While my knowledge of mathematics is limited I am intrigued by the subject and more so with statistics and its application. Of which I admit to know less. However, one statistical method of particular value is known as Baye’s rule or theorem.  It was derived to analyze evidence, change our minds as we get new information, and make rational decisions in the face of uncertainty. Today the rule is used to measure the likelihood of being correct when predicting one particular event. This ingenious and powerful mathematical formula was developed by an obscure eighteenth century minister in England, Thomas Bayes (1702-1761).  Unfortunately Bayes published no original mathematics during his life. His reason for not publishing  his essay can hardly have been fear of controversy. Perhaps he thought his discovery was useless; but if a pious clergyman like Bayes thought his work could prove the existence of God, surely he would have published it. Some thought Bayes was too modest. Others wondered whether he was unsure about his mathematics. Whatever the reason, Bayes made an important contribution to a significant problem—and suppressed it.  When he died, he bequeathed his papers to a friend, Richard Price. Price was an amateur mathematician, who developed Bayes’s ideas and probably deserves some of the glory.  After Price, however, Bayes’s theorem lapsed into obscurity until the illustrious French mathematician Pierre Simon Laplace extended and applied it in clever ways in the early 19^th century.

Reverend Bayes set out his ideas in an essay entitled “Essay towards solving a problem in the doctrine of chances.”   It was published posthumously in 1763 and outlined a radically new way to approach and compute probabilities.  Bayes method, however, was largely ignored by statisticians. However, that is not the case today.  According to Sharon McGrayne in her book, The Theory That Would Not Die, Bayes’s theory affects all of us. “ It’s a logic for reasoning about the broad spectrum of life that lies in the gray areas between absolute truth and total uncertainty. We often have information about only a small part of what we wonder about. Yet we all want to predict something based on our past experiences; we change our beliefs as we acquire new information. After suffering years of passionate scorn, Bayes has provided a way of thinking rationally about the world around us.” 

Statisticians find Bayes’s theorem mathematically easy to use but conceptually powerful. It turns out to have incredibly deep consequences and is applicable to almost every fiend of inquiry, from finance and genetics to political science and historical studies.  Bayes is particularly of value today during the coronavirus crisis for use in diagnostic testing.  A new disease like Covid-19 brings a number of uncertainties and there is growing need to discover the basic reproduction number, the rate at which new cases arise, the infection fatality rate and the proportion of people with the virus that it kills.  Bayesian reasoning is a means to integrate what you previously thought with what you have learned and come to a conclusion that incorporates them both.  Today, Bayesian frameworks and methods, powered by computer support are at the heart of various models of epidemiology and other scientific fields.

One spectacular earlier success occurred during the Second World War, when Alan Turing developed Bayes to break Enigma, the German navy’s secret code, and in the process helped to both save Britain and invent modern electronic computers and software. The U.S. Navy used it to search for a missing H-bomb and to locate Soviet subs; RAND Corporation used it to assess the likelihood of a nuclear accident; and Harvard and Chicago researchers used it to verify the authorship of the Federalist Papers.

John Allen Paulos, a famous mathematician, sums up the theory as he says “ simply.”  You begin with a provisional hypothesis about the world and assign to it an initial probability called the prior probability.  After actively collecting or happening upon some potentially relevant evidence, you use Bayes’s theorem to recalculate the probability of the hypothesis in light of the new evidence.  This revised probability is called the posterior probabililty.  Specifically Bayes’s theorem states that the posterior probability of a hypothesis is equal to the product of (a) the prior probability of the hypothesis and (b) the conditional probability of the evidence given the hypothesis, divided by (c) the probability of the new evidence. Simple enough.

Final thoughts

In discovering its value for science, many supporters underwent a near-religious conversion yet had to conceal their use of Bayes’s rule and pretend they employed something else. It was not until the twenty-first century that the method lost its stigma and was widely and enthusiastically embraced.