An Annotated Bibliography on the Foundations of Statistical Inference

•

AN ANNOTATED BIBLIOGRAPHY ON THE FOUNDATIONS OF STATISTICAL INFERENCE

• by Stephen L. George A cooperative effort between the Department of Experimental • Statistics and the Southeastern Forest Experiment Station Forest Service U.S.D.A.

Institute of Statistics Mimeograph Series No. 572 March 1968 The Foundations of Statistical Inference--A Bibliography

During the past two hundred years there have been many differences of opinion on the validity of certain statistical methods and no evidence that ,. there will be any general agreement in the near future. Also, despite attempts at classification of particular approaches, there appears to be a spectrum of ideas rather than the existence of any clear-cut "schools of thought. " The following bibliography is concerned with the continuing discussion in the statistical literature on what may be loosely termed ''the foundations of statistical inference." A major emphasis is placed on the more recent works in this area and in particular on recent developments in Bayesian analysis. Invariably, a discussion on the foundations of statistical inference leads one to the more general area of scientific inference and eventually to the much more general question of inductive inference. Since this bibliography is intended mainly for those statisticians interested in the philosophical foundations of their chosen field, and not for practicing philosophers, the more general discussion of inductive inference was deliberately de-emphasized with the exception of several distinctive works of particular relevance to the statistical problem. Throughout, the temptation to gather papers in the sense of a collector was resisted and most of the papers listed are of immediate relevance to the problem at hand. The bibliography covers principally the period from. the early 1940's to the summer of 1967, but contains in addition several classical papers from earlier times. No claim is made as to the completeness of the bibliography or the quality of the papers- especially the more recent ones. The form. of the bibliography is fairly standard with the following abbreviations used throughout: ~ - Journal of the American Statistical Association AMS - .Az:u'1als of Mathematical Statistics lSI - International Statistical Institute JRSS - Journal of the Royal Statistical Society 2

Anscoinbe, Frank. J., "Statistical Inference,";rns,s" (B), 15(1953), 30-76. Anscombe, F. J. and R. J. Aumann, "A Definition of Subjective Probability," AMS, 34(1963), 199-205. Armitage, P., "Consistency in Statistical Inference and Decision," JRSS, (B), 23(1961), 1-37. ---- A discussion to C. A. B. Smith

, "Sequential Medical Trials: Some Comments on F. J. Anscombe's --=Pa-p-e-r," JASA, 58(1963), 384-387· A reply to Anscombe's review of Armitage's book Sequential Medical Trials. Suggests that consequences of likelihood inferences may be misleading. Arrow, Kenneth J., SOcial Choice and Individual Values, Cowles Commission Monograph No. 12, New York, John Wiley and Sons, 1951, 2nd ed. 1963. Arrow, Kenneth J., David Blackwell, and M. A. Girsh1ck, "Bayes and Minmax Solutions of Sequential Decision Problems," Econometrica, 17(1949), 213-243. Ashby, W. Ross, "Induction, Prediction, and. Decision-Making in Cybernetic Systems," Induction: Some' Current Issues, K;y'burg and Nagel (eds.), 1963· Bahadur, Raghu Raj, "Sufficiency and Statistical ,Decision Functions," AMS" 25(1954), 423-462. Bahadur, Raghu Raj, and Herbert Robbins, "The Problem of the Greater Mean," AMS, 21(1950), 469-481 Baker, S. F., Induction and Hypothesis: A'Study of the Logic of Confirmation, Cornell University Press, Ithaca, N. Y., 1957. Barnard, G. A., Discussion of "On the Foundations of Statistical Inference," JABA, 57(1962), 308.

, Discussion of "Consistency in Statistical Inference and --D-e-c-i-sion," JRSS, (B), 23(1961), 25-27.

, "Fisher's Contribution to Mathematical Statistics," ---~J:'::"RS=S:--, (A), 126(1963), 162-166.

, "Logical Aspects of the Fiducial Argument," Bull. Int. --~St~a~t-. Inst., 40(1964), 2, 870-883. , "The Meaning of a Significance Level," Biometrika, -----3~4(~1-947), 179-182.

, "Sampling Inspection and Statistical Decisions," JRSS, (B), --~1~6~(1~954), 151-174. 3 Barnard, G. A., "Some Logical Aspects of the Fiducial Argument," JRSS (B), 25(1963), 111-114. ---- A discussion of two earlier papers in this volume.

