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Bayes’ Theorem – “The Signal and the Noise: The Art and Science of Prediction”, Nate Silver

November 21st, 2012 · No Comments · Policy, Prediction, Statistics

Bayes’s theorem is concerned with conditional probability. That is, it tells us the probability that a theory or hypothesis is true if some event has happened. Suppose you are living with a partner and come home from a business trip to discover a strange pair of underwear in your dresser drawer. You will probably ask yourself: what is the probability that your partner is cheating on you? The condition is that you have found the underwear; the hypothesis you are interested in evaluating is the probability that you are being cheated on. Bayes’s theorem, believe it or not, can give you an answer to this sort of question— provided that you know (or are willing to estimate) three quantities: First, you need to estimate the probability of the underwear’s appearing as a condition of the hypothesis being true—that is, you are being cheated upon. Let’s assume for the sake of this problem that you are a woman and your partner is a man, and the underwear in question is a pair of panties. If he’s cheating on you, it’s certainly easy enough to imagine how the panties got there. Then again, even (and perhaps especially) if he is cheating on you, you might expect him to be more careful. Let’s say that the probability of the panties’ appearing, conditional on his cheating on you, is 50 percent. Second, you need to estimate the probability of the underwear’s appearing conditional on the hypothesis being false. If he isn’t cheating, are there some innocent explanations for how they got there? Sure, although not all of them are pleasant (they could be his panties). It could be that his luggage got mixed up. It could be that a platonic female friend of his, whom you trust, stayed over one night. The panties could be a gift to you that he forgot to wrap up. None of these theories is inherently untenable, although some verge on dog-ate-my-homework excuses. Collectively you put their probability at 5 percent. Third and most important, you need what Bayesians call a prior probability (or simply a prior). What is the probability you would have assigned to him cheating on you before you found the underwear? Of course, it might be hard to be entirely objective about this now that the panties have made themselves known. (Ideally, you establish your priors before you start to examine the evidence.) But sometimes, it is possible to estimate a number like this empirically. Studies have found, for instance, that about 4 percent of married partners cheat on their spouses in any given year, so we’ll set that as our prior. If we’ve estimated these values, Bayes’s theorem can then be applied to establish a posterior possibility. This is the number that we’re interested in: how likely is it that we’re being cheated on, given that we’ve found the underwear?

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