Belief Movement , Uncertainty Reduction , & Rational Updating ∗

When a Bayesian learns new information and changes her beliefs, she must become concomitantly more certain about the state of the world on average. Consequently, it is rare for a Bayesian to frequently shift beliefs substantially while remaining relatively uncertain, or, conversely, consistently become very confident with relatively little belief movement. We formalize this intuition by developing specific measures of movement and uncertainty reduction given a Bayesian’s changing beliefs over time, showing that these measures are equal in expectation, and creating consequent statistical tests for Bayesianess. We then show intriguing connections between these two core concepts and four common psychological biases, suggesting that the test might be particularly good at detecting these biases. We provide support for this conclusion by numerically simulating the performance of our test and other martingale tests. Finally, we apply our test to datasets of individual, algorithmic, and market beliefs. ∗We thank Tristan Gagnon-Bartsch and Kevin He for valuable research assistance, and Roland Benabou, Doug Bernheim, Aaron Bodoh-Creed, Stefano DellaVigna, Emir Kamenica, Michael Katz, David Laibson, Eben Lazarus, David Levine, Ulrike Malmendier, Muriel Niederle, Devin Pope, Josh Schwartzstein, Carl Shapiro, Avner Shlain, Charles Sprenger, Joel Sobel, Alex Steiny, Philipp Strack, Tomasz Strzalecki, Neil Shephard, Itai Sher, Dan Svirsky, and Eric Zitzewitz, as well as seminar audiences at UC-Berkeley, Dartmouth, Harvard, Princeton, University of Chicago Booth, BEAM, Stanford, Brown, Yale SOM, USC, University of Pittsburgh, Penn State, and UCSD, for helpful comments. †Haas School of Business, University of California, Berkeley. [email protected] ‡Department of Economics and Business School, Harvard University. [email protected]