"Statistical Inference," JRSS (B), 11(1949), 115-149.

Barnard, G. A., G. M. Jenkins, and C. B. Winsten, "Liklihood, Inferences, and Time Series,1t JRSS (A), 125(1962), 321-372.

Barrett, W., "The Present State of the Problem of Induction, It Theoria, 6(1940).

Bartholomew, D. J., "A Comparison of Some Bayesian and Frequentist Inferences," Biometrika, 52(1965), 19-35.

Bartlett, M. S., itA Comment on D. V. Lindley's Statistical Paradox," Biometrika, 44(1957), 533-534. See Lindley's article in Biometrika (1957) for original article. , "Discussion of Professor Pratt's Paper," JRSS (B), 27(1965), --""""19".....7---=-198.

______, Essays on Probability and Statistics, London, Methuen, 1962. A collection of the author's own lectures given by invitation at various places from 1949 to 1956 with particular emphasis on the author's own philosophy of statistical inferences.

, "The Present Position of Mathematical Statistics," JRSS (A), ---~1~03~(~1940), 1-29·

, "Probability and Chance in the Theory of Statistics," ---~Pr-o-c-eedings of the Royal Society (A), 141(1933), 518.

_____-,.-_, "Probability, Statistics, and Time," Inaugural Lecture, University College, London.

, "R. A. Fisher and the Last Fifty Years of Statistical ---:M::-::'e-:t"':"'h-odology, It JASA, 60(1965), 395-409. Text of the R. A. Fisher Memorial Lecture given in December 1964. ------, "Statistical Probability," JASA, 31(1946), 553· Bayes, Thomas, "Essay Toward Solving a Problem in the Doctrine of Chances," The Philosophical Transactions, 53(1763), 370-418. Reprinted in Biometrika, 45(1958), 293-315· One of the most important publications in the history of statistics and probability. Contains a bibliographical note by G. A. Barnard.

Bell, E. T., Men of Mathematics, New York, Simon and Schuster, 1937·

Bergmann, Gustav, "The Logic of Probability," American Journal of Physics, 9(1941). 4 Berkson, J., "Some Difficulties of' Inte!'Tret.ation Encountered in the Application of the Chi-square irest, If JASA, 33(1938), 526-542.

Bernoulli, Jacob (James), Ars Conjectanr~, Basel, 1713. A German translation of this classic work is available.

Beveridge, W. I. B., The Art of Scientific Investigation, Melbourne, Heinemann, 1951.

Birnbaum, Allan, "The Anomalous Concept of Statistical Evidence, A Axioms, Interpretations in Elementary Exposition," Paper presented to Joint European Conference of Statistical Societies, Berne, Switzerland, Sept. 14, 1964.

_____.,..--, "Another View on the F'oundations of Statistics," Amer. Stat., 16(1962), 1, 17-21. Illustrates a viewpoint different from either the "Bayesian" or "Frequentist" positions.

Discussion of "On t.he Foundations of Statistical Inference," 57(1962), 322,

. ,"On the Foundat:i.ons of Statistical Inference," JASA, --~5=7~(1~962), 269-306. The author deduces the likelihood principle from his "principle of conditionality." A lively discussion follows the article.

__-=__, "On the Foundations of Statistical Inference: Binary Experiments," AMS, 32(1961), 414-435,

"A Unified Theory of Estimation, I, " AMS, 32(1961), 112-135.

Bizley, M. T. L., "Some Notes on Probability," Journal of the Institute of Actuaries Students' Society, 10(1951), 161-203.

Blackwell, David, "Comparison of Experiments," Proceedings of the Second (1950) Berkeley Symposium on Math. Stat, and Probability, ed. Jerzy Neyman, Berkeley, Univ, of Calif. Press, 1951, 93-102.

Blackwell, David and L. Dubi.ns, ''Merging of Opinions with Increasing Inforriiation," AMS, 33(1962), 882-886.

Blackwell, David and M. A. Girschick, The Theory of Games and Statistical Decisions, New York, John Wiley and Sons, 1954. Although the subject matter of this book is still considered open to discussion by most statisticians, the authors largely ignore philosophy and history in its development.

Bohm, David, Causality and Chance in Modern Physics, London, Routledge and Kegan Paul, 1957, Discusses the importance of probability theory in modern physics, Written for the educated layman. 5 Boole, G., Studies in Logic and Probability, LaSalle, Open Court, 1952.

