  We develop a theory of Bayesian price formation in electronic markets. We
formulate a stylised model in which market participants update their Bayesian
prior on an efficient price with a model-based learning process. We show that
exponential intensities for aggressive orders arise naturally in this
framework. The resulting theory allows us to derive simple analytic formulas
for market dynamics and price impact in the case with Brownian efficient price
and informed market takers. In particular we show that for small spreads there
is an asymptotic market regime. We illustrate our results with numerical
experiments.
