Bayesian in actuarial application

PRINCIPAL APPLICATIONS OF BAYESIAN METHODS

IN ACTUARIAL SCIENCE: A PERSPECTIVEUdi E. Makov*

ABSTRACT

Bayesian ideas were introduced into actuarial science in the late 1960s in the form of empiricalcredibility methods for premium setting. The advance of the Bayesian methodology was slow dueto its subjective nature and to the computational difficulties associated with the full Bayesiananalysis. This paper offers a brief survey of Bayesian solutions to some actuarial problems anddiscusses the current state of research.

1. INTRODUCTION

Bayesian ideas and techniques were introducedinto actuarial science in a big way in the late1960s when the papers of Buhlmann (1967, 1969)and Buhlmann and Straub (1970) laid down thefoundation to the empirical Bayes credibility ap-proach, which is still being used extensively inthe insurance industry.

Bayesian methodology is used in various areaswithin actuarial science. This paper does not aimat providing an extensive review of all applica-tions of this methodology, but rather seeks topoint out the main areas of application and iden-tify the main themes of contemporary research.

The most important areas of application are:

● Experience rating (including credibility the-ory), where premiums are set given the accu-mulated past claims in a portfolio.

● Experience reserving and compound claimmodeling. The former deals with the amount ofreserves to be kept by an insurance companyand the latter with the assessment of the accu-mulated claims and their impact on the compa-ny’s financial standing.

An account of Bayesian statistics in actuarial sci-ence can be found in a book by Klugman (1992)which concentrates on the Bayesian approach to

credibility. For recent review papers see Makov etal. (1996), Schmidt (1998), and Pacakova (1997).

Section 2 introduces the basic ideas of experi-ence rating and credibility theory. It then dis-cusses the broader theoretical and computationalissues relating Bayesian solutions to experiencerating, which are relevant to actuarial science asa whole. Section 3 discusses experience reservingand compound loss models. Concluding remarksand suggested research directions are given insection 4.

2. THEORETICAL AND COMPUTATIONAL

ISSUES: THE CASE OF EXPERIENCE

RATING

In experience rating, uij (i 5 1, . . . I, j 5 1, . . . J),the risk parameter, is regarded as the total char-acterization of the risk of contract i at time j.Given uij, the actual claims associated with con-tract i, Xi1, Xi2 . . . are stochastically independentand follow a distribution f(xijuuij) and the u’s arei.i.d. and follow a prior distribution U[, com-monly called structure distribution. The fair pre-mium is denoted by m(uij) 5 E[Xijuuij] and theactual premium is calculated by E[Xi,J11uD] 5E[m(uij)uD], where D is the available data, in thiscontext, Xij, (i 5 1, . . . I, j 5 1, . . . J). SinceE[m(uij)uD], the exact credibility, was typicallyhard to calculate, empirical Bayes techniqueswere employed to produce an estimate of theexact credibility by means of the famous credibil-ity formula zx# i 1 (1 2 z)m, where m is the meanof U[ and x# i the mean claim of the ith contract;

* Udi E. Makov, Ph.D., is Director, Actuarial Programme, Departmentof Statistics, University of Haifa, Haifa 31905, Israel, e-mail:[email protected].

53

z, the credibility factor, is chosen to produce thebest (m.s.e.) linear empirical Bayes estimator ofthe exact credibility. Typically z 5 aJ/(aJ 1 s2),where a 5 var[m(uij)], and s2 5 E{var[Xijuuij]} areunknown and estimated from the data.

The credibility model has been generalizedover the years and in most cases maintained itssimple linear formula and empirical Bayes flavor.For further reading on the various credibilitymodels, see a brief survey in Makov et al. (1996),Waters (1987), Goovaerts and Hoogstad (1987),Goovaerts et al. (1980), and Herzog (1994).

The empirical Bayes credibility model was asuccessful practical compromise at a time whenopposition to the Bayesian methodology was cen-tered on two major points: (1) opposition to thesubjective nature of Bayesian statistics and thesearch for a more objective tool, and (2) reserva-tions about its applicability, especially as closed-form analytical solutions were not widely avail-able.

The traditional credibility formula constitutes,in a way, a distribution-free methodology. Buthow accurate is it for particular claim distribu-tions? Herzog (1990) examined the compatibilityof the Bayesian and Buhlmann models andshowed that the Buhlmann credibility estimate isthe best linear approximation of the Bayesianestimate of the fair premium. Following Jewell(1974), it was shown that the credibility estimateis equal to the Bayesian estimate for a large classof problems (exponential family/conjugate pri-ors). See also Schmidt (1980), Goel (1982), andGerber (1995). Landsman and Makov (1998,1999a) established that the simple credibility for-mula is also correct, in a Bayesian sense, for claimdistributions belonging to the Exponential Dis-persion Model (EDF) (see Jorgensen 1987, 1992),thus allowing computationally simple estimationof the fair premium for members of this family.

Landsman and Makov (1999b, 1999c) estab-lished a relationship between stochastic approx-imation and credibility. Using this relationship, ageneralized sequential credibility was suggestedand an optimal stepwise gain sequence derived toproduce an estimate of the exact credibility forclaim distribution belonging to the location dis-persion family and to the symmetric locationfamily. This new methodology allows the deriva-tion of near exact credibility for relevant claimdistributions like the log-gamma and log-normal,

for which the traditional credibility methods pro-duced only suboptimal solutions. See also Taylor(1977).

The issue of prior specification has recentlybeen dealt with in various ways. Young (1997)suggested using kernel density estimation to esti-mate the prior distribution of the parameter ofinterest. To enhance accuracy, Young (2000) sug-gested employing a loss function, which is a linearcombination of a squared-error term and anotherterm designed to reduce divergence. Informationmeasures were used to establish a prior distribu-tion for the dispersion parameter l of the EDF;Landsman and Makov (1998) used the maximumentropy principle and Landsman and Makov(1999a) minimized the Fisher information. Thislatter criterion was also used in Landsman andMakov (2001) to establish a prior distribution forl in conjunction with knowledge on the probabil-ity that a claim exceeds a certain threshold, thusallowing for information on tail behavior to affectthe premium. Gomez-Deniz et al. (1999) carriedout robustness analysis with respect to the priordistribution by considering a contaminated classof prior distribution.

REMARK

All credibility models and their extensions men-tioned above (with the exception of Young 2000)assume, in effect, a squared error-loss functionand, hence, the choice of the predictive mean asexact credibility. Other loss functions can be con-sidered, each producing a different premiumprinciple. See Heilmann (1989), Kamps (1998),and Gomez-Deniz et al. (1999).

For many years, the implementation of Bayes-ian models was only possible for simple low di-mensional problems (see Klugman, 1992, for an-alytical approximations suitable for such cases).After all, the evaluation of E[Xi,J11uD] 5E[m(uij)uD] would typically require (I 2 1) (J 21)1 1 multiple integrals, an impossible task formost heterogeneous portfolios. During the lastdecade, however, there has been an increasingrealization that the computations required for fullBayesian analysis can be carried out effectivelyby means of simulation-based methods and, inparticular, Markov chain Monte Carlo (MCMC)methods (Gilks et al., 1996). In effect, the expe-rience-rating problem discussed above can befully investigated on a PC by means of an hierar-

54 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 5, NUMBER 4

chical Bayes model for a portfolio as large asneeded. For an introduction to MCMC methodsand their actuarial applications see Makov et al.(1996) and Scollnik (1996).

3. LOSS RESERVING AND COMPOUND

LOSS MODELS

Loss reserves are needed whenever losses remainunpaid at the end of a year, typically as a result ofclaims incurred but not yet reported (IBNR) orreported but not yet settled (RBNS).

Let the random variables Xij (i 5 1, . . . , I; j 51, . . . I 2 i 1 1) denote claim figures (or lossratios, claim frequencies, and so forth.) for the ith

year of origin at the jth development year. Thedata constitute a triangle (the so called runofftriangle), where the upper triangle is given andthe lower triangle is to be estimated. Since theproblem is crucial for insurance companies, a lotof research has been invested in developing effec-tive methodologies (Taylor 2000). However, rela-tively little was done from the Bayesian perspec-tive. Verrall (1990) carried a Bayesian analysis ofthe chain ladder model, which can be interpretedas the two-way model:

log~Xij! 5 m 1 aI 1 bJ 1 eij,

but no attempt was made to treat the unknownvariances in a fully Bayesian manner.

A complete hierarchical Bayes model was im-plemented by Hazan and Makov (2000), whereMCMC was used to estimate the parameters oftwo models: the chain ladder model and a switch-ing regression model that allows the delayedclaims to increase up to a point and then decreaseover time. MCMC was also employed in Ntzoufrasand Dellaportas (n.d.) where various models foroutstanding claims problems are discussed andwhere claim count uncertainty is incorporated.Bayesian methods were also used by Haastrupand Arjas (1996) for estimating claim counts andamounts in individual claim data and by Jewell(1989) and de Alba et al. (1997) for estimatingclaim counts. State-space models (or Kalman fil-ters), which are dynamic Bayesian models, werealso suggested for modeling loss reserves. See DeJong and Zehnwirth (1983).

The benefit of the Bayesian approach is in pro-viding the decision maker with a posterior pre-

dictive distribution for every entry in the lowertriangle of the runoff triangle. Such a distributionallows the assessment of the required reserves interms not only of point estimators but also ofBayesian confidence intervals and probabilisticindication on the chance that the amount of afuture claim exceeds a given threshold.

The common risk model for aggregate claimsS 5 Y11, . . . 1 YN, where Y1, Y2, . . . representthe amounts of successive claims (assumed i.i.d.),is a dual stochastic process with a claim distribu-tion f(yuu) and a count distribution g(nuw). Thetraditional practice is to fit distributions to N (forvarious count processes see Schmidt 1998) and toY and then to employ recursive algorithms tocalculate the compound distribution of S. Aspointed out by Dickson et al. (1998), the majordrawback of this approach is that the fitted dis-tributions are assumed to be known with cer-tainty and, thus, there is no account of the pa-rameter estimation error. For an example ofrecursive algorithms, see Panjer (1981), Schroter(1991), and Sundt (1992).

The Bayesian model (see, for example, Ryt-gaard 1990 and Hesselager 1993) is aimed at eval-uating the posterior distribution of u and w andthe predictive distribution of S, given historicaldata on claims and on the total number of claimsat various periods. The MCMC method could al-low, in principle, a full implementation of theBayesian model. No account of a thorough studyof this approach has yet been reported. For apreliminary study see Pai (1997).

4. CONCLUSION AND FURTHER WORK

Recent attempts to implement Bayesian methodsin actuarial science proved promising. It has longbeen appreciated by many that the Bayesian par-adigm offers a more complete framework of anal-ysis as it allows the uncertainty of the estimationerror to be incorporated into the learning pro-cess. The posterior distribution contains a wealthof information well beyond the point estimatorscommonly used in the traditional non-Bayesiananalysis.

The MCMC method, which has revolutionizedBayesian statistics, is still to be adopted moreextensively in actuarial science. There are newproblems to be tackled, and there is a need toadopt the more recent advances in MCMC, which

55PRINCIPAL APPLICATIONS OF BAYESIAN METHODS IN ACTUARIAL SCIENCE: A PERSPECTIVE

have appeared so far only in the statistical liter-ature (Roberts and Rosenthal 1998).

