The Reverent Thomas Bayes was an 18th century Presbyterian minister who is credited with developed the Bayesian Theorem statistical method (and who is buried opposite my old office).
“The essence of the Bayesian approach is to provide a mathematical rule explaining how you should change your existing beliefs in the light of new evidence. In other words, it allows scientists to combine new data with their existing knowledge or expertise.
The canonical example is to imagine that a precocious newborn observes his first sunset, and wonders whether the sun will rise again or not. He assigns equal prior probabilities to both possible outcomes, and represents this by placing one white and one black marble into a bag. The following day, when the sun rises, the child places another white marble in the bag. The probability that a marble plucked randomly from the bag will be white (ie, the child’s degree of belief in future sunrises) has thus gone from a half to two-thirds. After sunrise the next day, the child adds another white marble, and the probability (and thus the degree of belief) goes from two-thirds to three-quarters. And so on. Gradually, the initial belief that the sun is just as likely as not to rise each morning is modified to become a near-certainty that the sun will always rise.” ref.
“1% of women at age forty who participate in routine screening have breast cancer. 80% of women with breast cancer will get positive mammographies. 9.6% of women without breast cancer will also get positive mammographies. A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer?”
Most people (and 85% of doctors) think the answer is around 80% – but they are wrong. The answer is in fact 7.8%.
“The correct answer is 7.8%, obtained as follows: Out of 10,000 women, 100 have breast cancer; 80 of those 100 have positive mammographies. From the same 10,000 women, 9,900 will not have breast cancer and of those 9,900 women, 950 will also get positive mammographies. This makes the total number of women with positive mammographies 950+80 or 1,030. Of those 1,030 women with positive mammographies, 80 will have cancer. Expressed as a proportion, this is 80/1,030 or 0.07767 or 7.8%.”
The mistake is to ignore both the proportion of women with breast cancer, and the percentage of women who receive a false positives, and only consider the percentage of women with breast cancer who get positive results. Consider the following alternative scenario:
“To see that the final answer always depends on the chance that a woman without breast cancer gets a positive mammography, consider an alternate test, mammography+. Like the original test, mammography+ returns positive for 80% of women with breast cancer. However, mammography+ returns a positive result for only one out of a million women without breast cancer – mammography+ has the same rate of false negatives, but a vastly lower rate of false positives. Suppose a patient receives a positive mammography+. What is the chance that this patient has breast cancer? Under the new test, it is a virtual certainty – 99.988%, i.e., a 1 in 8082 chance of being healthy.”