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Library of Congress Cataloging-in-Publication Data
Klugman, Stuart A., 1949-
Bayesian statistics in actuarial science : with emphasis on
credibility I Stuart A. Klugman.
p. em. --:- (Huebner international series on risk, insurance,
and economic security)
Includes bibliographical references and index.
ISBN 0-7923-9212-4 (acid-free paper)
1. Insurance-Statistical methods. 2. Bayesian statistical
decision theory. I. Title. II. Series.
HG8781.K58 1991
368 '.01-dc20 91-25856
CIP
Copyright © 1992 by Kluwer Academic Publishers
All rights reserved. No part of this publication may be reproduced, stored in a retrieval
system or transmitted in any form or by any means, mechanical, photocopying, recording,
or otherwise, without the prior written permission of the publisher, Kluwer Academic
Publishers, 101 Philip Drive, Assinippi Park, Norwell, Massachusetts 02061, U.,:S.A.
Printed on acid-free paper.
Printed in the United States of America
TABLE OF CONTENTS
Preface ····························································································· xi
1. Introduction ................................................................................. . 1
2. Bayesian Statistical Analysis ......................................................... . 5
A. The Bayesian Paradigm ..................................................... . 5
B. An Example ..................................................................... . 7
C. Prior Distributions ............................................................ . 10
D. Model Selection and Evaluation ......................................... . 13
1. Graphical Analysis ............................................... .. 14
2. A Selection Criterion ............................................. . 15
3. Computational Aspects of Bayesian Analysis ................................... . 17
A. An Example ..................................................................... . 17
B. Numerical Integration ....................................................... .. 19
1. Adaptive Gaussian Integration .............................. .. 20
2·. Gauss-Hermite Integration .................................... .. 21
3. Estimating the Mean and Covariance .................... .. 24
4. An Example .......................................................... . 25 ..
C. Monte Carlo Integration. .................................................... . 29
D. Adjustments to the Posterior Mode .................................... . 32
E. Empirical Bayes Style Approximations ............................... . 33
F. Summary .......................................................................... . 35
4. Prediction with Parameter Uncertainty. ........................................... . 37
A. The Model ........................................................................ . 37
B. A Life Insurance Example .................................................. . 38
C. A Casualty Insurance Example. ......................................... .. 42
D. The Kalman Filter ............................................................ .. 46
E. Return of the Casualty Insurance Example ........................ .. 50
5. The Credibility Problem ................................................................ . 57
A. A Simple Model ............................................................... .. 58
B. Estimating the Class Mean ................................................ .. 58
C. Credibility Issues ............................................................ .. 62
6. The Hierarchical Bayesian Approach .............................................. .. 65
A. What It Is ........................................................................ . 65
B. An Example ..................................................................... . 67
C. The General Hierarchical Model.. ...................................... .. 71
D. Simplifying Assumptions.................................................... 76 9. Modifications to the Hierarchical Normal Linear Model.. .................. 151
1. Normality .............................................................. 76 A. Lognormal ......................................................................... 151
2. Linearity .......................... ....... .... ........ ..... ...... ... .... 78 B. Poisson ............................................................................. 152
3. Variance Independent of the Mean........................... 78 C. Non-normal Models Based on Parameter Estimates ............. 154
7. The Hierarchical Normal Linear Model............................................. 81 Appendix. Algorithms, Programs, and Data Sets ................................ .. 159
A. The Model ......................................................................... 81 A. The Simplex Method of Function Maximization ................ .. 159
B. Examples - Description .............. ...... .. .......................... ...... 82 B. Adaptive Gaussian Integration .......................................... .. 163
1. One-Way ............................................................... 82 C. Gauss-Hermite Integration ................................................ .. 165
2. Two-Way ............................................................... 83 D. Polar Method for Generating Normal Deviates .................. .. 166
3. Linear Trend ...................... ...................... .... ......... 85 E. GAUSS Programs .............................................................. . 166
4. Kalman Filter .. .. .. .. .. .. .. .. .. .. .. .. .... .. .. .. .. .. .. .. .. .. .. .. .. .. .. . 86 1. Simplex Maximization. .......................................... .. 166
5. Graduation .................. ....... ...... .. .. .. ....... .......... ... ... 89 2. Adaptive Gaussian Integration .............................. .. 170
C. Analysis of the Model .......... ...................... ...................... ... 89 3. Gauss-Hermite Integration .................................... .. 173
