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.