The attraction effect and other decoy effects are often understood as anomalies and modeled as departures from rational inference. We show how these decoy effects may arise from simple Bayesian updating. Our new model, the Bayesian probit, has the same parameters as the standard multinomial probit model: each choice alternative is associated with a Gaussian random variable. We show how, unlike any random utility model, the Bayesian probit can jointly accommodate similarity effects, the attraction effect and the compromise effect. We also provide a new definition of revealed similarity based only on the choice rule and show that in the Bayesian probit (i) signal averages capture revealed preference; (ii) signal precision captures the decision maker’s familiarity with the options; and (iii) signal correlation captures our new definition of revealed similarity. This link of parameters to observable choice behavior facilitates measurement and provides a useful tool for discrete choice applications. ∗Department of Economics, Washington University in St. Louis. E-mail: email@example.com. This paper is based on the first chapter of my doctoral dissertation at Princeton University. I wish to thank my advisor, Faruk Gul, for his continuous guidance and dedication. I am grateful to Wolfgang Pesendorfer for many discussions in the process of bringing this project to fruition. I also benefited from numerous conversations with Dilip Abreu, Roland Bénabou, Meir Dan-Cohen, Daniel Gottlieb, David K. Levine, Justinas Pelenis, and seminar participants at Alicante, Arizona State, Berkeley, D-TEA Workshop, IESE, IMPA, Johns Hopkins, Kansas University, Haifa, Harvard/MIT Theory Seminar, Hebrew University, NYU, Princeton, the RUD Conference at the Colegio Carlo Alberto, SBE Meetings, Toronto, UBC, Washington University in St. Louis and the D-Day workshop at Yale. All remaining errors are my own.