October 2011


Many former students have struggled through an undergraduate or graduate university course in statistics. Their only memories are that they are just thankful that the course was over and that they somehow got a passing grade.

Nevertheless, statistics has wide application to important problems in many different disciplines. Since the start of World War II, Bayesian statistics has enjoyed a renaissance with a wide variety of successful applications. In this brief article we focus on some analytic work that helped the Team USA men’s volleyball team win the Olympic gold medal in 2008 in Beijing. We also briefly discuss a successful application of Bayesian statistics[1] to cryptanalysis during World War II. In contrast, we describe a failure of classical statistical methods in Merck’s analysis of a clinical trial study involving its drug Vioxx.

• Team USA Men’s Volleyball[2]

The story starts with Carl McGown, a former Brigham Young University (BYU) volleyball coach and an assistant coach for the Team USA men’s volleyball team in the Athens 2004 Olympics. After reading the book Moneyball—the story of how the Oakland Athletics baseball team employed statistical methods to develop winning teams despite having a lower budget than most other major league baseball teams—McGown contacted his friend Gil Fellingham, a professor of statistics at BYU who holds both a doctorate in biostatistics and a master’s degree in physical education.

McGown gave Fellingham reams of data from international volleyball matches. These data covered every touch in every match of the top teams in the world over a single year. Fellingham and his colleague Shane Reese analyzed these data using Bayesian statistical procedures. The study, which cost $8,000, recommended that a larger proportion of scarce practice time be devoted to serving and suggested specific techniques for doing this. McGown and the U.S. men’s volleyball team followed the suggestions. Although the team was ranked 14th in the world, it managed to finish fourth in the 2004 Olympics. McGown attributed a lot of this success to the advice he got from professors Fellingham and Reece.

In the 2008 Olympics in Beijing, with further help from Fellingham and Reece, the Team USA men’s volleyball squad made it to the top, winning the gold medal.

Among other projects, Fellingham and Reece have since worked with Philadelphia Eagles coach Andy Reid, a former BYU football player, to try to help Reid and the Eagles improve their performance on the football field.

• Clinical Trial Studies

The purpose of clinical trial studies typically is to compare the efficacy of two or more different treatments, or, less typically, the side effects of such treatments. Both issues lead to statistical inference questions in which the task is to compare the proportion of successes to failures of the treatments. Over the years, there have been a number of clinical trial studies that have generated controversy.

One such study was conducted by Merck in 2000 to compare the side effects of its non-steroidal anti-inflammatory drug, Vioxx, to those of naproxen—the generic name of a competing non-steroidal anti-inflammatory drug produced under a variety of brand names such as Aleve. In this study, as reported in Ziliak and McCloskey (2008), eight of the people treated with Vioxx suffered heart attacks during the study versus only one from the group receiving naproxen. Merck used classical statistical testing procedures in its analysis. Ziliak and McCloskey (2008) castigated Merck for claiming that because of “the lack of statistical significance at the 5 percent level, there was no difference in the effects of the two” drugs. Merck would later be sued for such adverse side effects; its losses were in the billions of dollars.

The question is, what does all of this have to do with insurance?

• Applications in Insurance

It turns out that Bayesian statistics has wide application to insurance as well as to sports.

I have used Bayesian statistical procedures to model the Federal Housing Administration’s home equity conversion mortgage (HECM) program.[3] In fact, my article describing that work won the Actuarial Education and Research Fund’s 1989 Practitioner’s Award. Along with Professor Fellingham (2005) and his BYU statistics department colleague Dennis Tolley, I have authored an article on predicting future health claim costs.

Bayesian models can also be employed as part of the NAIC’s Solvency Modernization Initiative. For example, Glenn Meyers (2008) of ISO has written a series of articles in which he uses Bayesian models to predict future claim losses for property/casualty insurance. An advantage of this approach is that the result is an entire probability distribution, so the results are already calibrated. Of course, almost no statistical models are exactly correct and these Bayesian models are no exception. So, even the best models are unlikely to produce precisely calibrated results.

Going beyond insurance, Kanellos (2003) describes an application of Bayes’ Theorem to data searches in his article “18th Century Theory is New Force in Computing.” This should provide an idea of the wide range of problems that are amenable to solution by Bayesian methods.

• Brief History of Bayesian Statistics

Before concluding, let us try to sketch the origin of the Bayesian paradigm of statistics.

Bayes’ Theorem goes back to the Reverend Thomas Bayes, who was born in London around 1702. According to Stigler (1987), “Bayes was an ordained Nonconformist minister in Turnbridge Wells about 35 miles southeast of London.” When Bayes died in 1761, he left £100 and his scientific papers to his friend, Richard Price. After adding an introduction and an appendix, Price presented Bayes’ essay “Toward Solving a Problem in the doctrine of Chance” to the Royal Society.

The famous French astronomer, probabilist and mathematician Pierre Simon Laplace, who lived from 1749-1827, both championed and extended Bayes’ work. In his text Essai philosophie sur les probabilities (Philosophical Essay on Probabilities), Laplace (1825) described a mathematical framework for conducting statistical inference. This constituted the essence of Bayesian inference. During the later part of the 19th century, Bayesian statistics fell out of favor with most statisticians who, preferred the competing classical or frequentist paradigm of statistics instead.

However, Bayesian statistics has enjoyed a renaissance over the last 70 years. Its first major application during this period was its use by British mathematicians to break the German cipher machine known as “Enigma.” These mathematicians preferred their own theory based on Bayes’ Theorem to the competing statistical theory based on relative frequencies championed by their compatriot R.A. Fisher.[4] While it is a large stretch to say that Bayes’ Theorem won World War II for the British, it certainly helped their cause.

[1] The Bayesian paradigm entails the application of Bayes’ Theorem which, in turn, involves the concept of conditional probabilities.[Return]

[2] This section is based on Walch [2005].[Return]

[3] See DiVenti and Herzog (1991).[Return]

[4] See Hodges (1992; page 199).[Return]


DiVenti, T.R. and T.N. Herzog, “Modeling Home Equity Conversion Mortgages,” Transactions of the Society of Actuaries, Vol. XLIII (1991), 261-275.

Fellingham, G.W., H.D. Tolley and T.N. Herzog, “Comparing Credibility Estimates of Health Insurance Claim Costs,” North American Actuarial Journal, Vol. 9, No. 1 (2005).

Hodges, A., Alan Turing: The Enigma. London: Vintage, Random House (1992).

Kanellos, M. 18th-century theory is new force in computing. CNET News.com, February 18, 2003 (available at http://news.cnet.com/Old-school-theory-is-a-new-force/2009-1001_3-984695.html).

Laplace, P.S., Philosophical Essay on Probabilities, translated from the fifth French edition of 1825 by Andrew I. Dale. New York: Springer (1995).

Meyers, G., “Stochastic Loss Reserving with the Collective Risk Model,” Casualty
Actuarial Society E-Forum
, Fall, 2008.

Stigler, S.M., The History of Statistics: The Measurement of Uncertainty Before 1900. Cambridge: Harvard University Press, (1986).

Walch, T., “Y. statisticians may help Eagles’ game,” Deseret News, January 14, 2005 (available at http://www.deseretnews.com/article/600104761/Y-statisticians-may-help-Eagles-game.html).

Ziliak, S.T. and D.N. McCloskey, The Cult of Statistical Significance. Ann Arbor, Mich.: The University of Michigan Press (2008).

Copyright © 2011 NAIC, All rights reserved.