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Prediction Markets as Bayesian Inverse Problems: Uncertainty Quantification, Identifiability, and Information Gain from Price-Volume Histories under Latent Types

Juan Pablo Madrigal-Cianci, Camilo Monsalve Maya, Lachlan Breakey

TL;DR

The paper treats prediction markets as Bayesian inverse problems where a binary outcome $Y$ is inferred from price-volume histories using a log-odds observation model driven by a latent-type mixture of trader behavior. It develops a principled framework for posterior uncertainty quantification, identifiability via KL-projection gaps, and information-theoretic diagnostics (realized and expected information gain), with finite-sample guarantees under regularity. A concrete Gaussian latent-type instantiation with volume-gated mixtures demonstrates identifiability conditions and concentration bounds, and the authors provide computation strategies (SMC and VI) plus extensive synthetic experiments confirming theory. The approach supplies diagnostics for when market histories are informative versus ill-posed, and offers a pathway to designing mechanisms that enhance information capture in prediction markets.

Abstract

Prediction markets are often described as mechanisms that ``aggregate information'' into prices, yet the mapping from dispersed private information to observed market histories is typically noisy, endogenous, and shaped by heterogeneous and strategic participation. This paper formulates prediction markets as Bayesian inverse problems in which the unknown event outcome \(Y\in\{0,1\}\) is inferred from an observed history of market-implied probabilities and traded volumes. We introduce a mechanism-agnostic observation model in log-odds space in which price increments conditional on volume arise from a latent mixture of trader types. The resulting likelihood class encompasses informed and uninformed trading, heavy-tailed microstructure noise, and adversarial or manipulative flow, while requiring only price and volume as observables. Within this framework we define posterior uncertainty quantification for \(Y\), provide identifiability and well-posedness criteria in terms of Kullback--Leibler separation between outcome-conditional increment laws, and derive posterior concentration statements and finite-sample error bounds under general regularity assumptions. We further study stability of posterior odds to perturbations of the observed price--volume path and define realized and expected information gain via the posterior-vs-prior KL divergence and mutual information. The inverse-problem formulation yields explicit diagnostics for regimes in which market histories are informative and stable versus regimes in which inference is ill-posed due to type-composition confounding or outcome--nuisance symmetries. Extensive experiments on synthetic data validate our theoretical predictions regarding posterior concentration rates and identifiability thresholds.

Prediction Markets as Bayesian Inverse Problems: Uncertainty Quantification, Identifiability, and Information Gain from Price-Volume Histories under Latent Types

TL;DR

The paper treats prediction markets as Bayesian inverse problems where a binary outcome is inferred from price-volume histories using a log-odds observation model driven by a latent-type mixture of trader behavior. It develops a principled framework for posterior uncertainty quantification, identifiability via KL-projection gaps, and information-theoretic diagnostics (realized and expected information gain), with finite-sample guarantees under regularity. A concrete Gaussian latent-type instantiation with volume-gated mixtures demonstrates identifiability conditions and concentration bounds, and the authors provide computation strategies (SMC and VI) plus extensive synthetic experiments confirming theory. The approach supplies diagnostics for when market histories are informative versus ill-posed, and offers a pathway to designing mechanisms that enhance information capture in prediction markets.

Abstract

Prediction markets are often described as mechanisms that ``aggregate information'' into prices, yet the mapping from dispersed private information to observed market histories is typically noisy, endogenous, and shaped by heterogeneous and strategic participation. This paper formulates prediction markets as Bayesian inverse problems in which the unknown event outcome is inferred from an observed history of market-implied probabilities and traded volumes. We introduce a mechanism-agnostic observation model in log-odds space in which price increments conditional on volume arise from a latent mixture of trader types. The resulting likelihood class encompasses informed and uninformed trading, heavy-tailed microstructure noise, and adversarial or manipulative flow, while requiring only price and volume as observables. Within this framework we define posterior uncertainty quantification for , provide identifiability and well-posedness criteria in terms of Kullback--Leibler separation between outcome-conditional increment laws, and derive posterior concentration statements and finite-sample error bounds under general regularity assumptions. We further study stability of posterior odds to perturbations of the observed price--volume path and define realized and expected information gain via the posterior-vs-prior KL divergence and mutual information. The inverse-problem formulation yields explicit diagnostics for regimes in which market histories are informative and stable versus regimes in which inference is ill-posed due to type-composition confounding or outcome--nuisance symmetries. Extensive experiments on synthetic data validate our theoretical predictions regarding posterior concentration rates and identifiability thresholds.
Paper Structure (84 sections, 12 theorems, 111 equations, 4 figures, 1 table, 2 algorithms)

This paper contains 84 sections, 12 theorems, 111 equations, 4 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Principled uncertainty quantification.
  4. Explicit identifiability and well-posedness criteria.
  5. Information-theoretic metrics.
  6. Paper Organization
  7. Background and Related Work
  8. Prediction Markets and Information Aggregation
  9. Market Microstructure and Informed Trading
  10. Bayesian Inverse Problems
  11. Mixture Models and Mixture of Experts
  12. Learning from Strategic Data
  13. Model and Bayesian Formulation
  14. Outcome and Observed History
  15. Log-Odds Coordinates
  16. ...and 69 more sections

Key Result

Proposition 4.1

Fix $(y^\star,\theta^\star)$ and $v_{1:T}$. The separation gap satisfies $\delta_T(y^\star,\theta^\star)=0$ whenever any of the following hold:

Figures (4)

  • Figure 1: Posterior concentration under $Y=1$. Points show the median of $\log(1-\pi_T)$ across 1000 replications; the shaded region is the 10--90% quantile range. The dashed line has slope $-\widehat{\delta}_T$, where $\widehat{\delta}_T$ is an estimated KL-projection gap.
  • Figure 2: Identifiability under decreasing informed weight. Posterior accuracy is the proportion of replications assigning higher posterior probability to the true outcome (here $T=100$). Accuracy degrades sharply when $\omega_1$ is sufficiently small, consistent with a vanishing separation gap.
  • Figure 3: Stability under perturbations of the increment path. Points show $|\log \mathrm{BF}_T(h)-\log \mathrm{BF}_T(h')|$ as a function of noise level $\sigma$; the dashed line is the stability bound from Theorem \ref{['thm:stability']} evaluated with a high-probability truncation radius $R$.
  • Figure 4: Information gain dynamics. Solid line: mean realized $\mathrm{IG}(H_t)$ across replications; shaded region: 10--90% quantile range. Dashed line: upper bound $\log 2$ for $\pi_0=1/2$.

Theorems & Definitions (39)

  • Remark 3.1: Volume as design
  • Definition 3.1: Trader type taxonomy
  • Example 3.1: Gaussian latent-type model
  • Remark 3.2: Orientation and outcome--nuisance symmetries
  • Definition 4.1: KL projection gap
  • Definition 4.2: Outcome identifiability (at the truth)
  • Remark 4.1: Uniform vs. truth-relative identifiability
  • Proposition 4.1: Sufficient conditions for outcome non-identifiability
  • proof
  • Theorem 4.1: A design-based sufficient condition for outcome separation
  • ...and 29 more