I have written a piece about when it is rational not to update one’s beliefs in response to new evidence, in particular with respect to anecdotal evidence. The piece contains some equations, so I have uploaded it as a pdf here:
When NOT to Update Your Beliefs (in pdf form)
I argue that in cases with low prior probabilities and unreliable evidence (e.g. personal anecdotes), it is rational not to update one’s posterior probabilities at all in response to additional low quality evidence (e.g. an additional anecdote). I present my basic case with reference to Bayes’ Theorem, and then consider some rebuttals. I reject that rebuttal that updates should be small but non-zero on the grounds that such small updates are within the bounds of error of one’s probabilities. I reject the rebuttal that many anecdotes provide stronger cumulative evidence on the basis that anecdotes are not independent events. I conclude with a discussion about the differences between updating in abstract theory, and updating in the real world.