Bayes’ Theorem
Bayesian methods of statistical analysis stem from a paper published posthumously in 1763 by the English clergyman Thomas Bayes. In a Bayesian analysis, probability calculations require a prior value for the likelihood of an association, which is then modified after data are collected. When the prior probability isn’t known, it must be estimated, leading to criticisms that subjective guesses must often be incorporated into what ought to be an objective scientific analysis. But without such an estimate, statistics can produce grossly inaccurate conclusions.
For a simplified example, consider the use of drug tests to detect cheaters in sports. Suppose the test for steroid use among baseball players is 95 percent accurate — that is, it correctly identifies actual steroid users 95 percent of the time, and misidentifies non-users as users 5 percent of the time.
Suppose an anonymous player tests positive. What is the probability that he really is using steroids? Since the test really is accurate 95 percent of the time, the naïve answer would be that probability of guilt is 95 percent. But a Bayesian knows that such a conclusion cannot be drawn from the test alone. You would need to know some additional facts not included in this evidence. In this case, you need to know how many baseball players use steroids to begin with — that would be what a Bayesian would call the prior probability.
Now suppose, based on previous testing, that experts have established that about 5 percent of professional baseball players use steroids. Now suppose you test 400 players. How many would test positive?
• Out of the 400 players, 20 are users (5 percent) and 380 are not users.
• Of the 20 users, 19 (95 percent) would be identified correctly as users.
• Of the 380 nonusers, 19 (5 percent) would incorrectly be indicated as users.
So if you tested 400 players, 38 would test positive. Of those, 19 would be guilty users and 19 would be innocent nonusers. So if any single player’s test is positive, the chances that he really is a user are 50 percent, since an equal number of users and nonusers test positive.
Bayes watch
Such sad statistical situations suggest that the marriage of science and math may be desperately in need of counseling. Perhaps it could be provided by the Rev. Thomas Bayes.
Most critics of standard statistics advocate the Bayesian approach to statistical reasoning, a methodology that derives from a theorem credited to Bayes, an 18th century English clergyman. His approach uses similar math, but requires the added twist of a “prior probability” — in essence, an informed guess about the expected probability of something in advance of the study. Often this prior probability is more than a mere guess — it could be based, for instance, on previous studies.
Bayesian math seems baffling at first, even to many scientists, but it basically just reflects the need to include previous knowledge when drawing conclusions from new observations. To infer the odds that a barking dog is hungry, for instance, it is not enough to know how often the dog barks when well-fed. You also need to know how often it eats — in order to calculate the prior probability of being hungry. Bayesian math combines a prior probability with observed data to produce an estimate of the likelihood of the hunger hypothesis. “A scientific hypothesis cannot be properly assessed solely by reference to the observational data,” but only by viewing the data in light of prior belief in the hypothesis, wrote George Diamond and Sanjay Kaul of UCLA’s School of Medicine in 2004 in theJournal of the American College of Cardiology. “Bayes’ theorem is … a logically consistent, mathematically valid, and intuitive way to draw inferences about the hypothesis.” (See Box 4)
With the increasing availability of computer power to perform its complex calculations, the Bayesian approach has become more widely applied in medicine and other fields in recent years. In many real-life contexts, Bayesian methods do produce the best answers to important questions. In medical diagnoses, for instance, the likelihood that a test for a disease is correct depends on the prevalence of the disease in the population, a factor that Bayesian math would take into account.
But Bayesian methods introduce a confusion into the actual meaning of the mathematical concept of “probability” in the real world. Standard or “frequentist” statistics treat probabilities as objective realities; Bayesians treat probabilities as “degrees of belief” based in part on a personal assessment or subjective decision about what to include in the calculation. That’s a tough placebo to swallow for scientists wedded to the “objective” ideal of standard statistics. “Subjective prior beliefs are anathema to the frequentist, who relies instead on a series of ad hoc algorithms that maintain the facade of scientific objectivity,” Diamond and Kaul wrote.
Conflict between frequentists and Bayesians has been ongoing for two centuries. So science’s marriage to mathematics seems to entail some irreconcilable differences. Whether the future holds a fruitful reconciliation or an ugly separation may depend on forging a shared understanding of probability.
“What does probability mean in real life?” the statistician David Salsburg asked in his 2001 book The Lady Tasting Tea. “This problem is still unsolved, and … if it remains un solved, the whole of the statistical approach to science may come crashing down from the weight of its own inconsistencies.”
-sciencenews.org
