Predict first: A strong prior says the coin is close to fair, but the first 3 flips are all heads. Which estimate should move more, MLE or MAP?
Prior × likelihood → posterior
PriorLikelihood (scaled)PosteriorMAP
θ̂MLE—
θ̂MAP—
Posterior Beta—
Posterior mean—
When are MLE and MAP the same?
With a uniform prior, the prior contributes the same factor for every θ, so maximizing posterior is the same as maximizing likelihood. With informative priors, MAP trades off data fit and prior belief.
As the sample size grows, the likelihood becomes more concentrated and the prior’s influence usually becomes smaller.