1. Preliminaries .. .. .... ................................ .... .. ........... 89 4. Monte Carlo Integration ........................................ .. 179
2. The Analysis .......... ...................................... ......... 90 5. Tierney-Kadane Integration. ................................... . 182
D. Examples - Analysis .......................................................... 97 F. Data Sets ........................................................................ .. 184
1. One-Way ............................................................... 98 1. Data Set 1 ............................................................ . 184
2. Two-Way ............................................................... 100 2. Data Sets 2-4 .................................................... .. 184
3. Linear Trend ......................................................... 107
4. Kalman Filter ......................................................... 109 Bibliography ...................................................................................... . 229
't
5. Graduation ............................................................ 109
E. Prior Distributions ............................................................. 110 Index ································································································· 235
F. Model Selection and Evaluation. .......................................... 111
8. Examples ...................................................................................... 115
A. Data Sets .......................................................................... 115
B. Analyses ........................................................................... 118
1. One-way Model, Data Set !............................ ......... 118
2. One-way Model, Data Set 2..................................... 125
3. Empirical Bayes Style Approaches........................... 129
4. An Iterative Approach ............................................ 131
5. Other Priors .. .... .................................................... 133
6. Diagnostics .... .... ........................... ......... ........... ... .. 135
7. Two-way Model, Data Set 4 .................................... 143
8. Linear Trend Model, Data Set 3 .............................. 145
9. Kalman Filter, Data Set 3 ....................................... 146
10. Graduation .......................................................... 148
PREFACE
The debate between the proponents of "classical" and "Bayesian"
statistical methods continues unabated. It is not the purpose of the text to
resolve those issues but rather to demonstrate that within the realm of
actuarial science there are a number of problems that are particularly suited
for Bayesian analysis. This has been apparent to actuaries for a long time,
but the lack of adequate computing power and appropriate algorithms had
led to the use of various approximations.
The two greatest advantages to the actuary of the Bayesian
approach are that the method is independent of the model and that interval
estimates are as easy to obtain as point estimates. The former attribute
means that once one learns how to analyze one problem, the solution to
similar, but more complex, problems will be no more difficult. The second
one takes on added significance as the actuary of today is expected to
provide evidence concerning the quality of any estimates.
While the examples are all actuarial in nature, the methods
discussed are applicable to any structured estimation problem. In
particular, statisticians will recognize that the basic credibility problem has
the same setting as the random effects model from analysis of variance.
As the technological problem is the most difficult, this monograph
spends a great deal of time on computational issues. The algorithms are
presented in the text and computer programs are listed in the Appendix. In
addition, each model is illustrated with an example that demonstrates that
these algorithms can be applied to realistic data sets. The data sets are also
included in the Appendix so that the interested reader can duplicate the
results presented here as well as try other models.
This work had its origins when I spent the Fall of 1984 at the
National Council on Compensation Insurance in New York City. I learned
Bayesian Statistics in
Actuarial Science
a great deal about credibility while working on a rate making project there.
This project was headed by Gary Venter, from whom I learned a great deal
about credibility and with whom I had numerous conversations over the
following years. Gary also was kind enough to supply the data that appears
in Data Sets 2-4. Many other useful ideas came from conversations with
Glenn Meyers, particularly while we were faculty members at The
University of Iowa. I, of course, owe an enormous debt to the many
actuaries who from the beginning of the century have advanced the theory
of credibility and to the many statisticians who have moved Bayesian
analysis from theory to practice.
Finally, financial support for this project was given by the S. S.
Huebner Foundation. This provided the time and the incentive for
completing this project.
1.
INTRODUCTION
In the minds of most statisticians there are (at least) two mutually
exclusive approaches to data analysis. The "classical" or "frequentist"
theory consisting of confidence intervals and hypothesis tests is certainly the
most widely used and takes up the vast majority, if not all, of the typical
statistics text. On the other hand, "Bayesian" statistics, a mode of
inference based on Bayes' Theorem, has attracted a small group of
passionate supporters. The debate continues with papers such as "Why
Isn't Everyone a Bayesian?" (Efron, 1986) drawing numerous comments
and letters.
The purpose of the monograph is not to settle this issue or even to •
attempt to make a contribution toward its resolution. Rather, it is to
demonstrate that there is a well-deserved place for Bayesian statistics in the
analyst's arsenal. This is accomplished by showing that two important
actuarial problems can be successfully solved by the Bayesian approach.
