An Wells, Kent. There is not much

An
Interview with Ann Varela: Bayes and Bayes Theorem

Michael F. Shaughnessy

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

Thomas
Bayes is 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’s except that he was educated privately, possibly by Abraham de
Moivre, a French mathematician, who was known to be teaching in London during
that time frame.  De Moivre’s specialty
was in the area of analytic geometry and the theory of probability.

The
other 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 is about 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 of female, so you may be curious about
the probability that I have a female friend.
The results of my research found that 27% of men and 31% of women drink
tea in the United States.  I also found
that the U.S. population consists of about 49% males and 51% females.

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}

Research
results:              drink tea:  27% male
vs. 31% female

(Source:
What We Eat in America, NHANES 2007-2008,
Day 1 dietary intake data, weighted.)

U.S.
population:  49% male vs. 51% female

(Source:
U.S. 2010 Census Briefs issued May 2011)

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).

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 was a minister of the Mount Sion chapel for about eighteen years.

Although
Bayes has 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.