Mathematics

Bayes Theorum



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Bayes' Theorem (also known as Bayes' rule or Bayes' law) is a theorem of probability theory. The theorem was discovered among the papers of the English Presbyterian minister and mathematician Thomas Bayes and published posthumously in 1763. Bayes' theorem tells how to update or revise beliefs in light of new evidence.

Some recent scientific papers have applied Bayes theorem (or similar approaches) like odds ratios to such diverse subjects as the analysis of bullet lead, human weight gain, and yes even the in-season prediction of baseball batting averages.

Now, before some of the more mathematically inclined begin to feel a little uneasy, let me state what this article is not. It is not an indictment of Bayesian theory, but rather a commentary on its use. However, like all things useful, some with agenda driven motives may seek to misuse or corrupt that usefulness. By its very nature Bayes is susceptible to such manipulation by those with an inclination to abuse rather than use it.

Most Bayesians who use Bayes Law have no intention of misusing it deliberately. They could be mistaken in its use however. Of course, Bayes law can be used to make a case for something that is not true deliberately but such users are in the minority.

Frequentist or Bayesian?

Dr. Bradley Efron of Stanford University commented in his paper, "Bayesians, Frequentists and Scientists" that; "there seems to be at least two ways to do statistics, and they don't always give the same answers." Dr. Efron is of course referring to the debate which rages between the frequentists and the Bayesians. Dr. Efron continues; "The Bayesian-Frequentist debate reflects two different attitudes to the process of doing science, both quite legitimate. Bayesian statistics is well-suited to individual researchers, or a research group, trying to use all the information at its disposal to make the quickest possible progress. In pursuing progress, Bayesians tend to be aggressive and optimistic with their modeling assumptions. Frequentist statisticians are more cautious and defensive.

One definition says that a frequentist is a Bayesian trying to do well, or at least not too badly, against any possible prior distribution. The frequentist aims for universally acceptable conclusions, ones that will stand up to adversarial scrutiny. The FDA for example doesn't care about Pfizer's prior opinion of how well its new drug will work, it wants objective proof. Pfizer, on the other hand may care very much about its own opinions in planning future drug development."

Bayes and the Courts

One sector which seems the least enamored with Bayes is the court system. Indeed in case after case the courts have ruled against the introduction of the Bayes based approach in presenting scientific evidence to a jury.

In one noted case - (R v. Adams, Court of Appeal (Criminal Division) 26 April 1996) - the trial judge, during his summing up, described the use of Bayes' theorem at some length but told the jury that they were not obligated to use it. The Court of Appeal quashed the defendant's conviction, and ordered a retrial. The Court suggested that Bayes' theorem was inappropriate for use in jury trials, giving the following reasons:

The apparently objective numerical figures used when applying Bayes' theorem might conceal the element of judgment on which the calculation was depended.

Bayes' theorem required that items of evidence be assessed separately, but this was too rigid an approach for the jury. The cogency of evidence has, in part, to be assessed in the light of a chain of evidence.

Jurors evaluate evidence not by means of a formula, but by the joint application of their common sense and knowledge of the world to the evidence before them.

The jury would find it difficult to apply the theorem during their deliberations. Jurors might differ in the figures to be attached to each item of evidence, and any compromise would not adequately reflect the jurors' views. The jurors would not be able to reconcile the individual views about the evidence if they used Bayes' theorem.

The introduction of Bayes' theorem into a criminal trial plunges the jury into inappropriate and unnecessary realms of theory and complexity deflecting them from their proper task.

The problem the courts have here with Bayes essentially boils down to the use of "priors" or "a priori" information. Formulating an all encompassing definition of a prior is a difficult task. Bayesian thinking considers not only what the data has to say, but what your expertise tells you as well. The Bayesian view of probability is related to degree of belief. It is a measure of the plausibility of an event given incomplete knowledge. So, just what is an appropriate Bayesian prior? That's not very well defined; especially it seems to the courts. In another noted case, the courts opined thusly regarding expert testimony presented through the use of Bayes; "testimony based on their prior probabilities would not necessarily be relevant or appropriate, because the jurors might have different priors, and the choice of a prior is not a matter of expertise." See State v. Spann, 617 A.2d 247 (N.J. 1993) (improper for expert to testify to posterior probabilities using her own prior.



Bayes in Science

Quite obviously, those who deal in science would be lost without statistics, but of course not all scientists are well groomed statisticians, and one can be a statistician and not a scientist. So essentially, the scientist is compelled to examine the evidence in its totality. Relying too heavily on statistics or proceeding from an isolated mathematical perspective, while ignoring the principles that underlie and define the evidence, is indeed poor science.

Should we concern ourselves then with the misuse of statistical treatments of data in science? The answer is yes, we should. In a recent communication to this author, Dr. Erik Randich of the University of California Lawrence Livermore National Laboratory stated; "Some recent forensic papers are cleverly using certain statistical methods to answer the question the author wants to answer and perhaps not the correct question that should be asked."



So, just what is ailing Bayes? When used properly, Bayes Theorem is a very useful tool. Of course with all things controversial, we should always put our best foot forward and examine the true nature of the controversy.

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