Borel, Emile, Elements of the Theo of Probabilit , (translated by JohnE. Freund, Englewood Cliffs, New Jersey, Prentice-Hall, Inc., 1965. Contains philosophical discussion mixed in with the development of basic mathematical ideas. ____--:' Probability and Certainty, N. Y., Walker and Co., 1963. Discusses how probability is related to "practical" and "absolute" certainty. A slight change of thought from some of Borel's earlier work.

Probability and Life, New York, Dover Pub., 1962. , "The Theory of Play and Integral Equations with Skew ---::---Symmetric Kernels; On Games that Involve Chance and the Skill of the Players; On Systems of Linear Forms of Skew Symmetric Determinant and. the General Theory of Play (Translated by Leonard J. Savage)," Econometrica, 21(1953),97-124. Born, Max, Natural Philosophy of Cause and Chance, N. Y., Dover Publications, 1964. Contains the WaYnflete Lectures delivered at Oxford in 1948 together with a more recent essay, lISymbol and Reality." Concerned with the philosophical implication of the use of probability theory in scientific explanation. Box, G. E. P., Discussion of "On the Foundations of Statistical Inference," JASA, 57(1962), 311. Box, G. E. P. and N. R. Draper, "The Bayesian Estimation of Common Parameters from Several Responses," Biometrika, 52(1965), 355-365· Box, G. E. P., and G. C. Tiao, "A Further Look at Robustness via Bayes' Theorem," Biometrika, 49(1962), 419-432.

Box, G. E. P., and G. C. Tiao, '~Bayesian Approach to the Importance of Assumptions Applied to the Comparison of Variances," Biometrika, 51(1964), 153-167. Box, G. E. P. and G. C. Tiao, 'Multiparameter Problems from a Bayesian Point of View," AMS, 36(1965), 1468-1482. Braga-IlIa, Alvise, "A Simple Approach to the Bayes Choice Criterion: The Method of Extreme Probabilities," JASA, 59( 1964), 1227·

Braithwaite, Richard Bevan, "The Role of Values in Scientific Inference," Induction: Some Current Issues, Kyburg and. Nagel, . eds., 1963. 6

Braithwaite, Richard Bevan, Scientific Explanation~ A Study of the Function of Theory, Probability, and Law in Science, Cambridge, University Press, 1953." . Book written by a professor of moral philosophy. Chapters 5 to 7 deal with probability and statistics using a mathematical approach. Chapters 8 to 11 deal with philosophic matters concerning induction and causality.

__~~_' Theory' of Games as a Tool for the Moral Philosopher, Cambridge: 1955.

Bridgman, P. W., "Science: Public or Private," Philosophy of Science, 7( 1940), 36-48. Bri111nger, D. R., "Examples Bearing on the Definition of Fiducial Probability with a Bibliography," AMS, 33(1962), 1349-1355. Broad, C. D., "On the Relation Between Induction and Probability (I)," Mind, 27(1918). "On the Relation Between Induction and Probabil1ty (II)," ---:-:-:--=-'Mind, 29(1920) . Brodbeck, May, "An Analytic Principle of Induction?;' Journal of Philosophy, 49(1952), 747-750. Bross, r. D. J., "Statistical Dogma: A Challenge," Amer. Stat., 15, 3(1961), 14-15· Bross issues a challenge to Bayesian and "orthodox" statisticians to put their respective approaches to an empirical test. , "Linguistic Analysis of a Statistical Controversy," --Am"--e-r-. Stat., 17(1963), 1, 18-21.

--~~-' , Discussion of "On the Foundations of Statistical Inference," JASA, 57(1962), 309· Brown, G. Spencer, Probability and Scientific Inference, N. Y., Longmans, Green and Co., 1957. Burks, A. W., "The Presupposition Theory of Induction," Philosophy of Science, 20(1954). Carlsson, Gosta, "Sampling, Probability, and Causal Inference," Theoria, 18(1952). Carnap, Rudolf, "On the Application of Inductive Logic," Phi1os. and Phenom. Res., 8(1947-48). , "An Axiom System of Inductive Logic," (Mimeographed --N=o'""':t-e-s), UCLA, 1959-61. 7 Carnap, Rudolf, The Continuum of Inductive Methods, Chicago, University of Chicago Press, 1952. A chapter of the proposed second volume to Carnap's Logical Foundations of Probability. Gives methods for determining the (inductive) probability of a given hypothesis on the basis of a given body of evidence.