The reservations about the use of prior distri-butions can be answered by attempts to imple-ment existing theories relating to noninformativepriors and to robust Bayesian analysis in whichthe sensitivity to the choice of prior distributionis examined (Moreno 2000).

REFERENCES

BUHLMANN, H. 1967. “Experience Rating and Probability,” ASTINBulletin 4: 199–207.

———. 1969. “Experience Rating and Probability,” ASTIN Bul-letin 5: 157–65.

BUHLMANN, H., AND E. STRAUB. 1970. “Glaubwurdigkeit furSchaden-satze,” Mitteilungen der Vereinigung Schweiz-erischer Versicherungsmathe-matiker 70: 111–33.

DE ALBA, E., T. M. MORENO, AND M. JUARES. 1997. “Bayesian Esti-mation of IBNR Reserves.” Technical Report, I.T.A.M.,Mexico.

DE JONG, P., AND B. ZEHNWIRTH. 1983. “Claims Reserving StateSpace Models and the Kalman Filter,” Journal of the Insti-tute of Actuaries 110: 157–81.

DICKSON, D. C. M., L. M. TEDESCO, AND B. ZEHNWORTH. 1998. “Pre-dictive Aggregate Claims Distributions,” The Journal ofRisk and Insurance 65: 689–709.

GERBER, H. U. 1995. “A Teacher’s Remark on Exact Credibility,”ASTIN Bulletin 25: 189–92.

GILKS, W. R., S. RICHARDSON, AND D. J. SPIEGELHALTER eds. 1996.Markov Chain Monte Carlo in Practice. London: Chap-man & Hall.

GOEL, P. K. 1982. “On Implications of Credible Means BeingExact Bayesian,” Scandinavian Actuarial Journal 41–6.

GOMEZ-DENIZ, E., A. HERNANDEZ-BASTIDA, AND F. J. VAZQUEZ-POLO.1999. “The Esscher Premium Principle in Risk Theory: ABayesian Sensitivity Study,” Insurance: Mathematics &Economics 25: 387–95.

GOOVAERTS, M. J., AND W. J. HOOGSTAD. 1987. Credibility Theory.Rotterdam: National Nederlanden N.V.

GOOVAERTS, M. J., R. KASS, A. VAN HEERWAARDEN, AND T. BAUWELINCKX.1990. Effective Actuarial Methods. North Holland.

HAASTRUP, S., AND E. ARJAS. 1996. “Claims Reserving in Continu-ous Time; A Nonparametric Bayesian Approach,” ASTINBulletin 26: 139–64.

HAZAN, A., AND U. E. MAKOV. 2000. “Recent Advances in BayesianMethods in Loss Reserving.” Paper presented at the ASTINColloq., September, Porto Cervo.

HEILMANN, W. 1989. “Decision Theoretic Foundations of Credi-bility Theory,” Insurance: Mathematics & Economics 8:77–95.

HERZOG, T. N. 1990. “Credibility: The Bayesian Model VersusBuhlmann’s Model,” Transactions of the Society of Actu-aries 41: 43–88.

———. 1994. Introduction to Credibility Theory. Winsted: AC-TEX Publications.

HESSELAGER, O. 1993. “A Class of Conjugate Priors with Applica-tions to Excess-of-loss Reinsurance,” ASTIN Bulletin 23:77–93.

JEWELL, W. S. 1974. “Credible Means are Exact Bayesian forExponential Families,” ASTIN Bulletin 8: 77–90.

———. 1989. “Predicting IBNYR Events and Delays,” ASTINBulletin 19: 25–55.

JORGENSEN, B. 1987. “Exponential Dispersion Models (with dis-cussion),” Journal of the Royal Statistical Society Series B49: 127–62.

———. 1992. “Exponential Dispersion Models and Extensions:A Review,” International Statistical Review 60: 5–20.

KAMPS, U. 1998. “On a Class of Premium Principles Including theEsscher Principle,” Scandinavian Actuarial Journal 71–85.

KLUGMAN, S. A. 1992. Bayesian Statistics in Actuarial Science.Boston: Kluwer.

LANDSMAN, Z., AND U. E. MAKOV. 1998. “Exponential DispersionModels and Credibility,” Scandinavian Actuarial Journal89–96.

———. 1999a. “Credibility Evaluation for the Exponential Dis-persion Family,” Insurance: Mathematics & Economics24: 33–9.

———. 1999b. “On Stochastic Approximation and Credibility,”Scandinavian Actuarial Journal 1: 15–31.

———. 1999c. “Sequential Credibility Evaluation for Symmet-ric Location Claim Distributions,” Insurance: Mathemat-ics & Economics 24: 291–300.

———. 2001. “On Credibility Evaluation and the Tail Area ofthe Exponential Dispersion Family” Insurance: Mathemat-ics & Economics 27: 277–83.

MAKOV, U. E., A. F. M. SMITH, AND Y-H. LIU. 1996. “BayesianMethods in Actuarial Science,” The Statistician 45: 503–15.

MORENO, E. 2000. “Global Bayesian Robustness for Some Classesof Prior Distributions,” Robust Bayesian Analysis, pp. 45–70, Lecture Notes in Statistics. Vol. 152. New York:Springer.

NTZOUFRAS, I., AND P. DELLAPORTAS. (n.d.). “Bayesian Modelling ofOutstanding Liabilities Incorporating Claim Count Uncer-tainty. Submitted for publication.

PACAKOVA, V. 1997. “The Bayesian Inference in Actuarial Sci-ence,” Central European Journal of Operations Researchand Economics 5: 255–68.

PAI, J. S. 1997. “Bayesian Analysis of Compound Loss Distribu-tions,” Journal of Econometrics 79: 129–46.

PANJER, H. H. 1981. “Recursive Evaluation of Family of Com-pound Distributions,” ASTIN Bulletin 12: 22–6.

ROBERTS, G. O., AND J. S. ROSENTHAL. 1998. “Markov-Chain MonteCarlo: Some Practical Implications of Theoretical Results,”Canadian Journal of Statistics 26: 5–31.

RYTGAARD, M. 1990. “Estimation in the Pareto Distribution,”ASTIN Bulletin 20: 201–16.

SCHMIDT, K. D. 1980. “Convergence of Bayes and CredibilityPremium,” ASTIN Bulletin 20: 167–72.

———. 1998. “Bayesian Models in Actuarial Mathematics,”


Mathematical Methods in Operations Research 48: 117–46.

SCHRORER, K. J. 1991. “On a Family of Counting Distributions andRecursions for Related Compound Distributions,” Scandi-navian Actuarial Journal 161–175.

SCOLLNIK, D. P. M. 1996. “An Introduction to Monte Carlo MarkovChain and Their Actuarial Applications,” Casualty Actu-arial Society 158: 114–65.

SUNDT, B. 1992. “On Some Extensions of Panjer’s Class of Count-ing Distributions,” ASTIN Bulletin 22: 61–80.

TAYLOR, G. 1977. “Abstract Credibility,” Scandinavian Actuar-ial Journal 149–68.

———. 2000. Loss Reserving—An Actuarial Perspective: Bos-ton: Kluwer.

VERRALL, R. J. 1990. “Bayes and Empirical Bayes Estimation forthe Chain Ladder Model,” ASTIN Bulletin 20: 217–38.

WATERS, H. R. 1987. An Introduction to Credibility Theory.London: Institute of Actuaries.

YOUNG, V. R. 1997. “Credibility Using Semiparametric Models,”ASTIN Bulletin 27: 273–85.

———. 2000. “Credibility Using Semiparametric Models and aLoss Function with a Constancy Penalty,” Insurance:Mathematics & Economics 26: 151–56.

DISCUSSIONS

MARJORIE A. ROSENBERG*I would like to commend the author for writing anarticle in the actuarial literature about the use ofBayesian techniques in actuarial science. Agree-bly, Bayesian methods prior to the 1990s werenot as visible as they are today due to their com-putational difficulties of high-level integrals. Withthe increased power of computers and the ad-vancement of Bayesian statistical techniques, theuse of Bayesian methods has increased in the1990s. Approaches that once seemed too difficultnumerically because of complex computationsnow are commonplace. Simulation of posteriordistributions of parameters and of functions ofthe simulated parameters are easily produced.

The author states that his paper “does not aimat providing an extensive review of all applica-tions of this methodology, but rather seeks topoint out the main areas of application and toidentify the main themes of contemporary re-search.” He continues by stating that the “most

important areas of application are experience rat-ing . . . and experience reserving.” These may bethe most prevalent areas of Bayesian actuarialresearch due to their historical tradition in actu-arial work. While I agree that these are importantareas of research, I would not agree that these arethe only main themes of Bayesian applications inactuarial research.

A couple of years ago, the Society of ActuariesWeb site displayed material relating a definitionof “actuarial modeling.” This material is no longeravailable on the Web site, but it said: (1) “Anactuarial risk is a phenomenon that has economicconsequences and is subject to uncertainty withrespect to one or more of the actuarial risk vari-ables: occurrence, timing, and severity,” and (2)“An actuarial model is a stochastic model, to-gether with a present value model, if applicable,of actuarial risks, based on assumptions about theprobabilities that will apply to the actuarial riskvariables in the future, including assumptionsabout the future environment.”

Through this definition of actuarial modeling, Iview “actuarial research” in a very broad context.Use of Bayesian methods in health care-relatedresearch has also increased in the last few yearsfor the same reasons as mentioned above. I wouldlike to introduce readers to some of the literatureusing Bayesian modeling in the health care prac-tice area. A search on Medline, a health caresearch engine on the Internet, using the keyword“Bayesian” and covering the years 1966 to thepresent produced 2,120 citations.1

A sample of the journals mentioned in theBayesian citation include American Journal ofCardiology, American Journal of EpidemiologyAmerican Journal of Public Health, Health Eco-nomics, Health Services Research, Inquiry, Jour-nal of Clinical Oncology, Medical Care, Medical

* Marjorie A. Rosenberg, Ph.D., F.S.A., is a Professor at the Universityof Wisconsin, School of Business and Department of Biostatistics andMedical Informatics in the Medical School, 975 University Ave., Mad-ison, WI 53706, e-mail: [email protected].

1 “Medline is produced by the U.S. National Library of Medicine andgathers information from Index Medicus, Index to Dental Literature,and International Nursing, as well as other sources of coverage in theareas of allied health, biological and physical sciences, humanitiesand information science as they relate to medicine and health care,communication disorders, population biology, and reproductive bi-ology. More than 10.8 million records from more than 3,900 journalsare indexed, plus records formally indexed in Healthstar, Bioethicslineand AIDSline. Abstracts are included for about 51% of the records”(from the Medline database description).


Decision Making, Radiology, Statistical Methodsin Medical Research, and Statistics in Medicine.

I focused on a subset of these journals thatcontains articles in which actuaries may be inter-ested. A search using a combination of the key-word “Bayesian” and the journal title Statistics inMedicine yielded 178 citations. The earliestBayesian paper listed was by McPherson (1982)concerning a method to incorporate uncertaintyinto the size of the sample for uncertainties at thedesign stage of a clinical trial. These 178 articlesrange in content from clinical trial methodology,results, and economic evaluation to costs, clinicalpractice decision making, drugs, meta-analysis,surveillance, screening and disease-mapping, andsurvival models.

