Predictive Market Theory

Markets as Distributed Bayesian Inference Engines
Matthew Long · YonedaAI Collaboration · 4 Mar 2026
"The Stock Market (or any market on the planet) is a collective predictive mechanism that models perceived future outcomes as a Belief Function."

Papers

I Markets as Distributed Bayesian Inference Engines
q-fin.GN 2603.04 Price Formation, Information Aggregation, Thermodynamic Structure
Markets as distributed Bayesian inference engines. Precision-weighted consensus. Thermodynamic structure: energy, temperature, free energy. Power laws, crashes, bubbles, liquidity as entropy. Market-neural network correspondence.
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II Predictive Market Theory: A Bayesian-Probabilistic Foundation
q-fin.MF 2603.04 Predictive Measure, Five Axioms, Variational Principles
Inverts the framework: prediction then price. Five axioms of PMT. Three-measure decomposition (P, Q, P_pred). Variational free energy minimization. Recovers FTAP, CAPM, Black-Scholes as special cases.
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III Unified Predictive Market Framework: Synthesis & Implementation
q-fin.CP 2603.04 Legendre Duality, UPMF, Python Implementation
Legendre duality connects Parts I and II. UPMF Master Equation. Maps to REE, behavioral finance, econophysics, ML. Complete Python computational implementation with six modular components.
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IV HFT Belief Formation & Autonomous AI Market Agents
q-fin.TR 2603.04 High-Frequency Trading, AI Agents, Phase Transitions
How HFT systems form beliefs at microsecond timescales. Extension to autonomous AI agents (LLMs, RL). Belief space phase transition. AI Market Stability Theorem. Multi-agent belief ecosystems and collective intelligence.
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Key Results

Theorem I
Price is a sufficient statistic for fundamental value (Gaussian case)
Theorem II
EMH is the fixed-point condition of distributed Bayesian inference
Theorem III
Market equilibrium = free energy minimization
Theorem IV
Thermodynamic and predictive frameworks are Legendre duals
Theorem V
UPMF Master Equation: single stochastic PDE for all market dynamics
Theorem VI
HFT→AI transition is a phase transition in belief dimensionality
Theorem VII
AI Market Stability requires belief diversity above critical threshold
Theorem VIII
Collective intelligence exceeds any individual agent (superadditivity)

Framework Hierarchy

Part I (Foundation) ├── Bayesian inference engine ├── Thermodynamic structure ├── Empirical phenomena └── Market as neural network │ Part II (Prediction Layer) ├── Prediction → Price (inverted) ├── Five axioms ├── Three-measure decomposition └── Classical finance recovered │ Part III (Synthesis) ├── Legendre duality ├── UPMF Master Equation ├── Literature unification └── Python implementation │ Part IV (Belief Agents) ├── HFT microsecond beliefs ├── AI agent belief expansion ├── Phase transition theorem └── Stability & ecosystems

Three-Measure Decomposition

P (Physical)
Objective probabilities of actual outcomes
Q (Risk-neutral)
Pricing measure with risk adjustments
P_pred (Predictive)
Market's collective forecast, purged of risk distortions
Relationship
M(ω) = A(ω) × R(ω) — assessment × risk