The first is the general problem of model-based prediction and normally
involves three steps. Begin by collecting data on the past behavior of an
insurance system. This may, for example, be the ages at death of the
holders of a certain type of life insurance policy or the indemnity amounts
paid to the members covered by a group insurance policy. The second step
is to build a model describing the random process that yielded these
observations. Examples of such models abound in the actuarial literature
(e.g., Batten (1978), Hogg and Klugman (1984), Hossack, Pollard, and
Zehnwirth (1983), and London (1988)). The third step is to make
predictions of future outcomes based on the model. Of growing importance
to the actuary is not just the expected future outcome, but the variability of
the prediction. In measuring variability it is essential that as many sources
as possible be taken into account. The two most obvious are the natural
process variability that is inherent in the model and the variability because
the model is estimated from data. It will be seen that the Bayesian
2 Bayesian Statistics in Actuarial Science Introduction 3
approach provides an ideal mechanism for combining these two sources of
appear to be unrealistic in the insurance setting, using them leads to results
variability.
that match those currently being used. These assumptions are stated and
defended in Chapter 6. In Chapter 7 the basic Bayesian model is
introduced. It incorporates the assumptions set out in Chapter 6 and adds
The second problem comes under the general name of credibility.
the additional assumption of linearity. Many of the commonly used models
Since the first part of the twentieth century it has been recognized that
fit this framework. The formulas for doing a complete credibility analysis
when setting insurance premiums the best result is obtained when the pure
are derived and the formulas for a number of specific examples given. In
premium is somewhere between the actual experience of the insured and the
Chapter 8 several data sets are analyzed to demonstrate both the power and
overall average for all insureds. This can be justified in several ways.
feasibility of the Bayesian approach. Finally, in Chapter 9 the assumptions
Foremost is that an individual insured or group of insureds will rarely
introduced in Chapter 6 will be relaxed to accommodate more models. In
provide enough information to accurately estimate their potential for future
this chapter the power of the Bayesian approach will be fully realized. The
claims. As a result there will be wide spreads in the recorded experience
monograph will then close with an Appendix in which programs for
from year to year. By shrinking to the overall average these fluctuations
performing the various computations and a description of the algorithms are
will be tempered. Another advantage is that insureds with exceedingly given.
unfavorable experience in a given year will not be penalized to the full
extent indicated by this experience. (Of course this must be balanced by
Throughout the monograph a number of examples will be used to
giving a less than full reward to those with good experience.) This creates
illustrate the various methods. Their main purpose is to demonstrate that
less hostility among the policyholders. At the other extreme it is clearly
the proposed methods can be implemented. To avoid introducing a new
incorrect to set a pure premium of zero for an insured who has no claims
example for each model it will often be the case that a data set is analyzed
during the experience period. The difficult problem is the determination of
with a model that is clearly inappropriate. One benefit is that the model
the relative emphasis to place on the specific and overall experiences.
selection techniques introduced in Chapter 2 should be able to point this
Again, it will be seen that a Bayesian approach is the best way to resolve
out. Nevertheless, it should be recognized that model selection may well be \
this problem.
more important than the method used to analyze the model. Selection of a
good model depends on a thorough knowledge of the process that produced
The first step in solving these problems is to provide the specifics of
the losses.
the Bayesian approach to modeling and data analysis. This is done in
Chapter 2 where it is seen that the ideas are deceptively simple while the
All the computations were done on an MS-DOS based personal
implementation is decidedly complex. Most of the complexity centers on
computer (80286) with the numerical co-processor added. All the programs
the computations needed to complete the analysis. For the most part these
were written in the GAUSS programming language and run under that
turn out to be numerical integration problems. Solutions to the system. This language is similar to the SAS Matrix language.1 More
computation issues are provided in Chapter 3.
details are given in the Appendix.
The problem of model-based prediction is taken up in Chapter 4. It
begins with a simple model and an appropriately simple illustration. This
is followed by a more complex example. This Chapter also introduces the
Kalman filter, a powerful model for describing an environment in which
parameters change over time. This allows the introduction of a third kind
of uncertainty, that tomorrow's model might be different from yesterday's.
The credibility problem is attacked in Chapters 5 through 9. The
details of the problem are set out in Chapter 5 along with various issues
that the analyst must resolve. The advantages of a Bayesian analysis are I
detailed as part of this discussion. The next step is to simplify matters by il 1
MS-DOS is a trademark of Microsoft, Inc.; GAUSS is a trademark of Aptech Systems;
placing a number of assumptions on the claims process. While they may
,f Inc., SAS is a trademark of SAS Inc.
2.
