Week6-2Bayesian theorem

Bayes’ theorem

Formula for joint probability

P(A,B)=P(B∣A)P(A)=P(A∣B)P(B)

Therefore

P(A∣B)=P(B∣A)P(A)P(B)

Example

We have the following table.

What is p(disease∣positive)?

P(disease∣positive)=P(positive∣disease)P(disease)P(positive)P(¬disease∣positive)=P(positive∣¬disease)P(¬disease)P(positive)

We don’t care what value P(positive) is, as long as it is larger than 0.

P(disease∣positive)P(¬disease∣positive)=P(positive∣disease)P(disease)P(positive∣¬disease)P(¬disease)

So P(disease) is crucial for the final ratio. Suppose P(disease)=0.001, and we would have

P(disease∣positive)P(¬disease∣positive)=0.95×0.0010.05times0.999=0.019P(disease∣positive)+P(¬disease∣positive)=1

Very low! we cannot compensate too much the prior even if the test is positive, only from 0.1% to about 2%, still far from 50%.