As examples, O’Hagan et al. (2001) in a recentarticle develop a Bayesian computation of theincremental cost-effectiveness acceptability curvefor assessing the relative cost-effectiveness of twotreatments in health economics, where data onboth costs and efficacy are available from a clin-ical trial. Ashby and Smith (2000) wrote a reviewarticle concerning the emergence of evidence-based medicine and the growing use of Bayesianstatistics in medical applications. Gelfand andWang (2000) focused on quantifying the cumula-tive risk associated with false-positive results onrepeated screening procedures for medical condi-tions, both at the population and the individuallevel. They developed actuarial models for lifetable data and added a Cox regression to enableindividual level modeling. Daniels and Gatsonis(1997) formulated a hierarchical polytomous re-gression model and applied it to the analysis ofvariations in the utilization of alternative cardiacprocedures in a national cohort of elderly Medi-care patients who had an acute myocardial infarc-tion during 1987. They examined how the rates ofcardiac procedures depend on patient-level char-acteristics, including age, gender, and race, andwhether there exist interstate differences and re-gional patterns in the use of these procedures.

The journal Health Economics yielded seven ci-tations for Bayesian studies. One paper by Hamilton(1999) concerned health care costs of Medicareenrollees in health maintenance organizations(HMOs). He found that HMOs select individualswho are less likely to have positive health careexpenditures prior to enrollment, but he did not

find evidence that HMOs disenroll high-cost pa-tients. Claxton (1999) believed the current regula-tion of new pharmaceuticals was inefficient becauseit demanded arbitrary amounts of information, thetype of information demanded was not relevant todecision makers, and the same standards of evi-dence were applied across different technologies.He advocated the use of Bayesian decision theoryand an analysis of the value of both perfect andsample information to consider the efficient regula-tion of new pharmaceuticals.

The journal Medical Decision Making yielded36 citations for Bayesian studies. Parmigiani et al.(1997) developed a framework to quantify uncer-tainty about costs, effectiveness measures, andmarginal cost-effectiveness ratios in complex de-cision models such as in their example of astroke-prevention policy. Rosenberg et al. (1999)extended the work of Carlin (1992) to develop aBayesian method of computing a measure of pop-ulation health. Observed health-adjusted life ex-pectancy (HALE) is an indicator of populationhealth. HALE is an adjustment to traditional ac-tuarial life expectancy, which weights the year asbetween 0 and 1 to adjust for quality of life duringa year, rather than fixing the measure as 1 for lifeexpectancy. HALE can be used as a tool to helpdefine social policy, as well as a method to assessthe needs of a community.2

The journal Medical Care yielded five citationsof Bayesian studies. Cressie’s work (1993) in-volved the prediction of small-area incidencerates from spatially contiguous regions. Miller etal. (1993) estimated what they described as phy-sician “costliness” using outpatient data obtainedfrom a general medicine practice of an urbanhealth care facility. Bay et al. (1983) presentedmethods used for validating a patient classifica-tion system that was based on the concept oftypes of care (PCTC system). The PCTC systemwas developed to improve placement decisionsfor long-term care patients and also to provide

2 Interestingly, Bayesian methods and graduation has roots in theactuarial literature in the 1960s and later. Jones (1965), Kimeldorfand Jones (1967), and Hickman and Miller (1977) conducted studiesof actuarial methods in graduation. Carlin (1992) and Carlin andKlugman (1993) discussed the use of hierarchical Bayesian models ingraduation techniques. Graduation methods in actuarial science arementioned in Makov et al. (1996).


information required for planning in the field oflong-term care.

The journal Statistical Methods in Medical Re-search yielded 13 citations of Bayesian studies.Sheiner and Wakefield (1999) discussed the vitalrole that population (hierarchical) modeling canplay in the drug development process, in thatpopulation pharmacokinetic/pharmacodynamicmodels can provide reliable predictions of an in-dividualized dose-exposure-response relation-ship. “A predictive model of this kind can be usedto simulate and hence design clinical trials, findinitial dosage regimens satisfying an optimalitycriterion on the population distribution of re-sponses, and individualized regimens satisfyingsuch a criterion conditional on individual fea-tures, such as sex, age, etc.” (p. 183).

Clayton (1994) reviewed the work over the last20 years for the analysis of data with recurrentevents, while addressing subject-level heteroge-neity. DeAngelis et al. (1993), in their study onAIDS and public health issues were concernedwith the uncertainty in the three components ofthe back calculation method (knowledge of re-ported AIDS cases, information on the time be-tween HIV infection and onset of AIDS, and as-sumptions on the rate at which infections occurand the increasingly available information on HIVprevalence) that must be considered to providerealistic projections. Their paper discussed waysof acknowledging uncertainty and suggests aBayesian formulation.

One of my areas of research is in the statisticalapplication of methods to health care resourceutilization and health care policy. With nationalhealth expenditures in the United States growingfrom $73 billion (7% of the gross domestic prod-uct) in 1970 to $1.2 trillion (14% of GDP) in 1999(projected), extensive research is being con-ducted to model and predict health care spending(see HCFA 1999).

My dissertation focused on the development ofa Bayesian statistical model to predict whether aclaim was acceptable or not, nonacceptableclaims (NACs), from the viewpoint of generallyaccepted clinical protocols. A statistical systemwas proposed to monitor NACs (Rosenberg 1999;Rosenberg et al. 1999; and Rosenberg and Griffith2000) with subsequent research proposing twodifferent approaches to monitor for changes ofthe rate of nonacceptable claims (Rosenberg

2001a and Rosenberg 2001b). The premise of thestatistical system is that inexpensive methods totarget audit resources effectively would provide acontinuous monitoring of health care costs andhelp curb NACs.

This research was expanded to examine codingof the diagnosis-related groups (DRGs) for healthcare claims. Both Medicare and non-Medicare in-patient claims are affected, as private insurershave adopted a payment scheme that is based onthe DRG system (Rosenberg et al. 2000). My ini-tial research showed the viability of a statisticalsystem to determine whether an individual claimshould be audited to determine whether its DRGcoding is incorrect. The statistical system even-tually will have three parts: detection, adaptation,and control. At the heart of the statistical systemis a hierarchical Bayesian model to predictwhether the DRG coding is incorrect.

In conclusion, I wanted to introduce to theactuarial community the breadth of literature inother journals on actuarial-related Bayesian re-search in health care. Topics span from studiesrelating to estimating incidence and prevalencerates, meta-analyses of studies, longitudinal stud-ies with respect to modeling costs and decisionmaking in health care utilization.

REFERENCES

ASHBY, D., AND A. F. SMITH. 2000. “Evidence-Based Medicine asBayesian Decision-Making,” Statistics in Medicine 19(23):3291–305.

BAY, K. S., P. LEATT, AND S. M. STINSON. 1983. “Cross-Validation ofa Patient Classification Procedure: An Application of the UMethod,” Medical Care 21(1): 31–47.

CARLIN, B. 1992. “A Simple Monte Carlo Approach to BayesianGraduation,” Transactions of the Society of Actuaries 44:55–76.

CARLIN, B., AND S. KLUGMAN. 1993. “Hierarchical Bayesian WhitakerGraduation,” Scandinavian Actuarial Journal 183–96.

CLAXTON, K. 1999. “Bayesian Approaches to the Value of Infor-mation: Implications for the Regulation of New Pharma-ceuticals,” Health Economics 8(3): 269–74.

CLAYTON, D. 1994. “Some Approaches to the Analysis of Recur-rent Event Data,” Statistical Methods in Medical Research3(3): 244–62.

CRESSIE, N. 1993. “Regional Mapping of Incidence Rates UsingSpatial Bayesian Models,” Medical Care, 31(5 Suppl.):YS60–5.

DANIELS, M., AND C. GATSONIS. 1997. “Hierarchical PolytomousRegression Models with Applications to Health ServicesResearch,” Statistics in Medicine 16(20): 2311–25.


DE ANGELIS, D., N. E. DAY, S. M. GORE, W. R. GILKS, AND M. A.MCGEE. 1993. “AIDS: The Statistical Basis for PublicHealth,” Statistical Methods in Medical Research 2(1):75–91.

GELFAND, A. E., AND F. WANG. 2000. “Modelling the CumulativeRisk for a False-Positive Under Repeated ScreeningEvents,” Statistics in Medicine 19(14): 1865–79.

HAMILTON, B. H. 1999. “HMO Selection and Medicare Costs:Bayesian MCMC Estimation of a Robust Panel Data TobitModel with Survival,” Health Economics 8(5): 403–14.

HEALTH CARE FINANCING ADMINISTRATION (HCFA). 1999. “NationalHealth Care Expenditures Projections.” Online at http://www. hcfa.gov/stats/NHE-Proj/proj1998/proj1998.pdf.

HICKMAN, J., AND R. MILLER. 1977. “Notes on Bayesian Gradua-tion,” Transactions of the Society of Actuaries 29: 1–21.

JONES, D. 1965. “Bayesian Statistics,” Transactions of the Soci-ety of Actuaries 17: 33–57.

KIMELDORF, G., AND D. JONES. 1967. “Bayesian Graduation,”Transactions of the Society of Actuaries 19: 66–112.

MAKOV, U., A. F. M. SMITH, AND Y-H. LIU. 1996. “Bayesian Methodsin Actuarial Science,” The Statistician 45(4): 503–15.

MCPHERSON, K. 1982. “On Choosing the Number of Interim Anal-yses in Clinical Trials,” Statistics in Medicine 1(1): 25–36.

MILLER, M. E., S. L. HUI, W. M. TIERNEY, AND C. J. MCDONALD. 1993.“Estimating Physician Costliness: An Empirical Bayes Ap-proach,” Medical Care 31 (5 Suppl.): YS16–28.

O’HAGAN, A., J. W. STEVENS, AND J. MONTMARTIN. 2001. “BayesianCost-Effectiveness Analysis from Clinical Trial Data,” Sta-tistics in Medicine 20(5): 733–53.

PARMIGIANI, G., G. P. SAMSA, M. ANCUKIEWICZ, J. LIPSCOMB, V. HAS-SELBLAD, AND D. B. MATCHAR. 1997. “Assessing Uncertainty inCost-Effectiveness Analyses: Application to a Complex De-cision Model,” Medical Decision Making 17(4): 390–401.

ROSENBERG, M. 1998. “A Statistical Control Model for UtilizationManagement Programs,” North American Actuarial Jour-nal 2(2): 77–87.

———. 2001a. “A Statistical Method for Monitoring a Change inthe Rate of Nonacceptable Inpatient Claims,” North Amer-ican Actuarial Journal 5(4): 74–83.

———. 2001b. “A Decision-Theoretic Method for Assessing aChange in the Rate of Non-Acceptable Inpatient Claims,”Health Services and Outcomes Research Methodology,2(1): 19–36.

ROSENBERG, M., R. ANDREWS, AND P. LENK. 1999. “A HierarchicalBayesian Model for Predicting the Rate of Non-AcceptableIn-Patient Hospital Utilization,” Journal of Business andEconomic Statistics 17(1): 1–8.

ROSENBERG, M., D. FRYBACK, AND D. KATZ. 2000. “A Statistical Modelto Detect DRG Upcoding,” Health Services and OutcomesResearch Methodology 1(3–4): 233–52.

ROSENBERG, M., D. FRYBACK, AND W. LAWRENCE. 1999. “ComputingPopulation-Based Estimates of Health-Adjusted Life Ex-pectancy,” Medical Decision Making 19(1): 90–7.

ROSENBERG, M., AND J. GRIFFITH. 2000. “A Management Tool forControlling the Rate of Non-Acceptable Inpatient HospitalClaims,” Inquiry 36(4): 461–70.

