4.5: Bayes' Theorem

Question 1

What is Bayes’ Theorem used for?

Answer 1

Bayes’ Theorem is used to update prior probabilities to posterior probabilities upon receiving new information.

Question 2

How do you find the posterior probability P(HIV | POS)?

Answer 2

Use Bayes’ Theorem:
P(HIV | POS) = [P(HIV)P(POS | HIV)] / [P(HIV)P(POS | HIV) + P(HIV)P(POS | HIV)].

Question 3

What does Bayes’ Theorem say about mutually exclusive states of nature and experimental outcomes?

Answer 3

Bayes’ Theorem states that the posterior probability P(Si | E) is P(Si ∩ E) / P(E), which can be expanded based on the prior probabilities and the probability of E given Si.

Question 4

If you have a prior probability and an experimental outcome, how do you calculate the posterior probability?

Answer 4

The posterior probability is calculated as P(Si | E) = [P(Si)P(E | Si)] / Σ[P(Sj)P(E | Sj)] for all states Sj.