Somebody Bet on the Bayes

Imagine that you’ve just received a call from your doctor. A medical test has indicated you are HIV positive. If you don’t regard AIDS as a serious threat to you, then imagine that we’re discussing the disease you fear the most, whether it’s TB, ovarian cancer, or you name it. A somber subject to be certain, but not a reason to give up all hope. Medical tests are not infallible, and Bayes’ Theorem of probability can help determine what the chances are that you actually don’t have the disease.

Who was Bayes, and how do you pronounce his name? The title above serves to remind (me, mostly) that Bayes rhymes not with sighs, but with sleighs. The Reverend Thomas Bayes, a Brit, wrote a paper in 1763 that turned the science of probability on its head. I’m not saying that the then prevailing theories of French gamblers Rene DesCartes and Pierre Fermat were incorrect; rather I claim that they were not addressing the compelling questions. Fermat and penpal Blaise Pascal invented probability to help their gaming buddies determine their chances of winning in games of, well, chance.

Neither am I making a moral judgment when I say, “Who cares?” I’m saying that there are more useful questions that probability can and does answer. Bayesian probability, that is. Let’s bring these issues up to date. Do you honestly think that consumers use the numerical chance of winning a lottery (snowball’s) in making the decision to purchase tickets? Will they hit the jackpot; will she run the table; will he draw a royal flush? We can calculate those odds, and then we say good luck to them.

Bayes’ Theorem calculates probability in reverse. Something has happened, for instance, a positive HIV test, and there could be two or more precursors of that event, for example, you have the AIDS virus, or you don’t have it. If certain information is known, then Bayes’ Theorem will calculate the odds that it was one cause over the other. When a test wrongly indicates that you have a disease, we call it a false positive reading.

Yet another application of Bayesian probability is no less deadly serious: the mathematical disadvantage of minorities when subject to the normal course of law enforcement. Here is a fictional case in point. We visited the small burg of Bristow, with an adult male population of 7400, where a murder had taken place the previous night. The town’s population is divided among two racial groups: 90% Caucasian and 10% African American. A patrolman observed a man fleeing the vicinity of the crime and reported that it was a black man. Since it happened at night, we decided to put the policeman’s powers of observation to the test. In numerous trials with suspects of each race, and conditions that replicated the distance and lighting of the crime scene, we determined that the cop correctly identified the race of the subject in 4 of every 5 attempts. Let’s work the numbers, assuming that any of the 7400 could be guilty, and then see how this reliability factor applies to each type of report and to each race.

If the officer had reported observing a white suspect, then for every 6,660 reports (the number of white males in town) he would report he saw a white man 80% of the time, or in 5328 of those cases, and he would report incorrectly that a Caucasian was an African American in the other 1332 cases. For every 740 reports, one for each of the 740 black males, an incorrect report of a white male would be issued in 20% of the cases, or 148 times. So, given that he reported a white male, the officer would be correct 5328 times and wrong 148 times for an average accuracy of 97.3%. A prosecutor might use this to convince a jury that the testimony in a particular case was extremely reliable; however, we haven’t figured the accuracy of the officer when he reports seeing a black man.

There are 740 black males in the town. For every 740 reports of a black man, using the same reliability factor of 80%, we figure the officer is correct in 592 of those reports, but more significantly, he would be wrong in 20% of the 6,660 cases, that is 1332 times, when it was actually a white male, not a black, fleeing. So he would be reporting a black suspect 1924 times (592 plus 1332) out of every 7400 cases and be correct in only 592. His accuracy when reporting black suspects is less than 31%! Please check my math. Granted this analysis doesn’t consider any other factors, such as who of the town’s adults had opportunity and motive, but even so....

Now we will face our doctor, armed with Reverend Bayes’ Theorem, and with data concerning the reliability of the HIV test that was used. A positive reading has occurred. The first factor to consider is the size of the minority who suffer from the syndrome. Let’s say that 1000 of every million people have HIV. Next, how accurate is the test in correctly identifying sufferers? Let’s say it’s 90% accurate. Lastly, for non-sufferers, how often will the test give an incorrect reading, in other words, a false positive? Let’s say in only one (1) case of every 1000. Even with this degree of reliability, you will be surprised at the probability that a positive test is wrong.

Let’s use one million cases and look at the outcomes. Of the 1000 infected with HIV, the test identifies 900 correctly, and fails to identify 100 sufferers. Of the 999,000 non-sufferers, the test gives 998,001 negative readings (the good news), and 999 positive readings (the false bad news). Your job is to finish the math. From every one million tests, find the total of the positive readings and the percent of the time that a positive reading is wrong. (Hint: it’s wrong in a majority of the positive readings.)