"On Inductive Logic," Philos. of Science, 12(1945), 72-97.

____~~-, Induktive Logik und Wahrscheinlichkeit, Vienna, Springer Verlag, 1959.

____~~~, Logical Foundations of Probability, Chicago, University of Chicago Press, 1950. (2nd ed. -1962). ' Volume I of the proposed two-volume work, Probability and Induction. Defines two kinds of probability (l'l and P2). PI is a "degree of confirmation" and P2 is the "relatrve frequency" or usual statistical defini"tion.

____~~_, The Nature and Application of Inductive Logic, Chicago, University of Chicago Press, 1951. Selected sections of Logical Foundations of Probability.

, "Notes on Probability and Induction," (Mimeographed ----,N=o""":t-e-s), UCLA, 1955.

----~---, "Remarks on Induction and Truth," Philos. and Phenom. Res. , 6( 1945-46). , "Two Concepts of ProbabiIity, " Philos. and Phenom. Res., --5~(1"""'9""'44'-45).

Chatalian, George, "Induction and the Problem of the External World," Journal of Philos., 49(1952), 601-607. Chernoff, H. and L. E. Moses, Elementary Decision Theory, New York, Wiley, 1959. An introduction to statistics from the decision theory viewpoint. Churchman, C. West, Prediction and ° timal Decision Philoso hical Issues of a Science of Values , New York, Prentice-Hall, 19 1. An exposition of the fundamental problem of the science of values. , "Statistics, Pragmatics, Induction," Philos. of Science, -----".1-5Cr::I-="94;8) . ! , Theory of Experimental Inference, New York, Macmillan, -----".1~94~8:--.' A mixture of modern statistical inference and classical philosophy. See the review by John Tu~ey in JASA, 44(1949), 136-139· 8 Clunies-Ross, C. W., 'DiscusSi6nof "On the Foundations of Statistical Inference," JASA, 57(1962), 314.

Cohan, M. R. and. E. Nagel, An Introduction to' wg1c and Scientific Method, New York, Harcourt, Brace and Co., 1934. Cohen, John, Chance, Skill, and Luck, London, Pelican, 1960.

Copeland, Arthur H., ''Mathemati~al Proof and Experimental Proof," Philos. of Science, 33(1966), 303-316. Cornfield, JerOllle, "Bayes Theorem," ISI Review, 35(1967).

--~---='-:-, "A Bayesian Test of Some Classical lIypotheses--With Applications to Sequential Clinical Trials," JASA, 61(1966), 577-594. ---- One form of data reduction is provided by calculating one or more posteriordistributions--each correspondirig to a different possible prior. Cornfield argues that in clinical trials one should use Jeffrey's priors. (See Theory of Probability. )

--.,...... ,,...... -, Discussion of "On the Foundations of Statistical Inference," JASA, 57(1962), 309. __...",...... ,.-_' "Sequential Trials, Sequential Analysis, and the Likelihood Principle," Amer. Stat., April, 1966, 18-23. Cornfield accepts the likelihood principle ("all observations leading to the same likelihood function must lead to the same conclusion") as opposed to the orthodox view of proceeding in the light of the probability of rejecting a true hypothesis.

Cox, D. R. (editor), Foundations of Statistical Inference: A Discussion, wndon, Methuen and Co., 1962.

, "Some Problems Connected with Statistical Inference," ----=-.AMS=-==-,-' 29(1958), 357-372. Cox, R. T., The Algebra of Probable Inference, .Baltimore, Johns Hopkins, 1961. Craik, Kenneth J., The Nature of Explanation, Cambridge, Cambridge University Press, 1943. Cramer, Harold, Mathematical Methods of Statistics, Princeton, Princeton University Press, 1946. Davidson, Donald, Patrik Suppes and Sidney Siegal, Decision Making: An Experimental Approach, Stanford, Calif., Stanford University Press, 1957·

__~~-:-' "A-Firiit1sticAxicimation of Subjective Probability and Utility," Econometrica, 24(1956), 264-275· 9 Day, J. P., Inductive Probability, New York, Humanities Press, 1961.

de Finetti, Bruno, "Dans quel sens la theorie des decisions est;'elle et doit elle etre normative?" Colloque sur Decision, Paris, May (1959).

---:-...,.--, "Does it Make Sense to Speak of 'Good Probability II' Appraisers'?" The Scientist Speculates, I. J. Good, general editor, New York, Basic Books, 1962.

