부산대학교 도서관

The theme of this book is summarized in a footnote on p. xix: ``ourtopic is the optimal processing of incomplete information''. And again,``Our theme is simply: probability theory as extended logic'' [p.\xxii]. Roughly, and perhaps slightly inaccurately put, this work is adevelopment of probability and statistics from this point of view.\parWhile frequentists might easily dismiss this as another Bayesian text,statisticians of the Bayesian persuasion would have greater difficultyin seeing it as supporting more classical (Fisher, Neyman-Pearson)methods. Jaynes himself, however, has no hesitation, claiming that hismethod, based on the principle of maximum entropy, encompasses notonly Bayesian and frequency calculations but also calculations inneither group.\parThe book is in two parts: I. Principles and elementary applications,and II. Advanced applications. There are a number of challengingexercises, two indexes, a list of references and a bibliography.\parIn Chapter 1, ``Plausible reasoning'', Jaynes introduces the keystoneof his theory: a robot that will ``carry out useful plausiblereasoning, following clearly defined principles expressing anidealized common sense'' [p. 8]. Three basic desiderata are given,which show not only the inner workings of the robot's brain, but alsohow the robot relates (or, more accurately, should relate) to theexternal world.\parIn Chapter 2 the usual fundamental rules for probabilities are derivedfrom the aforementioned desiderata. The method follows that of R. T. Cox\ [{\it The algebra of probable inference}, The Johns Hopkins Press,Baltimore, Md, 1961; MR0130703 (24 \#A563)].\par``Elementary sampling theory'' is the subject of Chapter 3, whileChapter 4 is concerned with elementary hypothesis testing. The readerused to a more conventional approach may be surprised to find at thebeginning of this chapter a discussion of likelihoods, prior andposterior probabilities.\parIn Chapter 5, ``Queer uses for probability theory'', matters asdiverse as extra-sensory perception, the discovery of Neptune, horseracing and weather forecasting are considered.\parChapter 6 is concerned with elementary parameter estimation. In $\S6.20$Jaynes discusses the ``taxicab problem'' and mentions his inability tofind it in the orthodox literature. It is in fact discussed in [H. Jeffreys, {\it Theory of probability}, Third edition, Clarendon Press,Oxford, 1961; MR0187257 (32 \#4710)] (where it is concerned with tramcars) and in [C. P. Robert, {\it The Bayesian choice}, Translated and revised from theFrench original by the author, Springer, New York, 1994; MR1313727 (97d:62059)]. Notice,by the way,that while Jeffreys attributes the problem to Maxwell Herman AlexanderNewman, a gremlin---dare we say Maxwell's demon?---causes Robert toattribute it to [Jerzy?] Neyman!\parChapter 7 is concerned with the normal distribution,and Chapter 8 deals with sufficiency, ancillarity ``and all that''.\par``Repetitive experiments: probability and frequency'' is the title ofthe ninth chapter, one in which Jaynes examines what predictions hisrobot can make with very limited information.\parIn Chapter 10 Jaynes considers the fundamental difficulty associatedwith the notion of ``random'' experiments, particularly in relation tophysics. The entropy principle is introduced in Chapter 11, the firstin the advanced applications section. Chapter 12 is concerned withignorance priors and transformation groups.\parBertrand's paradox is discussed here, the recognition of an element ofrotational symmetry enabling Jaynes to arrive at a uniqueanswer. Jaynes had considered this paradox in 1973 [Found. Phys. {\bf 3}(1973), 477--492; MR0426227 (54 \#14173)],and it is perhapsa pity that he does not respond here to later relevant work byothers---e.g. K. S. Friedman [Found. Phys. {\bf 5} (1975), no. 1, 89--91]and A. Nathan [Philos. Sci. {\bf 51} (1984), no.~4, 677--684 781314 ].\parJaynes devotes his thirteenth chapter, ``Decision theory, historicalbackground'', to showing that the procedures he has so far introducedare optimal in terms of some clearly defined criterion. In Chapter 14the decision theory of the previous chapter is applied to thedetection of signals in noise and the detection of systematic drift inmachine characteristics.\parThe word ``paradox'' is ambiguous, and Jaynes advisedly gives aprecise definition, perhaps different from that in common acceptance,of his usage of the term at the start of his fifteenth chapter, onedevoted to some probabilistic paradoxes.\parThe next chapter is devoted to the historical background of orthodoxstatistics. Jaynes being a physicist, we can perhaps see his view onscientific inference, similar to that of Daniel Bernoulli, Laplace andother mathematical physicists, as tainted, or at least tinctured, byprofessional interest. Jaynes briefly discusses the methods introducedby the Fisher and Neyman-Pearson schools, and mentions the attacks anddefences of men like M. G. Kendall, Cramér, Feller, von Mises,``and even the putative Bayesian L. J. Savage'' [p. 493], etc.---thecertainty expressed by these savants in their discussions Jaynes findsalmost inexplicable ``since they were all quite competentmathematically'' [p. 493]. O mysterious condescension!\parIn Chapter 17, the ``Principles and pathology of orthodoxstatistics'', Jaynes investigates the consequences of not usingBayesian methods in some very simple problems. Cox's theorems, asdeveloped by Jaynes, facilitate the examination of how, and under whatcircumstances, orthodox and Bayesian results differ.\parIn Chapter 18, ``The $A_p$ distribution and rule of succession'',Jaynes investigates whether some machinery can be built into the robotthat will enable it to store general conclusions. This is effected bythe introduction of a new proposition $A_p$, interpreted as``regardless of anything else you may have been told, the probabilityof $A$ is $p$''.\parJaynes returns to the matter of physical measurement in Chapter 19,paying especial attention to the problem of the great inequality ofJupiter and Saturn. Chapter 20 is concerned with model comparison, and``Outliers and robustness'' is the subject of Chapter 21. The finalchapter presents an introduction to communication theory.\parThat both the references and bibliography are annotated is mostuseful, Jaynes showing in some of his remarks that he is indeedfortiter in both {\it re} and {\it modo}.\parThe anecdote about ``Given $2+2=5$: prove that I am the Pope''attributed in Appendix B to G. H. Hardy and J. E. McTaggart isascribed in [G. Birkhoff, {\it Lattice theory}, Corrected reprint of the1967 third edition,Amer. Math. Soc., Providence, R.I., 1979; MR0598630 (82a:06001)] to BertrandRussell. Further, note that Borel (1924) is in fact translated in thefirst edition (1964) of Kyburg \& Smokler, and is not in the second.\parJaynes notes that part of the problem between Jeffreys (ageophysicist) and Fisher (a geneticist) rose from their considerationof very different problems. One may well ask oneself whether there is{\it one} probabilistic/statistical methodology that is appropriatefor all scientific inference, as this book in fact suggests.\parThis is a work written by a scientist for scientists. As such it is tobe welcomed. The reader will certainly find things with which hedisagrees: but he will also find much that will cause him to thinkdeeply not only on his usual practice but also on statistics andprobability in general. {\it Probability theory: the logic of science}is, for both statisticians and scientists, more than just``recommended reading'': it should be prescribed.