SHEINER, L., AND J. WAKEFIELD. 1999. “Population Modelling in

Drug Development,” Statistical Methods in Medical Re-search 8(3): 183–93.

M. MENDOZA*I would like to congratulate the editors for pro-posing a paper on Bayesian methods in actuarialsciences for discussion. The author has presenteda timely paper that deals with specific applica-tions of the Bayesian paradigm to the solution ofsome actuarial problems, and I hope that its pub-lication will contribute to further development ofthis area of research.

To have a general idea of the potential contri-bution of the Bayesian methods to the actuarialsciences, it might be useful to start with twoquotations. From the book by Bernardo andSmith (1994) we have: “Bayesian Statistics offersa rationalist theory of personalistic beliefs in con-texts of uncertainty, with the central aim of char-acterising how an individual should act in orderto avoid certain kinds of undesirable behavioralinconsistencies. The theory establishes that ex-pected utility maximisation provides the key tothe ways in which beliefs should fit together inthe light of changing evidence” (p. 4).

More recently, Lindley (2000) states: “Infer-ence is only of value if it can be used, so theextension to decision analysis, incorporating util-ity, is related to risk and to the use of statistics inscience and law” (p. 293). The point here is thatBayesian statistics is not merely a collection ofmethods for the analysis of data. It is a theory formaking decisions under uncertainty, where bothcomponents—probability and utility—are equallyimportant and a full Bayesian analysis includesdata analysis, construction of probability models,assessment of the prior information and the util-ity function, and, finally, making decisions.

In the financial world, making decisions underuncertainty has been recognized to be one of themost important activities. Foccardi and Jones(1998) said: “Risk measurement implies thatthere is a model of the market that, applied todata, measures risk. But risk management is notlimited to the more or less scientific process ofmeasuring risk. Once measured, subjective judg-

* M. Mendoza is a Professor at the Instituto Technologico Autonomode Mexico, Rio Hondo 1, San Angel, Mexico DF01000, e-mail:[email protected].


ment is used to evaluate and make decisions uponthe measurement” (p. 56).

Calculation of an appropriate premium and areasonable level of reserves, the topics presentedin the paper, are financial decision problems thatcan be formulated within the Bayesian frameworkin a very general setting. Particular solutions canbe obtained corresponding to different choices forboth, the probabilistic representation of uncer-tainty and the specification of the utility function,and even the dependence on these choices mightbe evaluated by comparing the different particu-lar solutions. Many other actuarial problems canbe treated in a similar fashion.

As for the experience-rating problem discussedin the paper, it seems that for some time theproposed solutions had a Bayesian interpretationbut they were not proper Bayesian solutions. It isinteresting to notice that only recently the situa-tion has been changing. Most of the contributionsthat explore the assessment of the prior probabil-ities and the utility function appeared in 1997 orlater, and even the more general topic of robust-ness has been considered. It might be desirable tosee in the future many contributions reportingposterior predictive distributions describing theobserved experience and proposing utility func-tions that might be used to calculate premiumscorresponding to different strategies and levels ofprotection.

Even though the author clearly says that thepaper does not aim at providing an extensivereview of all applications of the Bayesian meth-odology, it is remarkable the absence of any com-ment in relation to the topic of Bayesian gradua-tion. This subject has been explored for manyyears and some recent contributions have ap-peared that make use of MCMC and other simu-lation methods. The Bayesian approach to theconstruction of mortality tables was first consid-ered in relation to the Whittaker graduation pro-cedure, which is not Bayesian but has a Bayesianinterpretation. This relationship lead to furtherstudy of the conjugate multivariate normal model(see Kimeldorf and Jones 1967 and Hickman andMiller 1977), which apart from some variations ofthe beta-binomial analysis, was identified forsome time as the Bayesian model for graduation.

It is interesting to notice that these models donot incorporate any ordering restriction on themortality rates and the final product is the poste-

rior distribution for the mortality rates and, moreprecisely, a Bayes estimate given by the posteriormean. Later, in a paper by Broffitt (1988) anexponential-gamma conjugate model was consid-ered, and different ways of imposing an increasingpattern among mortality rates were explored.

More recently, Dellaportas and Smith (1993)dealt with a closely related problem of makinginferences in a class of generalized linear modelsand proportional hazards models. Carlin (1992)discussed a Monte Carlo approach to Bayesiangraduation and, in particular, presented a simu-lation-based analysis of the model proposed byBroffitt. Dellaportas et al. (2001) concentrated ona Bayesian version of a model originally proposedby Heligman and Pollard (1980), which is in-tended to give a complete description of the mor-tality rates across ages, including the so-calledaccident hump. Mendoza et al. (2001) fitted alinear regression model to a set of transformeddeath rates and, using predictive arguments, pro-posed a margin-loaded table, which is mandatedfor reserving purposes by the current Mexicaninsurance regulations.

It is worth noticing that, in the usual actuarialpractice, interest is focused on the estimation oftrue mortality rates. Hence, most of the applicationsof the Bayesian paradigm to the problem of gradu-ation are intended to produce these estimates,which might be used as a mortality (or life) table. Incontrast, Mendoza et al. (2001) derived the jointposterior predictive distribution for the mortalityrates to be observed in the future for a specificinsured population and, on that basis, proposed astrategy to select an appropriate mortality table.

Finally, I would like to congratulate the authorfor this stimulating paper. As a last comment, anddespite the already important number of valuablecontributions, I would also have to stress that, inmy opinion, the Bayesian approach is far fromhave being used in its full power to solve even themost well-known actuarial problems, for exam-ple, experience rating and graduation. Not onlydo prior distributions and utility function assess-ments have to be explored in further detail, butactuaries also have to take advantage of the pos-sibility of producing posterior predictive distribu-tions describing the uncertainty that they willface in the future.

An interesting reference in this direction isCairns (2000). Other relevant references in the


Bayesian literature that might have an impact inthe future application of the Bayesian methods inthe actuarial sciences are Bernardo and Munoz(1993), where Bayesian analysis of demographicdata is discussed; Berger and Chen (1993), whichmakes use of a multinomial model to predict thepattern of a retirement process; Schluter et al.(1997), where a Bayesian procedure is developedfor ranking hazardous traffic accident locations;Denison et al. (1998), which deals with the problemof estimating a wide variety of curves using piece-wise polynomials, and Walker et al. (1999), where,among other results, a nonparametric approach isproposed to estimate predictive survival curves.

REFERENCES

BERGER, J. O., AND M. CHEN. 1993. “Predicting Retirement Pat-terns: Prediction for a Multinomial Distribution with Con-strained Parameters,” The Statistician 42: 427–43.

BERNARDO, J. M., AND J. MUNOZ. 1993. “Bayesian Analysis of Pop-ulation Evolution,” The Statistician 42: 541–50.

BERNARDO, J. M., AND A. F. M. SMITH. 1994. Bayesian Theory.Chichester, England: Wiley.

BROFFITT, J. D. 1988. “Increasing and Increasing Convex Bayes-ian Graduation,” (with discussion). Transactions of theSociety of Actuaries 40: 115–48.

CARLIN, B. 1992. “A Simple Monte Carlo Approach to BayesianGraduation.” Transactions of the Society of Actuaries 44:55–76.

CAIRNS, A. J. G. 2000. “A Discussion of Parameter and ModelUncertainty in Insurance,” Insurance: Mathematics &Economics 27: 313–30.

DELLAPORTAS, P., AND A. F. M. SMITH. 1993. “Bayesian Inference forGeneralized Linear and Proportional Hazards Models viaGibbs Sampling,” Applied Statistics 42(3): 443–59.

DELLAPORTAS, P., A. F. M. SMITH, AND P. STAVROPOULOS. 2001.“Bayesian Analysis of Mortality Data,” Journal of the RoyalStatistical Society (Series A) 164: 275–91.

DENISON, D. G. T., B. K. MALLIK, AND A. F. M. SMITH. 1998. “Auto-matic Bayesian Curve Fitting,” Journal of the Royal Sta-tistical Society (Series B) 60: 333–50.

FOCCARDI, S., AND C. JONAS. 1998. Risk Management: Framework,Methods and Practice. New Hope, PA: F. J. Fabozzi Asso-ciates.

HELIGMAN, L., AND J. H. POLLARD. 1980. “The Age Pattern of Mor-tality,” Transactions of the Society of Actuaries 29: 1–21.

HICKMAN, J. C., AND R. B. MILLER. 1977. “Notes on Bayesian Grad-uation,” Journal of the Institute of Actuaries 107: 49–80.

KIMELDORF, G. S., AND D. A. JONES. 1967. “Bayesian Graduation,”Transactions of the Society of Actuaries 19: 66.

LINDLEY, D. V. 2000. “The Philosophy of Statistics,” The Statis-tician 49: 293–337.

MENDOZA, M., A. M. MADRIGAL, AND E. GUTIERREZ-PENA. 2001. “Pre-dictive Mortality Graduation and the Value at Risk: A

Bayesian Approach,” Technical Report DE-C01.5, InstitutoTecnologico Autonomo de Mexico.

SCHLUTER, P. J., J. J. DEELY, AND A. J. NICHOLSON. 1997. “Ranking andSelecting Motor Vehicle Accident Sites by Using a Hierarchi-cal Bayesian Model,” The Statistician 46: 293–316.

WALKER, S. G., P. DAMIEN, W. L. PURUSHOTTAM, AND A. F. M. SMITH.1999. “Bayesian Nonparametric Inference for Random Dis-tributions and Related Functions,” Journal of the RoyalStatistical Society (Series B) 61: 485–527.

FRANCISCO JOSE VAZQUEZ-POLO*The paper provides a suitable framework forthinking about important aspects of Bayesian ac-tuarial analysis. Most of the methodological andtheoretical topics in which Bayesian methodologyis developing were discussed, and I congratulatethe author. The paper illustrates the utility ofBayesian methods in solving practical issues ofcredibility theory. Knowing that Bayesian infer-ence can be meaningfully conducted, I will focusmy comments on prior sensitivity analysis of thepremium principles.

According to robust Bayesian methodology, un-certainty in the prior can be modeled by specify-ing a class G of priors instead of a single one.Bayesian robustness analysis has received sub-stantial attention, and numerous authors haveproposed solutions to this problem (Berger 1985,1990; Berger and O’Hagan 1988; Lavine 1991;Gustafson 1996; Gustafson and Wasserman 1995;Sivaganesan 1988, 1993; and others). An excel-lent survey of this topic can be found in Berger(1994). However, relatively few papers have beendevoted to Bayesian robustness in credibility the-ory (see Makov 1995 and more recently Gomez etal. 1999, Gomez et al. 2000, and Young 2000).

Robust Bayesian Methodology

Since prior specification is typically imprecise,recent attention focuses on local sensitivity,which measures the effect of perturbations of theinputs on the final answer. Our approach is basedon the assumption that the practitioner is unwill-ing or unable to choose a functional form of thestructure function, p0, but that he may be able torestrict the possible prior to a class that is suitable

* Francisco Jose Vazquez-Polo is Associate Professor, Department ofQuantitative Methods in Economics and Management, University ofLas Palmas de Gran Canaria, 35017 Las Palmas de Gran Canaria,Spain, e-mail: [email protected].


for quantifying the actuary’s uncertainty. There-fore it is of interest to study how the premium forpriors in such a class behaves. We use the e-con-tamination class of priors (see Berger 1985, 1994;Sivaganesan 1991; Sivaganesan and Berger 1989;and Ying-Hsing and Ming-Chung 1997; amongothers), Ge 5 {p 5 (1 2 e)p0 1 eq, q [ 4}, wherep0 is the base elicited prior, 4 is the class ofallowed contaminations and e [ [0, 1] measuresthe uncertainty of the base prior p0.

