Susan either likes George or dislikes him. His prior belief is that there is a 50% chance that she likes him. He also believes that if she does, there is an 80% chance she will smile at him, and if she does not, there is a 60% chance. She smiles at him. What should he think of that?

The Frequentist approach says that George should choose the answer which has the greatest likelihood given the data, and so he should believe that she likes him.Click here to read more

It warns, however, that if he follows this plan, and she really doesn't like him, then he will come to the wrong conclusion with 60% probability. Thus, he can't confidently reject the null hypothesis that she dislikes him. (Though he also could not confidently reject the null hypothesis that she likes him-- that plan would lead him to mistakenly rejecting the null 80% of the time!)

The Bayesian answer is that he should do this computation:

Prob(likes|smiles) = Prob (smiles|likes)Prob(likes)/Prob(smiles)

Prob(likes|smiles)= .8*.5/(.8*.5+.6*.5) = .4/(.4+.3) = 4/7.

If he must choose a binary action-- say, to invite her out or not-- then if the losses from each kind of mistake are symmetric, he should act as if she likes him, knowing that with probability 3/7 he will be making a mistake.

Now let's think about where prior information enters. It enters via the Prior, of course-- the 50% chance that she likes him. But it also enters via the Likelihood Function-- the 80% and 60% figures. If we ask how sure George is of his 50% prior, it is in the Likelihood Function that we need to look for the answer. Suppose we want to keep George's belief at 50% but make him very sure of 50%, as if he'd had lots of evidence on each side rather than not knowing Susan at all and guessing based on his experience with other women. How does that show up here?

It shows up as this one smiling incident having very little effect on his belief. If his prior is solid, then he thinks that the data conveys little information. Maybe she would smile with probability 80% if she likes him and 79% if she dislikes him. Moreover, the next time he meets her, her second smile or frown will convey about the same, small, amount of information.

If his prior were loose, the first incident would have a big impact and the second would have a noticeably smaller impact. Maybe she would smile with probability 80% if she likes him and 20% if she dislikes him. Moreover, the next time he meets her, her second smile or frown will convey less information-- maybe the numbers would shift to 80% and 40%.

But my last two paragraphs do not satisfy me. I must go on to other things, so I will leave this post now.

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