When a Bayesian learns new information and changes her beliefs, she must on average become concomitantly more certain about the state of the world. Consequently, it is rare for a Bayesian to frequently shift beliefs substantially while remaining relatively uncertain, or, conversely, 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 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 simulating the performance of our test and other martingale tests. Finally, we apply our test to data sets of individual, algorithmic, and market beliefs.

Fixed differences loom smaller when compared to large differences. We propose a model of relative thinking where a person weighs a given change along a consumption dimension by less when it is compared to bigger changes along that dimension. In deterministic settings, the model predicts context effects such as the attraction effect but predicts meaningful bounds on such effects driven by the intrinsic utility for the choices. In risky environments, a person is less likely to sacrifice utility on one dimension to gain utility on another that is made riskier. For example, a person is less likely to exert effort for a fixed monetary return if there is greater overall income uncertainty. We design and run experiments to test basic model predictions and find support for these predictions.