The idea of robustness is as follows. Suppose anactuary (decision maker) is looking for a priordistribution of a risk parameter u, but is unable tochoose a single prior. We may say therefore thathe is indifferent to the choice of p [ Ge. A naturalgoal of a robustness study is to find the range ofthe posterior quantity P*p(m) when p varieswithin Ge. Thus, attention will be focused onlower and upper bounds for Bayesian premium:infp[Ge

P*p(m) and supp[GeP*p(m), respectively.

These classes have been used in several situa-tions to measure the sensitivity of quantities thatcan be expressed in terms of the posterior expec-tation of parameter functions. Nevertheless, as inother areas of applied statistics, empirical prob-lems raise technical challenges to the state of theart, impinging on the development of new tech-niques. This is the case of Esscher and variancepremium principles. When Esscher or variancepremium principles are used, the quantity of in-terest can be expressed as the ratio of two partic-ular posterior expectations.

In Gomez et al. (1999, 2000), we presentedbasic results to study the range of the posteriorratio quantities in the form Pp 5 (*Q

g1(u)p(u)du)/(*Q g2(u)p(u)du), as p varies overan e-contamination class

Ge 5 $p~u ! 5 ~1 2 e!p0~u ! 1 eq~u !uq [ 4%, (1)

where e reflects the amount of probabilistic un-certainty in a base prior p0 and 4 is a class ofallowable contaminations. For 41 5 {All probabil-ity distributions}, and 42 5 {All unimodal distri-butions} we determine the range of Pp as p variesover Ge.

The variational problems involved are reducedto finding the extremes of functions of one vari-able to be solved numerically and the explicitsolutions of supp[G Pp(m) and infp[G Pp(m) to beobtained. A complete development of these tech-

niques can be found in Gomez et al. (1999, 2000),with several illustrations. Among other conclu-sions, these papers show that unimodality is veryconvenient, and easy to elicit, for modeling priorinformation about the risk parameter. It is usefulto focus here on the types of actuarial statisticalproblems in which Bayesian sensitivity analysiscan be applied. One future direction presentedhere is the bimodal structure function.

Bayesian Robustness Analysis in BimodalStructure Function

Assume that the Bayesian approach is applied tothe estimation of net premium. Let Q be a ran-dom variable, and XiuQ 5 u, i 5 1, . . . , t, theclaims or loss amount in subsequent years. Weassume that, given u, the X9i’s are conditionallyindependent and identically distributed randomvariables.

Suppose that we are interested in studing thesituation where the collective has two types ofrisks; a% are good risks with a low probability of aclaim or loss amount, and the other b% are badrisks with a high claim or loss amount probability(see Hewitt 1996; Hewitt and Lefkowitz 1979; andVenter 1991; among others) that can be modeledby two structure functions (prior distributions)p1(u) and p2(u). Therefore, our prior distributionof u is given by

p0~u ! 5 Oi51

2

wipi~u !, (2)

with w1, w2 [ R, w1 1 w2 5 1.In experience ratemaking, the actuary takes a

claim experience } 5 m from the random vari-ables X1, X2, . . . , Xt and uses this information toestimate the unknown fair premium 3(u). Theposterior distribution of u, given the likelihood m,is given by

p0~uum! 5 Oi51

2

w9ipi~uum!, (3)

where

w9i 5wip~mupi!

¥i512 wip~mupi!

and p(mupi) 5 *Q f(muu)pi(u)du is the marginaldistribution of } with respect to prior pi.


The Bayesian net premium is obtained by

3*p0~m! 5 EQ

SEx

xf~ xuu ! dxDp0~uum!du

5 Oi51

2

w9i3*pi~m!.

Since in our model there are two distinct claimor loss amount generating processes—wheresome claims or losses are regular and may bedescribed by a p.d.f. p1(u), while others are nui-sance small claims or losses that may be de-scribed by a p.d.f. p2(u)—our e-contaminationclass is given by

Gej 5 Hp 5 ~1 2 e! O

i51

2

wipi

1 e Oi51

2

biqi, qi [ 4iJ , (4)

with j 5 1, 2, b1, b2 [ R, b1 1 b2 5 1.For 4i

1 5 {All probability distributions}, we de-termine the range of Bayesian net premium as pvaries over Ge. Now, if we want the model toinclude distributions with shapes similar to theprior distributions, we can consider the contam-ination class 4i

2 5 {qi(u) : qi is unimodal with thesame mode ui, as that of pi}.

It is straightforward to rewrite the Bayesianpremium under class Ge

1, as

under Ge2 class. Now, we can easily obtain the ranges

for Bayesian premiums using slight modifications ofthe theorems in Sivaganesan and Berger (1989) andBerger and Moreno (1994). Extensive developmentsof such results can be found in Gomez et al. (2001).

The Poisson-Gamma Mixture Model

Assume that the number of claims follows a Pois-son distribution with parameter u, while theamount of the individual claim is taken as fixed.Suppose that the prior density of u is a mixture oftwo gammas:

p0~u ! 5 w1Gamma~a1, b1! 1 w2Gamma~a2, b2!,

where a1, a2, b1, b2 are positive hyperparameters.Papers using the simple Poisson-gamma are Eichen-auer et al. (1988), Klugman (1992), and Freifelder(1974), among others. The following propositiongives lower and upper bounds for Bayesian net pre-mium under the class Ge

1. Analogously, proposition2 gives the bound under the class Ge

2. The proofs canbe found in Gomez et al. (2001).

Proposition 1

Under the class Ge1, the lower (upper) bound for

the Bayesian net premium is given by

infu[U

~sup!513*p0~m! 1 52~u!

51 1 53~u!,

where:

51 5 ~1 2 e! Oi51

2

ai

aibi

~bi 2 1!!~bi 1 tm 2 1!!

~ai 1 t!bi1tm ,

52~u ! 5 eutm11e2tm,

53~u ! 5 52~u !/u,

and

3*p0~m! 5 Oi51

2

a9ibi 1 tm

ai 1 t.

3*p~m! 5~1 2 e!$¥i51

2 aip~mupi!%3*p0~m! 1 e *Q g~u ! f~muu !q~u ! du

~1 2 e!$¥i512 aip~mupi!% 1 e *Q f~muu !q~u ! du

, ~5!

and

3*p~m! 5~1 2 e!$¥i51

2 aip~mupi!%3*p0~m! 1 e ¥i512 bi *0

` Hqi~zi! dF~zi!

~1 2 e!$¥i512 aip~mupi!% 1 e ¥i51

2 bi *0` Hi~zi! dF~zi!

, ~6!


Proposition 2

Under the class p [ Ge2, the lower (upper) bound

for the Bayesian net premium is given byinfz1,z2$0(sup)5(z1, z2), being

and

5~0, 0! 5513*p0~m! 1 ¥i51

2 bi52~ui!

53~u ! 1 ¥i512 bi53~ui!

,

where 51, 52(u), 53(u), and 3*p0(m) are as in

Proposition 1.To illustrate the above ideas, note the following

numerical illustration. We shall use bi 5 wi, i 5 1,2. Also, we have included a measure that does notdepend on the premium measurement units. Thisis the relative sensitivity (RS) index (Sivaganesan1991):

RSj 51

23*p0~m!@sup

p[G ej

3*p0~m! 2 infp[G e

j

3*p0~m!#

3 100%, ~ j 5 1, 2!.

EXAMPLE

Let Xuu have a Poisson distribution with parame-ter u and p0(u) 5 0.8Ga(2, 4) 1 0.2Ga(3, 30).With this elicitation, the actuary knows that thetwo modal values are around 1.5 and 10 (i.e., u1 51.5 and u2 5 10), and that claims above 5 are lessfrequent than smaller claims.

Let m 5 4, and m 5 12. Table 1 contains the

standard Bayesian premium. This particular situa-tion corresponds to e 5 0, that is, no errors in theprocess of elicitation. The bounds on the Bayesiannet premium are given in Figure 1 for the classes Ge

1

and Ge2. Figure 2 displays their RS factor.

Concluding Remarks

A basic assumption of credibility theory is that thevalues of the parameters of the probability distribu-tion of loss are unknown. In this case, the premiumthat the company charges is the Bayesian premium.This premium requires that the decision maker, theactuary, can define a probability distribution for thevalues of the unknown parameters of this loss pro-cess, the prior distribution.

Nevertheless, there will clearly be many priordistributions other than p0 which are also com-patible and, hence, could be used in place of p0.

Table 1Standard Poisson-Gamma Model

m 3*p1

(m) 3*p2

(m) a*1 a*1

3*p(m) 5

¥i 5 12 a9i3*

pi(m)

4 3.666 5.384 0.998 0.002 3.67012 10.333 11.538 0.001 0.999 11.538

5~ z1, z2! 5513*p0~m! 1 ¥i51

2 ~1/zi!bi *ui

ui1zi 52~u !du

51 1 ¥i512 ~1/zi!bi *ui

ui1zi 53~u !duif z1, z2 . 0,

5~ z1, 0! 5513*p0~m! 1 ~1/z1!b1 *u1

u11z1 52~u !du 1 b252~u2!

51 1 ~1/z1!b1 *u1

u11z1 53~u !du 1 b253~u2!if z1 . 0,

5~0, z2! 5513*p0~m! 1 b152~u2! 1 ~1/z2!b2 *u2

u21z2 52~u !du

51 1 b153~u1! 1 ~1/z2!b2 *u2

u21z2 53~u !duif z2 . 0,

Figure 1Ranges of Bayesian Net Premium:Poisson-Gamma Mixture Model.


Although bimodal models are common in actuar-ial practice, there has been relatively little re-search from a Bayesian point of view.

This discussion has presented a simple ap-proach to develop bimodal Bayesian sensitivityanalysis. The mixture of two unimodal densitiesappears appropriate when the researcher agreesthat a bimodal structure function is necessary.This is justified in our model, where prior p0 isgiven by a convex sum of two prior distributions,p1 and p2. This leads to the question of Bayesianrobustness, which has been treated in this discus-sion using the e-contamination class. We haveseen that bimodality effects are very important inmodeling subjective beliefs about risk parameter.Finally, all theorems and results presented herecan be used for alternative premium calculationprinciples (Heilmann 1989) such as exponential,Esscher, and variance, among others.

ACKNOWLEDGMENT

Research partially supported by the DGUI delGob. Aut. de Canarias under grant PI2000/061,Canary Islands, Spain.

REFERENCES

BERGER, J. O. 1985. Statistical Decision Theory and BayesianAnalysis. New York: Springer.

———. 1990. “Robust Bayesian Analysis: Sensitivity to thePrior,” Journal of Statistical Planning and Inference 25:303–28.

———. 1994. “An Overview of Robust Bayesian Analysis” (withdiscussion), Test 3: 5–124.

BERGER, J. O., AND E. MORENO. 1994. “Bayesian Robustness inBidimensional Models: Prior Independence,” Journal ofStatistical Planning and Inference 40: 161–76.

