Example 3: Why Lab Tests Must be Very Accurate

A patient either has a certain form of cancer or not. A biopsy will return either $\oplus$ meaning that the patient is sick, or $\ominus$. However, the biopsy only has 98% accuracy in identifying $\oplus$ and a 97% accuracy in identifying $\ominus$. Also, we know that the prior probability that a random person has this disease is 0.008. What is the posterior probability that a person for whom the test returns $\oplus$ has the disease?

We have

$$\begin{eqnarray*}P(cancer) &=& 0.008 \qquad P(\neg cancer) = 0.992 \\ P(\oplus\... ...eg cancer) &=& 0.03 \qquad P(\ominus\vert\neg cancer) = 0.97 \\ \end{eqnarray*}$$

We seek to calculate $P(cancer\vert\oplus)$. By Bayes rule,

$$\begin{eqnarray*}P(cancer\vert\oplus) &=& \frac{P(\oplus\vert cancer) \cdot P(ca... ...=& 0.21\\ P(\neg cancer\vert\oplus) &=& 1-0.21 \\ &=& 0.79\\ \end{eqnarray*}$$

Alternately, using Eqn 2,

$$h_{MAP} = \underset{h \in \{cancer, \neg cancer\}}{\rm argmax}\quad P(\oplus\vert h) P(h)$$

which gives $h_{MAP} = \neg cancer$.

Compute how accurate the lab test should be before a diagnosis of cancer can be made with more than 50% accuracy.

Read the given lucid discussion by E. T. Jaynes on why the prior is so important, esp re the quote from Laplace about people who recite miracles.