Reverend Thomas Bayes (1702–1761) was an English clergyman who happened to be great at mathematics, which was undoubtedly his first love. He formulated a method (“theorem”) bearing his name, which allows the mathematical calculation of probabilities of outcomes given certain baseline population characteristics. Bayes’s numerical formula remains pertinent today and is used by contemporary health professionals, psychologists, economists, physicists, and engineers. Although the mathematical formula* is complex, the underlying concept is understandable and fairly easy to apply.
Most methods of detection employ means that are not 100% accurate, meaning any given test will not accurately identify whether a given condition is present or absent. Thus, we must constantly assess the likelihood of presence if a given test result is positive, as well as the likelihood that a given condition is absent if a test result is negative. The idea that Bayes introduced was conditional probability, i.e., the likelihood of an event occurring given that another event has already occurred. In medical issues, Bayes’ Theorem provides a mathematical means to derive the actual probability of a disease after a given test is applied. The times that actual diseases are erroneously called absent, false negatives, and when called positive in the absence of disease, they are called false positive. Bayes’ mathematical formula provides a means to derive the actual probability of an outcome after these two variables are applied.
First let’s apply this idea to flying in a commercial airliner. Statistics show that such air travel is quite safe—approximately 60 times safer than car travel. In real terms, the odds of a major accident are one for every 1.2 million flights worldwide is extremely low, i.e, 0.00000819, a number that is virtually zero. So why are so many of us more fearful of airplanes than cars? The answer can be described in terms of conditional probabilities. So, according to Bayes’s concept, the chance of mortality from air travel is a result of the product of these two variables—the chance of a crash, which is almost zero, times the chance of death, nearly 100%. So, when the multiple is calculated, the answer remains virtually zero (0.00000818), meaning that safe arrival is virtually assured. This very high conditional probability should provide much comfort to all air travelers.
Firearm Deaths
Using the Bayes’ approach, let’s now evaluate mass firearm deaths resulting from mental disorders, a popular political belief that detection and treatment of such disorders can provide a major means to reduce such deaths.. According to the National Institute of Health, the prevalence of major mental illness in the U.S. is approximately 4.2% of the entire population, meaning that about 10.4 million people suffer from serious mental illness. We then must determine whether a given individual harboring a mental disorder is capable of firearm violence. Let’s then make a dubious second assumption that mental health professionals could predict with a 95% accuracy which ones are capable of mass violence. During a recent period of six years, there were 43 individuals responsible for mass firearm attacks. Let’s assume that this entire group of 43 was all mentally ill—which is unlikely—it would constitute an infinitesimally small percentage (000004) of all those suffering from mental illness. In order to predict the likelihood that a given individual selected from this huge baseline group of mentally ill is capable of mass murder, we can then apply Bayes’ formula, which raises accuracy of the selection process again to a negligible 0.000076, which provides no practical means to detect such murderers. Moreover, this likelihood is probably even less than this tiny number, since mental health professionals freely admit that it is far less than the 95% employed in this formula. What this means is that, given these extremely small numbers, detection and treatment of those with mental illness in the effort to ward off gun violence is a virtual impossibility, notwithstanding the pronouncements by many politicians. It all boils down to a simple bottom line: Major efforts must be aimed primarily at sensibly limiting everyone—whether or not mentally ill—from obtaining firearms capable of mass destruction. This type of analysis constitutes another victory for reverend Bayes!
In clinical medicine, we usually evaluate the likelihood that a given individual does or does not have the disease in question, which allows us to calculate these likelihoods based upon the two important factors: 1) The accuracy of the test and 2) The composition of the population undergoing the test (“pre-test probability”). The most important factor in this calculation is the composition of the population being tested, i.e., the percentage already possessing the disease in question. If one tests a population containing a low rate of a given disease, a “positive” test result will usually contain a large percentage of individuals that do not have the disease in question, i.e., they will be “false positives”. In all situations, Bayes’ principle allows us to raise the likelihood of disease after a positive result and, conversely, to lower it after a negative result. Illustrating this concept in a medical situation where this type of analysis is important, is that of X-ray mammography screening for detection of breast cancer. If we evaluate a population of women between ages 30 to 40, current estimates place the prevalence of cancer at approximately 1%, which is obviously quite low. We know the approximate accuracy of mammography (false positive rate of approximately 6%). So given those test characteristics and such low pre-test population figures (1%), we apply Bayes formula, and if a woman tests positive, the likelihood she has cancer is raised from 1% to a post-test likelihood of only an unlikely 11.2%, but such a result usually requires additional (expensive) testing. . On the other hand, however, what reassurance does a negative result provide? Applying the Bayes formula again, we now come up with a result of 99.7%, that is, the likelihood she does not have cancer. For all practical purposes, therefore, she has little or no chance of having this disease. Thus, given these underlying numbers, a negative result is far more useful and reassuring than is a positive result.
Bayes’s concept also applies interestingly to other non-medical situations such as airport screening. Even if the various preliminary screening devices had a very high accuracy (probably less than 1% false positives), the fact the groups subjected to this screening have such an infinitesimally small rate of dangerous or explosive devices, then a “positive” test response will almost certainly be a “false positive”. This conclusion is confirmed by the monotonous regularity with which no real threats turn up in response to the secondary screening methods, i.e., “wanding”, pat-downs, etc. I suspect that if Reverend Bayes were here today, he would be both surprised and gratified by the worldwide acceptance of his theorem, except when he would need to undergo a “pat-down” prior to entering a “new-fangled” airplane.
So we conclude by confirming the considerable utility of Bayes approach, useful for over 300 years and still in use!
* Formula noted in Tavel ME. Snake Oil is Alive and Well, 2012, Brighton Publishing Co, p. 245.
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