BERGER, J. O., AND A. O’HAGAN. 1988. “Range of Posterior Proba-bilities for the Class of Unimodal Priors with SpecifiedQuantiles.” In Bayesian Statistics 3, J. M. Bernardo, M. H.DeGroot, D. V. Lindley and A. F. M. Smith, eds. New York:Oxford University Press.

EICHENAUER, J., J. LEHN, AND S. RETTIG. 1988. “A Gamma-MinimaxResult in Credibility Theory,” Insurance: Mathematicsand Economics 2: 49–57.

FREIFELDER, L. 1974. Statistical Decision Theory and CredibilityTheory Procedures, proceedings of the Berkeley ActuarialResearch Conference on Credibility, Credibility Theory,and Applications. Berkeley: Academic Press.

GOMEZ, E., A. HERNANDEZ, AND F. VAZQUEZ-POLO. 1999. “The Es-scher Premium Principle in Risk Theory: A Bayesian Sen-sitivity Study,” Insurance: Mathematics and Economics25: 387–95.

———. 2000. “Robust Bayesian Premium Principles in ActuarialScience,” Journal of the Royal Statistical Society (SeriesD, The Statistician) 49(2): 241–52.

GOMEZ, E., A. HERNANDEZ, J. M. PEREZ, AND F. VAZQUEZ-POLO. 2001.“Premium Sensitivity Using Bimodal Prior Distributions,”Poster presented at SAMO, Madrid, Spain.

GUSTAFSON, P. 1996. “Local Sensitivity of Inferences to PriorMarginals,” Journal of the American Statistical Associa-tion 91: 774–81.

GUSTAFSON, P., AND L. WASSERMAN. 1995. “Local Sensitivity Diag-nostic for Bayesian Inference,” The Annals of Statistics24(6): 2153–67.

HEILMANN, W. 1989. “Decision Theoretic Foundations of Credi-bility Theory,” Insurance: Mathematics and Economics 8:77–95.

HEWITT, C. 1966. “Distribution by Size of Risk” (with discussion),Proceedings of the Casualty Actuarial Society 53: 106–17.

HEWITT, C., AND B. LEFKOWITZ. 1979. “Methods for Fitting Distri-butions to Insurance Loss Data,” Proceedings of the Casu-alty Actuarial Society 66: 139–60.

KLUGMAN, S. 1992. Bayesian Statistics in Actuarial Science.Boston: Kluwer Academic Publisher.

LAVINE, M. 1991. “Sensitivity in Bayesian Statistics: The Priorand the Likelihood,” Journal of the American StatisticalSociety 86: 400–3.

MAKOV, U. 1995. “Loss Robustness Via Fisher-Weighted Squared-Error Loss Function,” Insurance: Mathematics and Eco-nomics 16: 1–6.

SIVAGANESAN, S. 1988. “Range of Posterior Measures for Priorswith Arbitrary Contaminations,” Communications in Sta-tistics and Theory Methods 17(5): 1591–612.

———. 1991. “Sensitivity of Some Posterior Summaries When

Figure 2RS Factor: Poisson-Gamma Mixture Model.


the Prior is Unimodal with Specified Quantiles,” The Ca-nadian Journal of Statistics 19(1): 57–65.

———. 1993. “Robust Bayesian Diagnostic,” Journal of Statis-tical Planning and Inference 35: 171–88.

SIVAGANESAN, S., AND J. O. BERGER. 1989. “Ranges of PosteriorMeasures for Priors with Unimodal Contaminations,” TheAnnals of Statistics 17(2): 868–89.

VENTER, G. 1991. “Effects of Variations from Gamma-PoissonAssumptions,” Proceedings of the Casualty Actuarial So-ciety 78: 41–56.

YING-HSING, L., AND Y. MING-CHUNG. 1997. “Posterior Robustnessin Simultaneous Estimation Problem with ExchangeableContaminated Priors,” Journal of Statistical Planning andInference 65: 129–43.

YOUNG, V. R. 2000. “Credibility Using Semiparametric Modelsand a Loss Function with a Constancy Penalty” Insurance:Mathematics and Economics 26(1): 151–6.

JAMES C. HICKMAN* AND

DONALD A. JONES†

We are grateful to the author for providing a per-spective on current activity in applying Bayesianmethods in actuarial science. The author’s enthu-siasm for Markov chain Monte Carlo (MCMC)methods for implementing Bayesian methods isshared by all of us who are impressed by thecoherence of Bayesian methods but have beenfrustrated by the daunting amount of integrationrequired to obtain results from many practicalmodels.

The purpose of this discussion is to add histor-ical background to the contemporary view of theauthor. We will review the birth of Bayesian sta-tistics and the crucial role played by an actuary inthis event. Next will come sections on the earlyrole of Bayesian methods in graduation and cred-ibility. These two topics illustrate, in a naturalfashion, how Bayesian methods provide a frame-work for combining new information with exist-ing beliefs. The discussion will end with the sug-gestion that Bayesian time series analysis canbecome an important tool for actuaries, espe-cially in the design and management of health,pension, and social insurance systems.

CreationRichard Price (1723–91) was a remarkable gen-tleman of the age of enlightenment. By profes-sion he was a dissenting minister but as ascholar he had an impact on philosophy, statis-tics, public finance, demography, political sci-ence, and actuarial science. From 1768 to 1791,Price served as a consultant to the Society forEquitable Assurances on Lives and Survivor-ships. His first assignment was to calculate theprobability of survival of a complicated statusinvolving two women and one man of knownages. He constructed the Northampton Life Ta-ble and wrote Observations on ReversionaryPayments: On Schemes for Providing Annu-ities for Widows and for Persons of Old Age; Onthe Method of Calculating Values of Assur-ances on Lives; and On the National Debt(Price 1771). This two-volume publicationseems to have left no subject untouched. Ben-jamin Franklin was, however, sufficiently im-pressed to call it “the foremost production ofhuman understanding that this century af-forded us.” The book went through seven edi-tions and was the principal actuarial textbookuntil well into the 19th century. The story ofPrice’s contribution to actuarial science andFranklin’s good opinion of Price’s book can befound in Ogborn (1962, chap. 7).

With the passage of about 240 years, we knowlittle of the partnership of Price and ThomasBayes in examining the fundamental issues instatistics. Bayes was also a dissenting ministerand was probably about 20 years older than Price.Price mentions an argument developed by Bayesin his 1758 essay on morals, “A Review of thePrincipal Questions and Difficulties in Morals,”but there is little evidence of earlier joint work. Itis possible that both men had studied mathemat-ics with John Eames, a friend of Isaac Newton, atdifferent times in a dissenting academy. Price’sessay on morals and a critical review of his phi-losophy can be found in Hudson (1970).

Bayes, who died in 1761, left £100 and hisscientific papers to Price. In his will, Bayes re-ferred to his legatee as “Richard Price, now Isuppose, a preacher at Newington Green.” Bayesdied only four months after executing his will. Hisaddress for Price was correct. These events aredescribed by Stigler (1986, chap. 3).

* James C. Hickman is Emeritus Professor and Dean at the Universityof Wisconsin, Grainger Hall, 975 University Ave., Madison, WI 53706.† Donald A. Jones is a Professor at Oregon State University, Depart-ment of Mathematics, 368 Kidder Hall, Corvallis, OR 97331-4605,e-mail: [email protected].


Price decided that the most promising paper leftby Bayes was an essay, “Toward Solving a Problemin the Doctrine of Chance.” It was read by Price tothe Royal Society on Dec. 23, 1763, more than twoand one-half years after Bayes’ death. Bayes hadbeen elected a Fellow of the Royal Society (FRS) in1742, despite an apparent absence of any technicalpublications. Price was elected FRS in 1765 becauseof his role in presenting Bayes’ paper after adding anintroduction and appendix. In the appendix, Priceworked on the problem of evaluating the incom-plete beta function. There are other aspects of thepaper that have been attributed to Price, but thisissue will remain forever shrouded.

The paper had almost no immediate impact onscience. About 20 years later it was rediscoveredin connection with Laplace’s work on the sameproblem. Yet today what are called “Bayesian”methods are applied in most scientific fields, andthe field of statistics is divided into camps calledthe Bayesians and the “frequentists.”

The consequence of Bayes’ work was a mathe-matical procedure by which an investigator canchange existing (a priori) beliefs about a proposi-tion in light of new evidence (data) to produce arevised (posterior) estimate of the degree of beliefin the proposition. Although the original beliefs oftwo investigators might differ, if they bothadopted Bayesian methods and processed thesame new evidence, their posterior views of theproposition under review would move closer to-gether. Ultimately Bayes’ ideas prompted a re-evaluation of the fundamental scientific conceptsof evidence and causality.

Bayes’ paper is difficult for those of our gener-ation to read because, like Newton, he adopted ageometric mode of exposition. Yet to Price, andothers of the age of enlightenment, this was thestandard way for science to proceed. The analysisshowed a clear understanding of the philosophi-cal principles on which it was built, and it con-tained some subtle mathematics.

Despite Price’s critical role, the resulting schoolof thought is called Bayesian and not Prician, yetPrice was present at its creation and contributedto its propagation.

GraduationIn recent decades, graduation lost its position as aleading topic in actuarial education and research.

Yet to gain a balanced perspective on Bayesianmethods in actuarial science, we must return tograduation. Just as Richard Price played a criticalrole at the creation, E.T. Whittaker’s role inBayesian methods of graduation is central. Whit-taker’s (1920) address to the Faculty of Actuarieson the topic “On Some Disputed Questions ofProbability” generated a spirited discussion.These questions centered on the interpretation ofprobability. Some of the questions remain dis-puted and Whittaker’s insights remain valuable.

Whittaker’s graduation method involved theminimization of the loss function L 5 F 1 hS. Inthis function, F is a measure of the lack of fit, S ameasure of lack of smoothness, and h a positiveconstant. Whittaker’s ideas have motivated gen-erations of research-minded actuaries and otherapplied mathematicians. To many students, it hasbeen presented as a mathematical programmingproblem with the positive constant h under thecontrol of the investigator. But Whittaker pro-vided a remarkable and complete Bayesian moti-vation of L, and the constant h is the ratio of twovariances. The Bayesian development is perhapsmost easily found in Whittaker and Robinson(1944).

Later Kimeldorf and Jones (1967) extendedWhittaker’s model to include a prior distributionthat incorporates a vector of prior means and aprior covariance matrix. A stream of modifica-tions to Bayesian graduation methods followed.One of these contributions deserves special no-tice. Carlin (1992) reported on Bayesian gradua-tion and in performing what would have beentedious approximate integrations, he used theGibbs sampler, a Monte Carlo integration methodthat belongs to the MCMC class. We believe thiswas among the first actuarial applications ofMCMC.

CredibilityThe author correctly identifies Buhlmann’s(1967) credibility paper as the beginning of aflood of Bayesian-based research in actuarial sci-ence. Yet approximately a quarter of a centurybefore 1967, there was startling Bayesian-basedresearch on credibility. In a sequence of paperspublished between 1942 and 1950, Arthur Baileydeveloped the rudiments of a Bayesian founda-tion for credibility. Bailey’s (1950) ultimate paper


summarized his ideas under the encompassingtitle “Credibility Procedures: Laplace’s Generali-zation of Bayes’ Rule and the Combination ofCollateral Knowledge.” In this paper, he presentscogent criticisms of existing statistical methods,quotes Price’s introduction to Bayes’ paper anddiscusses Laplace’s extension. He also appliesthese ideas to credibility. Most of the credibilityideas developed in the 1960s and 1970s havevisible roots here. For example, the use of leastsquares regression lines to approximate posteriormeans is a thread running through much of theanalysis. The fact that these least squares areexact for several common examples is illustrated.All of this was put forth over 20 years beforeEricson (1970) and Jewell (1974) establishedthat, when a conjugate prior distribution (theprior and posterior distributions are of the sametype) is used with a likelihood (data distribution)that is a member of the linear exponential familyof distributions, then the least squares linear ap-proximation to the posterior mean is exact.

