Bayes Theorem

Short version: \[P(A|B) = \frac{P(B|A)P(A)}{P(B)}\]

Full version: \[P(A|B) = \frac{P(B|A)P(A)}{P(A)P(B|A)+P(B|\neg A)(1-P(A))}\]

This can be understood as: \[Posterior = \frac{Prior \times TruePositive}{Prior \times TruePositive + NotPrior \times FalsePositive}\]

Example: breast cancer from N. Silver

A positive breast cancer exam could sound like catastrophic news when we know the rate of false positives of the mammogram is around 10%. But for a young patient this is not always intuitively sound as the prior probability of being afflicted by cancer is low.

A woman has roughly 1.4% chance of devlopping breast cancer in her forties, this is the prior \(P(A)\). If she does a mammogramm, there is a 10% rate of false positives \((P(B|\neg A)\), and 75% percent of true positives \(P(B|A)\).

Applying Bayes Theorem, \[P(A|B) = \frac{P(B|A)P(A)}{P(A)P(B|A)+P(B|\neg A)(1-P(A))}= \frac{0.75 \times 0.014}{0.014 \times 0.75 + 0.1 \times (1-0.014)} = 9.62\%\]

The chance of that a woman in her forties has breast cancer given she’s had a positive mammogram is still low, around 10%.

Independence example from N. Silver

Each pool contains 5 securities that each have a 5% chance of default. The probability of loosing the bet depends drastically on the assumption of correlation between the securities.

	n = [1:5]';
	p = 0.05;
	res = (p.^n .* (1-p).^(5-n)) .* factorial(5)./factorial(max(5-n,n))./max(1,min(5-n,n));
	sprintf('%10f',res*100)

References and Further Reading

N. Silver. The Signal and the Noise: Why So Many Predictions Fail but Some Don’t.