BAYESIAN STATISTICAL ANALYSIS
A. THE BAYESIAN PARADIGM
As discussed in Chapter 1, it is not the intention of this monograph
to provide a convincing philosophical justification for the Bayesian
approach. Excellent discussions of these matters can be found, for example,
in Berger (1985, Chapters 1 and 4) and Lindley (1983). In the latter paper
the Bayesian paradigm is described in its simplest form. Of interest is a
quantity B whose value is unknown. What is known is a probability
distribution 1r( B) that expresses our current relative opinion as to the
likelihood that various possible values of B are the true value. For
additional discussion of the merits of expressing uncertainty by probability ,
see Lindley (1982 and 1987). This is called the prior distribution as it
represents the state of our knowledge prior to conducting the experiment.
The second item is the probability distribution f(x I B). It describes
the relative likelihood of various values of x being obtained when the
experiment is conducted, given that B happens to be the true parameter
value. This is called the model distribution and is the one element that is
common to both Bayesian and classical analyses. It should be noted that it
is possible for both x and B to be vectors, the former being the several items
from the sample and the latter a set of unknown parameters.
The next step is to use Bayes Theorem to compute
*(B I x) = f(x I B)1r(B)
11' J f(x I B)1r(B)dB (2.1)
the posterior distribution2 of B. It represents our revised opinion about (J
having seen the results of the current experiment. This density contains all
6 Bayesian Statistics in Actuarial Science Bayesian Statistical Analysis 7
of our current knowledge about the unknown parameter, just as the prior The Bayesian Central Limit Theorem (Berger, 1985, p. 224) indicates that
density used to. under suitable conditions the posterior density can be approximated by the
normal distribution and so the credibility interval is approximately
The final step is to use the posterior density to draw whatever
conclusions are appropriate for the problem at hand. We will concentrate
E(e; I x) ± zl-a/2~Var(e; I x). (2.5)
on four specific items. The first is a point estimate of one of the parameters
in the vector e. Writing it in vector form, we have fJ = (e1, ... ,ek)' and the
posterior distribution of ei is As with the usual Central Limit Theorem, the approximation improves as
the number of observations increases.
(2.2)
Of greater interest is the value of a future observation. Suppose
the density of this observation is g(y I e). Note that the model for this
A viable point estimate of ei is then some measure of the center of this density need not match the model that produced the observations x, but it
e.
posterior density. The two most common are the mean and the mode. must depend on the same parameter The predictive density is then
Justification for these choices can be obtained by choosing appropriate loss
functions and applying Bayesian decision theory, but it is sufficient to recall f*(y 1 x) = J g(y 1 e)7r*(e 1 x)de (2.6)
that the posterior is the sum of our knowledge and the mean and mode are
useful summary values. The second item is the Bayesian version of the
confidence interval. A 100(1-a)% credibility (not to be confused with and it represents all of our knowledge about a future observation. Both
actuarial credibility as discussed in Chapters 5 through 9) interval for ei is point and interval estimates for this value can be constructed as was done
any pair (l, u) that satisfies for the parameter itself.
There are three major problems in implementing the Bayesian
Pr(l < ei < u) = J ~ 1r*(ei I x)dei = 1-a:.3 (2.3) paradigm. One is the selection of the prior distribution. This is the mof!t
often criticized facet of Bayesian analysis with the objection being made
Perhaps the most desirable choice for the end points is that pair for which that the personal nature of this distribution removes the aura of objectivity
u-1 is the smallest. For a unimodal density the solution is the pair that that is expected to surround scientific explorations. The second is the
satisfies both (2.3) and selection of the model. The problems are not much different from those
faced by all analysts. These two issues will be discussed in detail later in
this Chapter. Once the two elements are in place the analysis then consists
1r*(ei = ll x) = 11'*(e; = u I x). (2.4) of evaluating the appropriate expressions as given above. While the
formulas are simple in appearance, the required integrals are often
extremely difficult to do. Being able to perform high dimension numerical
integration is essential for the Bayesian analyst. Several techniques for
doing this are presented in the next Chapter.
2
Throughout this monograph all densities will be written as if the random variable is
absolutely continuous. For discrete variables the integrals should be replaced by the
appropriate summation. B. AN EXAMPLE
3
The often used convention of using upper case symbols for random variables and lower
At this point it is appropriate to introduce an example. Suppose
case symbols for observed values will not be used here. Only one version of the symbol
we are observing the number of claims from one randomly selected driver
will appear and its role should always be obvious from the context. As well, the words
from the collection of all insured drivers. Assume that the number of
estimate and estimator will be used interchangeably. The Bayesian analyst must be
e
claims in one year has a Poisson distribution with parameter and that the
comfortable with the concept that both parameters and observations are random
numbers of claims in different years are independent. If we observe t years
variables at times.