Because he was a self-taught Bayesian, andmany of our present day concepts and notationwere in the future, Bailey is hard to read. Never-theless, his work demonstrates that Bayesianideas were an important ingredient of credibilitytheory before 1967.

Time SeriesWe recommend Bayesian time series analysis,especially to actuaries working with health, pen-sion, and social insurance systems. The analysisof time series data of health costs, wage rates, andthe size of the labor force is required in theseapplications. The conventional advice is to ana-lyze the data but combine this analysis with col-lateral information and prior knowledge. Bayes-ian time series analysis provides a disciplinedprocedure for this mixing of information. Rosen-berg and Young (1999) provide a primer onBayesian time series analysis. They use MCMC tocomplete the analysis and illustrate the methodswith unemployment data.

REFERENCES

BAILEY, A. L. 1942. “Sampling Theory in Casualty Insurance,”Proceedings of the Casualty Actuarial Society 29(59): 50–93.

———. 1945. “A Generalized Theory of Credibility,” Proceed-ings of the Casualty Actuarial Society 32(62): 13–20.

———. 1950. “Credibility Procedures: LaPlace’s Generalizationof Bayes’ Rule and the Combination of Collateral Knowl-edge with Observed Data,” Proceedings of the CasualtyActuarial Society 37(67): 7–23.

———. 1950. “Discussion,” Journal of the American Teachersof Insurance (now the Journal of Risk and Insurance) 17:17–24.

CARLIN, B. P. 1992. “A Simple Monte Carlo Approach to BayesianGraduation.” Transactions of the Society of Actuaries 44:55–76.

ERICSON, W. A. 1970. “On the Posterior Mean and Variance of aPopulation Mean,” Journal of the American StatisticalAssociation 65(330): 649–52.

JEWELL, W. S. 1974. “Credible Means are Exact Bayesian forExponential Families,” ASTIN Bulletin 8: 77–90.

HUDSON, W. D. 1970. Reason and Right: A Critical Examinationof Richard Price’s Moral Philosophy. London: Macmillan.

KIMELDORF, G. S., AND D. A. JONES. 1967. “Bayesian Graduation,”Transactions of the Society of Actuaries 19(1): 66–127.

OGBORN, M. E. 1962. Equitable Assurances: The Story of LifeAssurance in the Experience of the Equitable Life Assur-ance Society, 1762–1962. London: Allen and Unwin.

PRICE, R. 1771. Observations on Reversionary Payments: OnSchemes for Providing Annuities for Widows and for Per-sons in Old Age; On the Method of Calculating the Valuesof Assurance on Lives; and On the National Debt. London:T. Cadell and W. Davis.

ROSENBERG, M. A., AND V. R. YOUNG. 1999. “A Bayesian Approachto Understanding Time Series Data,” North American Ac-tuarial Journal 3(2): 130–43

STIGLER, S. M. 1986. The History of Statistics. Cambridge, MA:The Belknap Press.

WHITTAKER, E. T. 1920. “On Some Disputed Questions of Proba-bility,” Transactions of the Faculty of Actuaries 8: 163–206.

WHITTAKER, E. T., AND G. ROBINSON. 1944. The Calculus of Obser-vations. 4th ed. London and Glasgow: Blackie and Son.

ENRIQUE DE ALBA*I would first like to thank the author for his stim-ulating paper that deals with two of the mainapplications of Bayesian methods in actuarial sci-ence. Since Bayesian methods can essentially beapplied whenever there is a statistical inferenceproblem, there has always been a Bayesian optionin actuarial science. Its infrequent use in the past

* Enrique de Alba is a Professor at ITAM, Division de Actuaria, Esta-distica y Matematicas, Rio Hondo No. 1, Mexico, D.F. 01000, Mex-ico, and at the University of Waterloo, Department of Statistics andActuarial Science, 200 University Ave. West, Waterloo, ON N2L 3G1,Canada, e-mail: [email protected].


has had to do with the general situation existingin statistics over the last decades as well as withthe fact that its application has been, until re-cently, computationally cumbersome or down-right impossible.

In fact, I believe that the Bayesian contents ofBuhlmann (1967, 1969) have largely been mini-mized. Some authors only talk about the “struc-ture” distributions and use Bayesian results toderive formulas for estimation of credibility fac-tors. Once the formulas are obtained, the Bayes-ian framework is largely forgotten. Credibility fac-tors are then estimated by empirical Bayes (EB)or other methods. Available information can andshould be used to specify the prior distribution.This is different from using the data to estimatethe hyperparameters or some features of the priordistribution, or the decision rule. This is what EBdoes.

I would like to make some observations aboutEB methods that I believe are not generally takeninto account when using them, specifically in ac-tuarial science. The EB formulation generallyused in credibility falls under the parametric em-pirical Bayes label of Morris (1983), where theprior distribution of the risk parameter u, sayf(uuf), is assumed to be in some parametric classwith unknown hyperparameters f. In a fullyBayesian approach these would be known, per-haps elicited from available prior information. Al-ternatively, a hierarchical model may be used anda distribution specified for f. However, in EBthese hyperparameters are typically estimated bysome classical (non-Bayesian) method and sub-stituted into the prior distribution, assuming theyare the known fixed values (see Berger 1985).

The analysis then proceeds using the priorf(uuf). This, however, ignores the fact that thehyperparameters were estimated and the errorsintroduced by this are not considered in any ofthe conclusions. Klugman (1992) mentions thatnone of the existing credibility analyses allows forthe extra variability. Hence, I do not totally agreewith the author that EB was a successful compro-mise, since its true meaning and properties usu-ally are not fully taken into account. Further-more, EB methods have the problem that theiroptimality properties are essentially asymptotic.Yet, their small sample properties, if known, areusually not considered.

In addition, they essentially provide point esti-

mates. In general, there are no interval estimatesfor the credibility premiums. When they are ob-tained they are usually “naıve” intervals (see Car-lin and Louis 1996), where the extra variabilitydue to estimation of the parameters is not incor-porated. Theoretically, this will yield intervalsthat are too short, and adjustments may be nec-essary. Klugman (1992) does provide an examplewhere he shows that the difference is negligible.The use of full Bayesian hierarchical models au-tomatically accounts for the added uncertainty.Hence, strictly speaking, EB is not Bayesian be-cause it does not admit a distribution for f, thehyperparameters of the prior distribution f(uuf) ofthe risk parameter (O’Hagan 1994).

There are two points of Bayesian methods thatI believe are not stressed sufficiently in this pa-per. One is that actuarial science is a field wherevery frequently one has considerable prior infor-mation, be it in the form of global or industry-wide information (experience) or in the form oftables. In this respect, it is indeed surprising thatBayesian methods have not been used more in-tensively up to now. This availability of informa-tion would make one question the author’s state-ment that they have not been used moreintensely because of their subjective nature.There is a wealth of “objective” prior informationavailable to the actuary.

Another advantage of Bayesian methods that Ifeel is not emphasized sufficiently in the paper isthe possibility of obtaining complete posterior orpredictive distributions. Actuarial science is afield in which adequate understanding and knowl-edge of the complete distribution is essential. Inaddition to expected values, we are looking atcertain characteristics of probability distribu-tions, for example, probability of ruin, value atrisk (VAR), and so on. Some of the usual approx-imations currently in use ignore the possibleskewness of the resulting distributions. It is auto-matically incorporated when using Bayesianmethods, whether analytically or numerically.

One advantage of full Bayesian methods is thatthe posterior distribution for the parameters isessentially the exact distribution; that is, it is truegiven the specific data used in its derivation. Theyautomatically account for all the uncertainty inthe parameters.

I agree with the author that one cause of thelow use of Bayesian methods has been the fact


that closed analytical forms had not been avail-able and numerical approaches had been toocumbersome to carry out. However, things havechanged drastically in recent years. The availabil-ity of software that allows one to obtain the pos-terior or predictive distributions by direct MonteCarlo methods or by Markov chain Monte Carlo(MCMC) has opened a broad area of opportunitiesfor the applications of these methods in actuarialscience. He mentions that no attempt has beenmade to use Bayesian models to evaluate theposterior distribution of the parameters in thefrequency and severity distributions of theclaims. In addition to the reference given in thepaper, this is done in de Alba (2001) by usingdirect Monte Carlo methods.

One area of application of Bayesian methods inactuarial science that has totally been left out ofthe paper is graduation. This topic has received agreat deal of attention from actuaries, includingBayesians; see Kimeldorf and Jones (1967) andHickman and Miller (1977). Indeed, London(1985), in a literature review of this topic, de-scribes the use of Bayesian methods and providesexamples. He deals explicitly with Bayesian grad-uation and indicates how prior information maybe used from standard published tables. Londonmentions the case where

. . . [W]e needed mortality rates for making pen-sion calculations for a large group of employees,but had no data derived from the recent experi-ence of this group. We would be likely to chooserates from a published table which was based onthe experience of another group, with character-istics as similar to our group as possible. If then, afew years later, we did a study of mortality in ourgroup, and chose to graduate it . . . , these stan-dard table rates would be logical candidates forthe prior mean vector . . . (1985, p. 79).

He is clearly outlining the “updating” or learn-ing process that emerges naturally in Bayesianmethods. He later points out that a statisticalmodel for mortality can be summarized in theform of a probability distribution, with any num-ber of “distinct” mortality tables obtained from it.A procedure along these lines has been applied inMendoza et. al. (2001), who use a Bayesian pre-dictive approach to graduation that incorporatesa desired level of protection against deviations inmortality. It can be interpreted in terms of VAR.

Among a host of other applications, Scollnik(2001) illustrates how the Kimeldorf-Jones(Kimeldorf and Jones 1967) graduation modelcan be implemented via MCMC. The main advan-tage here over the traditional analysis is that, inaddition to means, percentiles associated withthe posterior distribution can easily be deter-mined.

Another area of growing interest in actuarialscience is the use of generalized linear models(Habermann and Renshaw 1996). Specifically,these authors consider survival modeling andclaims reserving, among other applications.These applications can actually be carried outfrom the Bayesian point of view, as has been donein Verrall (1990) and de Alba (2001) in the case ofclaims reserving. Mendoza et. al. (2001) applythem in their graduation models.

Thus, in addition to the principal applicationsof Bayesian methods in actuarial science men-tioned in the paper, there many potential uses ofthese procedures that can contribute to a betterunderstanding of actuarial problems. Actuarialscience can use them to exploit the wealth ofavailable prior information in a formal way.

REFERENCES

BERGER, J. O. 1985. Statistical Decisions and Bayesian Analysis.2nd ed. New York: Springer-Verlag.

BUHLMANN, H. 1967. “Experience Rating And Probability,” ASTINBulletin 4: 199–207

———. 1969. “Experience Rating And Probability,” ASTIN Bul-letin 5: 157–65

CARLIN, B. P., AND T. A. LOUIS. 1996. Bayes and Empirical BayesMethods for Data Analysis, New York: Chapman & Hall.

