# An Interview with Ann Varela: Bayes and Bayes Theorem

Michael F. Shaughnessy –

1) The name Bayes is almost synonymous with math. Who was Thomas Bayes, and what exactly is Bayes’s Theorem?

Thomas Bayes was an Englishman who was born in the early 1700’s. His father was a nonconformist Presbyterian minister. The younger Bayes spent the last three decades of his life as a Presbyterian minister in Tunbridge Wells, Kent. There is not much information published about Bayes except that he was educated privately, possibly by Abraham de Moivre, a French mathematician, who was renowned for teaching in London during that period. De Moivre’s specialty was in the area of analytic geometry and the theory of probability.

Another theory involving the origins of Bayes’s education takes place at the Fund Academy in Tenter Alley. Since this school was the only nearby option to obtain a liberal arts education for the ministry, it seems to be a likely possibility.

Bayes attended the University of Edinburgh, as Nonconformists were not permitted to attend Oxford or Cambridge at that time. While at Edinburgh, he studied both logic and theology. It is unclear as to whether he also studied mathematics there as well; however, he alluded to that possibility in one of his writings.

Bayes’s Theorem is used to deduce causes from effects. In other words, it involves conditional probability. The question Bayes was seeking to answer was how a probability of a future event could be calculated based on knowing how many times it had occurred or not occurred previously. Why not examine a scenario to see how Bayes’s Theorem works?

Suppose I have a friend who drinks tea. I have not mentioned whether the friend is male or female, so you may be curious about the probability that I have a female friend. The results of my research found that 27% of males and 31% of females drink tea in the United States. (Source: *What We Eat in America, NHANES 2007-2008, Day 1 dietary intake data, weighted*). I also found that the genders of the U.S. population consist of about 49% males and 51% females. (Source: *U.S. 2010 Census Briefs issued May 2011*).

What is the probability that my friend is female given that the friend drinks tea? This is where Bayes’s Theorem is useful.

First, we must define the events. A: {female} B: {drink tea}

Apply Bayes’s Theorem: P(A|B) = [P(B|A)P(A)]/P(B)

P(A|B) = [(0.31)(0.51)] / [(0.31)(0.51) + (0.27)(0.49)]

P(A|B) = 0.1581 / (0.1581 + 0.1323) = 0.1581 / 0.2904 = 0.5444 = 54.44%

Because we have prior knowledge about tendencies to drink tea, we are able to infer that there is a 54.44% chance that I have a female friend. Does that probability seem correct? Since females constitute slightly more than half of the U.S. population, I would expect the probability of a female friend to be a higher probability than that of a male friend.

2) Probability and statistics are almost inter-linked. What was Bayes’s work in this area?

Bayes’s work in the field of statistics formed the foundation of Bayesian data analysis. The important concept behind Bayesian analysis is that one can use prior information along with observed data to make better decisions. Bayesian techniques have been employed for numerous events. Among them are: combining Bayesian statistics with the Enigma machine to decipher German codes during World War II, determining insurance rates, linking patients’ health histories with cause of disease, gambling, and even detecting spam emails.

3) What is probability inference? Moreover, how does it relate to math today?

Probability inference is the method of calculating the probability that an event will occur in the future, based on the frequency with which the event has occurred in previous trials. According to this Bayesian view, all quantities are one of two kinds: known an unknown to the person making the inference. It follows that quantities are defined by their known values. Unknown quantities are expressed by a joint probability distribution.

4) Inverse probability seems to be linked with Bayes- what is the connection?

The term, “inverse probability” is no longer used. Presently, it is referred to as inferential statistics, which deals with determining an unobserved variable. Back in the seventeenth and eighteenth centuries, astronomers and biologists used inverse probability to estimate a star’s position in the sky on a particular date and time.

Since the 1950’s, inferential statistics has been used to assign a probability distribution to an unobserved variable, which is called Bayesian probability. In Bayesian probability it is important to note that probability is interpreted as reasonable expectation or strength of beliefs, and not frequency (numerical count) or propensity (tendency) of an event occurring.

5) Much of his work seems to have been recognized after his death. Was this due to his involvement in theology or something else?

Upon graduating from the University of Edinburgh, Bayes occupied his time assisting his father in London at St. Thomas’s Meeting House from about 1722-1734. Bayes moved back to Tunbridge Wells, Kent, where he continued his ministry at the Mount Sion chapel for about eighteen years.

Although Bayes had two manuscripts published during his lifetime, one mathematical in nature and the other theological, it seems that he lived under the radar, except for perhaps being recognized for his paper on “fluxions”. Bayes’s election into the Royal Society may be, in part, due to the sophistication and content of his “fluxions” paper. In addition, Bayes only lived until the age of fifty-nine, which means he did not have the luxury of time to spend on mathematical pursuits. I suppose one could argue that he could ponder statistical theories while being a minister.

6) What have I neglected to ask?

Bayes’s Theorem was applied as far back as 1951 by Jerome Cornfield to answer a public health question about the chances of a person contracting lung cancer. His paper assisted epidemiologists to make the connection between a patient’s medical history and measuring the link between a disease and its potential cause. Cornfield was successful in associating smoking and lung cancer, which was substantiated by separate studies conducted in both England and the United States.