Spam or not? A Bayesian posterior from one word

Question

At an inbox, 30 percent of incoming email is spam. The word "free" appears in 60 percent of spam messages but only 10 percent of legitimate (ham) messages. A new email arrives containing the word "free."

Using Bayes' rule, what is the probability that this email is spam?

Accepted Answer

Intuition first. The word "free" is six times more common in spam than in ham, so seeing it should raise your suspicion. But spam is only 30 percent of mail to begin with, so we must combine that base rate with the clue. Bayes' rule does exactly this weighing. The recipe. Multiply each hypothesis' prior by the chance it produces the clue, then normalize. The two weighted paths: 	ext{spam path} = P(	ext{spam})\cdot P(	ext{free}\mid	ext{spam}) = 0.30	imes 0.60 = 0.18, 	ext{ham path} = P(	ext{ham})\cdot P(	ext{free}\mid	ext{ham}) = 0.70	imes 0.10 = 0.07. Normalize the spam path by the …

Spam or not? A Bayesian posterior from one word

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