DE ALBA, E., AND M. JUAREZ. 2001. “Bayesian Estimation of Out-standing Claims Reserves,” Research Report 01-01, Insti-tute of Insurance and Pension Research, University of Wa-terloo.

HABERMAN, S., AND A. E. RENSHAW. 1996. “Generalized LinearModels and Actuarial Science,” The Statistician 45(4):407–36.

HICKMAN, J. C., AND R. B. MILLER. 1977. “Notes on Bayesian Grad-uation,” Transactions of the Society of Actuaries 29: 1–21.

KIMELDORF, G. S., AND D. A. JONES. 1967. “Bayesian Graduation,”Transactions of the Society of Actuaries 19, 66-112.


LONDON, R. L. 1985. “Graduation: The Revision of Estimates.”Winsted, CT: ACTEX Publications.

MENDOZA, M., A. M. MADRIGAL, AND E. GUTIEERREZ-PENA. 2001. “Pre-dictive Mortality Graduation and the Value at Risk: A


Bayesian Approach.” Working Paper DE-C01.5, ITAM,Mexico.

MORRIS, C. 1983. “Parametric Empirical Bayes Inference: Theoryand Applications,” Journal of the American StatisticalAssociation 78: 47–65.

O’HAGAN, A. 1994. Kendall’s Advanced Theory of Statistics, Vol.2B, Bayesian Statistics. London: Edward Arnold.

SCOLLNIK, D. P. M. (2001) “Actuarial Modeling with MCMC andBUGS,” North American Actuarial Journal 5(2): 96–124.

VERRALL, R. J. 1990. “Bayes and Empirical Bayes Estimation forthe Chain Ladder Model,” ASTIN Bulletin 20: 217–38.

DAVID P. M. SCOLLNIK*I’d like to thank the author for his summary re-view of Bayesian methods appearing in actuarialscience. I would also like to take this opportunityto draw attention to two recent papers absentfrom this summary.

The first of these is by Promislow and Young(2000) and relates to the discussion of experiencerating in Section 2. This paper continues previouswork of these authors relating to credibility esti-mators developed in accordance with a principleof equity. They make the case that a person whoshould be charged 1 unit but is actually charged10 units is being treated more unfairly than aperson who should be charged 1,001 units but isactually charged 1,010 units.

This situation can arise when credibility pre-miums are developed using a loss function likethe squared error, which emphasizes the absolutedifference between the charged premium and thetrue premium. Their solution is to develop equi-table credibility premiums using an entropy lossfunction, instead, so that a measure of the relativedifference between the charged premium and thetrue premium is minimized in place of the usualsquared error. They develop simple credibilityformulas and demonstrate that these are exact forsome interesting cases when the claim distribu-tion belongs to the linear exponential family.

The second paper is by Scollnik (2001) andrelates to the discussion in Sections 3 and 4. Thispaper describes how a number of different actu-arial models, including models of the sort de-scribed in Klugman (1992); Makov, Smith, andLiu (1996); Pai (1997); and Rytgaard (1990), can

be implemented and analyzed in accordance withthe Bayesian paradigm using Markov chain MonteCarlo (MCMC) via the BUGS (Bayesian inferenceUsing Gibbs Sampling) suite of software packages.BUGS is a specialized software package for imple-menting MCMC-based analyses of full probabilitymodels in which all unknowns are treated as ran-dom variables. Its programming language is easilyunderstood and allows the user to make astraightforward specification of the full probabil-ity model under consideration. The Windows ver-sion of BUGS is known as WinBUGS, and it pro-vides a graphical interface to the BUGS language.Scollnik (2001) also includes many additional ref-erences to recent papers in actuarial science in-corporating Bayesian methods.

REFERENCES


MAKOV, U. E., A. F. M. SMITH, AND Y-H. LIU. 1996. “BayesianMethods in Actuarial Science,” The Statistician 45: 503–15.

PAI, J. S. 1997. “Bayesian Analysis of Compound Loss Distribu-tion,” Journal of Econometrics 79: 129–46.

PROMISLOW, S. D., AND V. R. YOUNG. 2000. “Equity and ExactCredibility,” ASTIN Bulletin 30(1): 3–11.

RYTGAARD, M. 1990. “Estimations in the Pareto Distribution,”ASTIN Bulletin 20: 167–72.

SCOLLNIK, D. P. M. 2001. “Actuarial Modeling with MCMC andBUGS,” North American Actuarial Journal, 5(2): 96–124.

Additional discussions on this paper can be submit-ted until April 1, 2002. The author reserves the rightto reply to any discussion. Please see the SubmissionGuidelines for Authors on the inside back cover forinstructions on the submission of discussions.

AUTHOR’S REPLY

I would like to thank the discussants for theirvaluable comments and additional references,and to commend the editors of the North Ameri-can Actuarial Journal (NAAJ) for their decisionto turn my contribution into a discussion paper.The end result is an in-depth review of the im-portant role of Bayesian statistics in actuarial sci-ence. I share the hope expressed by Mendoza thatit will contribute to a further development of thisfield of research. After all, as de Alba put it,“Bayesian methods can essentially be applied

* David P.M. Scollnik, A.S.A., Ph.D., is Associate Professor, Depart-ment of Mathematics and Statistics, University of Calgary, Calgary,Alberta, Canada, T2N 1N4, e-mail: [email protected].


whenever there is a statistical inference prob-lem.”

When I singled out the principal applications ofBayesian methods in actuarial science, my aimwas to point out some areas in which Bayesianmethods have been applied extensively (credibil-ity and loss reserving). In so doing, I could notcover all areas of significant importance. Thanksto the discussants, additional areas and new di-rections for future research are now exposed tothe readers of the NAAJ.

Hickman and Jones placed the Bayesian in-roads into the actuarial profession in a fascinatinghistorical perspective and rightly emphasized thepioneering contribution of Bailey. Rosenberg fo-cused on Bayesian modeling in the importantarea of health care. She scanned major journalsand exposed current fields of research. Whilemany of the thousands of articles that emergedfrom her search of the word “Bayesian” in Med-line are not directly related to actuarial practice,the many recent publications she quoted bearwitness to a lively research area reported in jour-nals unfamiliar to many actuaries.

Another area of application is that of gradua-tion, which has attracted considerable Bayesianattention over the years. Hickman and Jones, deAlba, and Mendoza outlined milestones of Bayes-ian graduation from the Bayesian interpretationof Whittaker graduation to Markov chain MonteCarlo (MCMC) solutions.

Mendoza took a wider look at the Bayesianapproach by supplementing probabilistic model-ing with utility functions as part of the broaderdecision theory context. Consequently, the endproduct of the analysis is not merely the evalua-

tion of posterior distributions (or estimation ofthe model’s parameters), but the taking of a de-cision that maximizes the expected utility. Thisapproach requires both the assessment of priordistributions and utility functions, an importantarea for future research. Mendoza’s suggestionthat actuaries have to take advantage of the pos-terior predictive distribution should attract seri-ous consideration. This is also emphasized by deAlba.

Vazquez-Polo provided us with a lucid intro-duction to Bayesian prior robustness. He furtherdeveloped Bayesian robustness analysis in bi-modal structure function and obtained interest-ing results, which are suitable whenever the het-erogeneity of the portfolio (and that of thestructure parameter) can be expressed by meansof a mixture model. In so doing, he laid out thedirection future research is likely to follow.

Hickman and Jones recommend using Bayesiantime series analysis in the fields of health, pen-sion, and social insurance systems. De Alba re-ports the growing interest in the use of Bayesiangeneralized linear models. These methodologiesshould be seriously considered, especially sincecomputational difficulties are dramatically re-duced with the advancement of the MCMC meth-odology. This methodology allows us, for in-stance, to do away with empirical Bayes methods,correctly criticized by de Alba, and to carry fullBayesian analysis via hierarchical models.

Finally, Scollnik brought our attention to hisrecent paper on MCMC, which referred to theBUGS software as well as many additional recentresearch works in actuarial science incorporatingBayesian methods.


Defined Benefit Plans in CanadaAlthough the prevalence of DB plans has not de-clined in Canada to the same extent as in theUnited States, the decline nonetheless is signifi-cant. As shown in Table 1 of Brown and Liu, thenumber of DB plans decreased from 8,305 to6,795 from 1988 to 1998.

Moreover, for many reasons, most employerswho convert to a DC plan allow existing membersof the DB plan to retain their entitlements in theDB plan, without transferring their entitlementsto the DC plan. These plans, although they areclosed to new members, would not be consideredterminated, and would therefore not appear inthe statistics quoted by Brown and Liu. With theclosure of DB plans to new participants, the de-cline in DB assets, benefit payments, and partic-ipants will be slow and prolonged.

Ostaszewski challenges the often-used explana-tion that government involvement is a cause ofthe shift from DB to DC plans (Excess RegulationTheory). The data in Canada support Ostaszew-ski’s questioning of this explanation, at least fromthe perspective of the impact of increased regu-lation on the cost of pension plan administration.Statistics Canada reports that, although the ad-ministration costs of trusteed pension plans havemore than tripled from $338 million to $1.164million from 1988 to 1998 (Statistics Canada1998, p. 15), the average administration expensesin 1998 amounted to only 0.22% of plan assets.Given the double-digit returns of pension fundsduring the decade, the support for the ExcessRegulation Theory from a cost perspective is notcompelling.

REFERENCES

CANADIAN INSTITUTE OF ACTUARIES. 2001. Report on Economic Sta-tistics 1924–2000. Ottawa, ON: Canadian Institute of Ac-tuaries.

STATISTICS CANADA. 1998. Trusteed Pension Plans, Financial Sta-tistics, 1998. Ottawa, ON: Statistics Canada.

———. 1999. Pension Plans in Canada, Statistical Highlightsand Key Tables. January 1. Ottawa, ON: Statistics Canada.

AUTHOR’S REPLY

I appreciate Mr. Satanove’s insight into the hy-pothesis about the relationship between definedbenefit and defined contribution plans that I hadproposed. His idea looks particularly appealingfrom the perspective of behavioral finance. How-ever, because it also results in fewer data pointsbeing available for analysis (as he only considersrecent rates of return), it becomes more difficultto prove empirically.

Of course, when working with economic data,empirical proofs become quite a challenge. Themessage I am trying to convey is: Rates of returndo matter. I concur with Mr. Satanove that theymay be perceived rates of return. — Krzysztof M.Ostaszewski

“Principal Applications ofBayesian Methods in ActuarialScience: A Perspective”Udi E. Makov, October 2001

ROBERT F. LINK*This article notes that Bayesian methods wereintroduced into actuarial science in the late1960s. For the historical record, I recall readingan Equitable Life Assurance Society office memoon this subject in the early 1950s. It was writtenby Howard H. Hennington, FSA, MAAA, a closeassociate at the time. In it, Mr. Hennington pre-sents a derivation of a group insurance credibilityformula explicitly based on Bayes’s Theorem. Theformula had a form that was familiar at the time,and it was used in Equitable’s group coverages.I’m not aware that anyone at the time thought ofthis as a pioneering breakthrough, though I didadmire the elegance of the work. It is doubtfulthat anyone could find a copy of the memo at thislate date.

* Robert F. Link, FSA, MAAA, is a retired actuary, 309 Ferry Rd., OldLyme, CT 06371, e-mail: [email protected].

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Bayesian in actuarial application