Priors likelihoods and posteriors

Changing your mind in a fair way

Bayes' rule is a recipe for changing your mind when you see evidence. You start with a prior: what you believed before seeing the data. Then you ask how likely the data would be under each possible truth. That is the likelihood. Multiply the two, rescale so that everything adds up to $1$, and you get the posterior: your belief after the evidence. The number used to rescale is the evidence, the total probability of the data over all possible truths.

Take a strategy that is either good, winning $60\%$ of days, or bad, winning $50\%$ of days. Before any data you give a $30\%$ chance that it is good. It then wins $7$ days out of $10$. That result is more likely under good ($0.215$) than under bad ($0.117$), but not by much, and good was unlikely to begin with. The posterior chance of good is only $44\%$. With $70$ wins in $100$ days, the same win rate gives $99.5\%$. More data lets the likelihood outweigh the prior.

• Prior, likelihood, posterior: what you believed, how well each truth explains the data, what you believe now.
• Evidence $P(D)$ is only the rescaling number that makes the posterior add up to $1$.
• Data beat the prior: the more data, the less the prior matters.

Lab · what's graded

• A likelihood15 pts
• The evidence15 pts
• The posterior15 pts
• Posterior odds15 pts
• More data10 pts
• Match the vocabulary15 pts
• Why ten days is not proof15 pts