathematical Proof of Fitness Beats Truth (Rapid Communication)}
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\author[1]{Antigravity}
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\address[1]{Institute for Advanced Cybernetic Physics}
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\begin{abstract}
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Evolutionary epistemology, particularly the "Fitness Beats Truth" (FBT) theorem, asserts that biological perception is tuned strictly to utility rather than objective reality. In this Letter, we provide a formal, rigorous mathematical proof of FBT using the framework of Bounded Rational Decision Making and the Information Bottleneck method. We define the objective world as a Riemannian manifold $\mathcal{M}$ endowed with a prior probability measure $\mu(x)$. By defining biological distortion directly as the expected utility loss under an optimal action policy, we formulate perception as a joint optimization over the perceptual encoder $p(y|x)$ and the actor policy $a(y)$ subject to a strict Shannon channel capacity bound $I(X;Y) \le C$. We mathematically prove that for generic fitness landscapes where the level sets of fitness do not align with the distance balls of the metric $g$, the optimal perceptual channel must actively destroy structural isomorphism to minimize the Lagrangian cost.
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\end{abstract}
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\begin{keyword}
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Evolutionary Game Theory \sep Information Bottleneck \sep Perception \sep Bounded Rationality
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\end{keyword}
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\end{frontmatter}
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\section{Introduction}
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Standard Rate-Distortion theory assumes an objective distortion metric $D(x,y)$ independent of the perceptual channel. However, biological perception is a decision-theoretic problem. The true biological cost of a perception depends entirely on the action $a(y)$ the organism subsequently takes. Thus, subjective inference directly defines the biological cost.
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\section{Formal Definitions and The Joint Optimization Model}
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\begin{definition}[State Space and Measure]
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Let $\mathcal{M}$ be a compact Riemannian manifold representing objective world states, endowed with metric $g$ and a prior probability measure $\mu(x)$ absolutely continuous with respect to the volume form. Let $\mathcal{Y}$ be a finite set of perceptual states. Let $\mathcal{A}$ be the space of actions.
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\end{definition}
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\begin{definition}[Fitness Landscape]
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Let $F: \mathcal{M} \times \mathcal{A} \to \mathbb{R}$ be a smooth fitness function mapping a world state and an action to a biological payoff.
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\end{definition}
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The organism possesses a bounded channel capacity $I(X;Y) \le C$. The optimal action policy maximizes expected fitness given the perceptual posterior:
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\begin{equation}
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a^*(y) = \arg\max_{a \in \mathcal{A}} \int_{\mathcal{M}} F(x, a) p(x|y) d\mu(x)
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\end{equation}
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The organism minimizes the Lagrangian functional $\mathcal{L}$:
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\begin{equation}
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\mathcal{L}[p(y|x), a(y)] = \int_{\mathcal{M}} \sum_{y} p(y|x) [-F(x, a(y))] d\mu(x) + \frac{1}{\beta} I(X;Y)
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\end{equation}
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\section{Minimizing Distortion Destroys Isomorphism}
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\begin{lemma}
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For a generic smooth fitness landscape $F(x, a)$, the level sets of $F$ do not align with the distance balls defined by the Riemannian metric $g$. Therefore, there exist points $x_1, x_2 \in \mathcal{M}$ separated by a large geodesic distance such that $a^*(y_1) = a^*(y_2)$ maximizes fitness.
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\end{lemma}
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\begin{theorem}
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Given a strict capacity bound $C < H(X)$ and a generic fitness landscape $F$, the encoder $p(y|x)$ minimizing $\mathcal{L}$ must violate structural isomorphism.
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\end{theorem}
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\begin{proof}
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Suppose $p(y|x)$ strictly preserves structural isomorphism. By Lemma 1, if distant points $x_1$ and $x_2$ share identical optimal actions $a^*$, distinguishing them requires allocating mutual information $\Delta I > 0$. Because the actions are identical, the expected fitness $\mathbb{E}[F]$ remains constant whether they are distinguished or clustered. However, distinguishing them strictly increases the channel cost $\frac{1}{\beta} I(X;Y)$. To minimize $\mathcal{L}$, the optimal encoder must actively collapse topologically distant points in $\mathcal{M}$ that share fitness level sets, obliterating structural isomorphism.
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\end{proof}
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\bibliographystyle{elsarticle-num}
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\begin{thebibliography}{10}
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\bibitem{Hoffman2015} D. D. Hoffman, M. Singh, C. Prakash, The interface theory of perception, Psychonomic Bulletin \& Review 22 (2015) 1480-1506.
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\bibitem{Ortega2013} P. A. Ortega, D. A. Braun, Thermodynamics as a theory of decision-making with information-processing costs, Proceedings of the Royal Society A 469 (2013) 20120683.
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\end{thebibliography}
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\end{document}
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