1.6. Standard medical example by applying Bayesian rules of probability#
Goal: Use the Bayesian rules to solve a familiar problem. \(\newcommand{\pr}{\textrm{p}} \)
This illustrates how to avoid the Base Rate Fallacy.
Bayesian rules of probability as principles of logic#
Notation: \(p(x \mid I)\) is the probability (or pdf) of \(x\) being true given information \(I\)
Sum rule: If set \(\{x_i\}\) is exhaustive and exclusive,
\[ \sum_i p(x_i \mid I) = 1 \quad \longrightarrow \quad \color{red}{\int\!dx\, p(x \mid I) = 1} \]cf. complete and orthonormal
implies marginalization (cf. inserting complete set of states or integrating out variables - but be careful!)
\[ p(x \mid I) = \sum_j p(x,y_j \mid I) \quad \longrightarrow \quad \color{red}{p(x \mid I) = \int\!dy\, p(x,y \mid I)} \]Product rule: expanding a joint probability of \(x\) and \(y\)
\[ \color{red}{ p(x,y \mid I) = p(x \mid y,I)\,p(y \mid I) = p(y \mid x,I)\,p(x \mid I)} \]If \(x\) and \(y\) are mutually independent: \(p(x \mid y,I) = p(x \mid I)\), then
\[ p(x,y \mid I) \longrightarrow p(x \mid I)\,p(y \mid I) \]Rearranging the second equality yields Bayes’ Rule (or Theorem)
\[ \color{blue}{p(x \mid y,I) = \frac{p(y \mid x,I)\, p(x \mid I)}{p(y \mid I)}} \]
See Cox for the proof.
Answer the questions in italics. Check answers with your neighbors. Ask for help if you get stuck or are unsure.#
Suppose there is an unknown disease (call it UD) and there is a test for it.
a. The false positive rate is 2.3%. (“False positive” means the test says you have UD, but you don’t.)
b. The false negative rate is 1.4%. (“False negative” means you have UD, but the test says you don’t.)
Assume that 1 in 10,000 people have the disease. You are given the test and get a positive result. Your ultimate goal is to find the probability that you actually have the disease. We’ll do it using the Bayesian rules.
We’ll use the notation:
\(H\) = “you have UD”
\(\overline H\) = “you do not have UD”
\(D\) = “you test positive for UD”
\(\overline D\) = “you test negative for UD”
Before doing a calculation (or thinking too hard :), does your intuition tell you the probability you have the disease is high or low?
In the \(\pr(\cdot | \cdot)\) notation, what is your ultimate goal?
Express the false positive rate in \(\pr(\cdot | \cdot)\) notation. [Ask yourself first: what is to the left of the bar?]
Express the false negative rate in \(\pr(\cdot | \cdot)\) notation. By applying the sum rule, what do you also know? (If you get stuck answering the question, do the next part first.)
Should \(\pr(D|H) + \pr(D|\overline H) = 1\)? Should \(\pr(D|H) + \pr(\overline D |H) = 1\)? (Hint: does the sum rule apply on the left or right of the \(|\)?)
Apply Bayes’ theorem to your result for your ultimate goal (don’t put in numbers yet). Why is this a useful thing to do here?
Let’s find the other results we need. What is \(\pr(H)\)? What is \(\pr(\overline H)\)?
Finally, we need \(\pr(D)\). Apply marginalization first, and then the product rule twice to get an expression for \(\pr(D)\) in terms of quantities we know.
Now plug in numbers into Bayes’ theorem and calculate the result. What do you get?