Bayesian Statistical Modelling

implementing these tools. Supporting developments are given in Part II. The printed tables and access to the CD-ROM are given in Part III as needed to implement the methods. Detailed case studies are developed in Part IV, illustrating the range of data analyses supported by the tables. The stated objectives—to offer statistical methodology for use by laymen outside the grasp of supporting principles—are achieved commendably by the authors. The tables, both printed and electronic, are easily accessed by the novice through a self-paced study following step-by-step examples, especially as given in Part IV. At the same time, knowledgeable users deserve some explanation as to the statistical principles on which the methodology rests. Comments on these issues constitute much of the remainder of this review. Let X be the sample space containing outcomes X4n5 D 6X11 : : : 1Xn7 of independent Bernoulli trials having parameter M p 2 601 17 with value p to be determined, and let XS D X1 C X2 C ¢ ¢ ¢ C Xn. Procedures are based in principle on either X4n5 or XS , but in practice on the latter. The authors trace developments back to Jacob Bernoulli, and draw heavily on the foundations of Jerzy Neyman for “scientiŽ c statistics.” The authors break ranks with conventional statistics on two essential grounds: (1) the range of M p and (2) the use of asymptotics in a practice often typiŽ ed by small to moderate samples. Regarding (1), they conŽ dently assert that users can accurately stipulate a proper subset 6p1 N p7 of 601 17, called the measurement space, wherein M p is known to lie with certainty. Accordingly, they seek solutions to problems of inference that conform to the measurement space. Classical estimation using XS=n is faulted heavily here in giving often nonconforming values. With regard to (2), the computerintensive developments on which the tables rest are essentially exact for small samples. In rough analogy with scales for weighing mice and elephants, to each measurement space there corresponds a variety of data—analytic tools, listed as (a)–(d) in the second paragraph of this report, that are supported through the printed tables and the CD-ROM. Technical developments begin with the dual issues of ‚-measurement intervals for M p in 6p1 N p7 and ‚-prediction regions in X, with conŽ dence level ¶ ‚ as a gauge of their reliability. These constitute the “‚-measurement & prediction space.” Other methods build on these. Point estimation focuses on ‚-estimators in 6p1 N p7 depending on the sample size as well as on the conŽ dence level ‚. These include (a) the “minimum MSE ‚-estimator,” designed to minimize the conditional mean squared error, given the ‚-measurement & prediction space, and (b) the “midpoint ‚-estimator” as the midpoint of the ‚-measurement interval for M p in 6p1 N p7. The aim of exclusion, in lieu of hypothesis testing, is “to show that the actual value p of M p is different from any value in 6p1 N p7 6p1 N p7, where the reliability of the exclusion procedure is speciŽ ed by the signiŽ cance level .” Thus H0 2 M p 2 6p1 N p7 is excluded from 6p1 N p7. There appear to be no errors of the second kind, because M p always belongs to 6p1 N p7, and thus no concept of the power of an exclusion procedure to exclude. The extensive tables are the result of computer-intensive optimization algorithms seeking optimal precision for each nominal reliability level, while reducing excess reliability arising from discreteness of the problem. Procedures based on X4n5 are shown by inclusion to be superior to ones based on XS . Nonetheless, the available tables use XS owing to apparent practical constraints. In particular, typical input variables are the measurement range 6p1 N p7, the sample size n, the conŽ dence level ‚, the realization XS , and allied information pertaining to exclusion, for example. Output in turn consists of ‚-measurement intervals and other quantities of use in assessing the data. Principles undergirding the analyses are ostensibly non-Bayesian. Nonetheless, M p does become a random variable during the course of the authors’ developments, essentially through the assignment of a Bayesian uniform prior over 6p1 N p7. Despite the careful development of these methodologies, and extensive tables for their implementation, this reviewer sees serious impediments to their effective use. These reservations focus largely on the assumption that the measurement range 6p1 N p7 can itself be stipulated accurately by users. This concern pervades every stage of the scientiŽ c method. New experiments, unless strictly conŽ rmatory, do chart new paths, so that past experience regarding earlier measurement spaces need not carry over without modiŽ cation. At issue are problems with misspeciŽ cation of the range, the consequences of such misspeciŽ cation, and possible robustness of procedures to such misspeciŽ cation. The authors essentially remain mute on these critical issues. For if the parameter range is cast too wide, then the authors’ objections to classical methods (based on M p 2 601175 apply verbatim to their own methods, but now with regard to nonconformity with the actual (now smaller) measurement space. Consequences of prescribing too narrow a range remain to be studied. On the other hand, if the range supported by prior user knowledge is sufŽ ciently narrow, then all statistical procedures become moot. Early in their monograph the authors appear to subscribe to the following point of view: “ If statistics is an applied Ž eld and not a minor branch of mathematics, then more than ninety-nine percent of the published papers are useless exercises.” Apparently, Binomial Distribution Handbook for Scientists and Engineers represents their efforts to be included in the other 1%. I must leave it to the experience of other users to judge how well this objective has been met.