Fall 2011 Homework 3: Question 2
The odds of an event with probability p are defined to be $\frac{p}{1-p}$, e.g., an event with probability 3/4 is said to have odds of 3 to 1 in favor (or 1 to 3 against). We are interested in a hypothesis H (which we think of as a event), and we gather new data as evidence (expressed as an event D) to study the hypothesis. The prior probability of H is our probability for H being true before we gather the new data; the posterior probability of H is our probability for it after we gather the new data. The likelihood ratio is defined as $\frac{P(D|H)}{P(D|H^{c})}$.
(a) Show that Bayes' rule can be expressed in terms of odds as follows: the posterior odds of a hypothesis H are the prior odds of H times the likelihood ratio.
(b) As in the example from class, suppose that a patient tests positive for a disease affecting 1% of the population. For a patient who has the disease, there is a 95% chance of testing positive (in medical statistics, this is called the sensitivity of the test); for a patient who doesn't have the disease, there is a 95% chance of testing negative test (in medical statistics, this is called the specificity of the test). The patient gets a second, independent test done (with the same sensitivity and specificity), and again tests positive. Use the odds form of Bayes' rule to find the probability that the patient has the disease, given the evidence, in two ways: in one step, conditioning on both test results simultaneously, and in two steps, first updating the probabilities based on the first test result, and then updating again based on the second test result.
Solution: (a) Use Bayes' rule to obtain the desired equation. (b) To go from odds back to probability, we divide odds by (1 plus odds). One-update method: (1/99)(0.95^2/0.05^2) = 361/99, which corresponds to probability of ~ 0.78. Two-update method: Gives us same result with [(1/99)(0.95/0.05)](0.95/0.05) where the posterior odds become the new prior odds. (both conditionally and unconditionally).
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."