What is Bayes’s Theorem?
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Bayes’ Theorem is P(A|B) = P(B|A)P(A) / P(B).
What it’s talking about is the scenario where you get up in the morning thinking that A (whatever that might be) has a certain chance of being true, and then you get some new information, called B. The theorem tells you *how much* you should adjust your belief, now that you’ve heard B.
For instance, you might wake up with a 99.997% confidence that your girlfriend is faithful (her being faithful is A), and then B is the new information: Lying Joe tells you that she’s cheating on you. Should you instantly start believing that your girlfriend’s cheating?
In general terms, if you think Lying Joe is completely full of shit, your opinion of your girlfriend shouldn’t change one iota. But if you know he tells the truth occasionally, your 99.997% might turn into a 98.7% or something, and if Lying Joe is the most reliable person in the universe (?!?) then your confidence should fall substantially. The formula lets you decide that, based on P(B|A) and P(B) which are indicators of Lying Joe’s reliability.
Bayesian reasoning is good because when you get new information, it makes you slow down and think! It asks you, basically, how confident were you in your belief before this (they call that your ‘prior’), how believable is this new claim, and how “well connected” is this new claim to reality.
One of the standard examples is, imagine the doctor’s office calls a week after you get some blood tests, and tells you that you have kuru, an ultra-rare disease associated with New Guinea that you usually get from eating brains. (Imagine for this example that you live very far from New Guinea, have never been there, and have never knowingly eaten brains.) Some work with Bayes’ Theorem can tell you that “mistake in the lab” or “office called the wrong person” are both more likely than “you suddenly got this ultra-improbable disease”.
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