Recent conversations with friends & colleagues about probability theory reminded me of conversations with a friend of mine in graduate school about the supposed virtues of making one's own reasoning in one's daily life more systematic through Bayesian inference. The basic idea, in rough qualitative terms, is that one's belief in a hypothesis can be quantified through a prior probability, and when one observes some data related to that hypothesis, one can use the probabilities of observing that data when that hypothesis does or does not hold to update one's belief in (becoming the posterior probability of) that hypothesis based on the data. An example of quantitative & qualitative explanations can be found on the site LessWrong [LINK]. However, even in graduate school and again more recently, I realized that it is very easy for one to talk oneself into believing that one is using systematic Bayesian reasoning while actually just rationalizing one's own prior beliefs & changes in beliefs after the fact. This can be illustrated mathematically in a few ways that are not exhaustive. Follow the jump to see more.