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

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.