Bayes' Theorem

For programmers

Joke

● What's the difference between a mathematician, an engineer and a programmer?

Punchline

● Mathematicians use natural log (base e)● Engineers use decibels (10 times log base

10)● Programmers use bits (log base 2)

Useful functions● odds(p)=p/(1-p)

– Gambler talk: 1/3 → “1-to-2”

● logit(p)=log(odds(p))– Remember: logs are base 2, or bits

● expit(p)=exp(p/(1+p))– Inverse of logit

What is belief?

● Belief(X) = logit(Probability you assign to X)– Measured in bits

● Fun fact: Belief(not X)=-Belief(X)

Examples

● Belief(X)=0: probability 0.5, zero knowledge

● Belief(X)=1: probability is 2/3● Belief(X)=-1: probability is 1/3● Belief(X)=5: probability about 0.97

● Belief(X)=10: “I’m 99.9% certain about this!”

● Belief(X)=-10: “There’s a 0.001 chance of that!”

● Belief(X)=infinity: probability 1, or “The religious belief”…

More examples

Accuracy of belief

● Overconfidence: >>1-expit(B) of beliefs of strength >B are wrong (for some B>0)

● Underconfidence: <<1-expit(B) of beliefs of 0<strength<B are wrong (for some B>0)

● Well-calibrated: Neither overconfident nor underconfident

Evidence

● Event E happened. Is X true?● E is helpful only when P(E given X) != P(E

given not X). But how much?● Likelihood(E given X) = P(E given X)/P(E

given not X)● Evidence(E about X) = log(Likelihood(E

given X))● Evidence is measured in bits!

THE FORMULA

Belief(X after seeing E) = Belief(X)+Evidence(E about X)

Bayes' Theorem

● “If you are well-calibrated, and update beliefs according to THE FORMULA, you remain well-calibrated”

● Corrolary: If you sometimes count evidence twice, or sometimes only weakly, you FALL OUT OF CALIBRATION!

Remember, Kids!

Bayes’ Theorem is math, not a suggestion.If you care about being right,you can’t afford to ignore it!