Hidden Participation and the Timing of Price Discovery: Exact Bayesian Inference and a Dynamic Linear-Projection Benchmark

This paper studies how hidden, stochastic continuation of informed participation shapes price discovery in a two-period Kyle-type market. Hidden participation separates two analytical questions that coincide in the standard Gaussian-linear model. Conditional on a specified linear continuation order, we first derive the exact Bayesian posterior mean and variance of a mixture comprising an informed-trading regime and a noise-only regime; this is a conditional-inference result, not a full nonlinear equilibrium. We then derive an equilibrium under a constrained best-linear-pricing protocol in which market makers use the minimum-mean-square-error affine projection and the insider optimizes pointwise against linear prices. The exact Bayesian posterior responds nonlinearly because order flow reveals both residual value and the likelihood of informed participation, while moderate flows can preserve substantial regime uncertainty. In the projection benchmark, a lower continuation probability accelerates first-period information revelation, shifts insider rents toward the initial round, and creates opposing early- and late-learning effects. A dimensionless analysis characterizes how inference varies with continuation probability and the informed-to-noise variance ratio and establishes scale invariance for the benchmark’s normalized comparative statics. The paper thus isolates a participation margin in price discovery and states precisely which results concern exact inference and which concern a constrained equilibrium.