, "Fondamenti logici del ragionaments probabilistico," -----B~o~l~l-. Un. nat. Italo, 1930, 9(Sec. A), 258-261.

__-=-~:--' "Foundations of Probability," . Philosophy in the Mid-Century, La Nuova Italia Editrice, Florence, 1958, 140-147.

--~--=~., "Foresight: Its Logical Laws, Its Subjective Sources," Studies in Subjective Probability, Kyburg and Smokler (eds.), 1964. -----:-, "La notion de 'destribution d'opinin' conune base d'un essai d' interpretation de la statistique, It Publications· de l'Institut de Statistique de l'Universite de Paris, 1(1952), 1-19. , "La Prevision; Ses lois Logiques, Ses Sources SUbjectives," ---~An'-n-a~les de l'Institut Henri Poincare, 7(1937), 1-68. English translation in Studies in Subjective Probability (Kyburg and SmokIer, eds.). , "Recent Suggestions for the Reconciliations of Theories ---o-:f:-::P~ro·bability," Proceedings of the Second (1950) Berkeley Symposium on Math. Stat. and Probability, ed., Jerzy Neyman, Berkeley, Univ. of Calif. Press, 1951, 217-226. "Le vrai et Ie probable," Dialectica, 3(1949), 78-93·

de Finetti, Bruno, and Leonard J. Savage, "Sul modo di scegliere Ie probabilita Iniziali," Biblioteca del "metron, " Sec. C., Vol. 1, Sui fondamenti della statistica, Rome, University of Rome, 1962.

De Morgan, Augustus, An Essay on Probabilities, 1878. Dempster, A. P., Discussion of "On the Foundations of Statistical Inference," JASA, 57(1962), 318. , "Further Examples of Inconsistencies in the Fiducial ---'7'Ar-gu-ment," AMS 34(1963), 884-891-

, "On Direct Probabilities," JRSS, (B) 25(1963), 100-110. -----Introduction to what the author calls "direct" probabilities- related to fiducial probability. 10

Dempster, A. P., "On the Difficulties Inherent in Fisher's Fiducial Argument," JASA, 59(1964), 56-·66. Discusses two modes of interpreting probability and discusses them ''lith special emphasis on the confidence argument (Neyman-Pearson) and the fiducial argument (Fisher),

Diamond, Solomon, The World of Probability: Statistics in Science, New York, Basic Books, 1964. See review in JASA, 60(1965), 1207.

Doob, J. L., lIprobability and Statistics," Transactions of the American Mathematical Society, 36(1934),

, "Statistical Estimation," Transactions of the American ---:'If.'="".a-:"t"':""h-ematical Society, 39 (1936), 410-421.

Ducasse, C. J., "Causation: Perceivable? Or Only Inferred?" Philos. and Phenom. Res., 26(1965), 173-179.

Ducasse, C. J., "Some Observations Concerning the Nature of Probability," Journal of Philosophy, 38(1941), 393-403.

Dunsmore, I. R., "A Bayesian Approach to Classification," JRSS (B), 28(1966), 568-577. ----

Edwards, W., "Subjective Probabilities Inferred from Decisions," Psychol. Review, 1962, 69, 109-135 (b).

Edwards, Ward, Harold Luidman, and L. J. Savage, "Bayesian Statistical Inference fop Psychological Research," Psychological Review, 70(1963), 193-242. Ellis, Leslie, liOn the Foundations of the Theory of Probabilities,lI Transactions of the Cambridge Philosophical Society 8(1944).

Emmerich, David S. and J. G. Greeno, lISome Decision Factors in Scientific Investigation," Philos. of Science, 33(1966), 262-270.

Evans, I. G., '~ayesian Estimation of the Variance of a Normal Distribution," --JRSS (B), 26(1964),. 63-68. Feibleman, James, "Pragmatism and Inverse Probability," Philos. and Phenom. Res" 5(1944-45)· Feigl, H., "On the Vindication of Induction," Philos. of Science, . 28(1961).

Feller, W., "Statistical Aspects of E. S. P.," Journal of Parapsycho1., 1 4(1940), 271-298.

, lISur les axiomatiques du calcul des probabilites et leurs ---r-e~l-a~tions avec les experiences," Actualites scientifiques et industrielles, no. 735(1938), 7· 11 Fisher, Ronald A., NCommeht on the Notes by Neyman, Bartlett, and Welch in This Journal," JRSS (B), 19(1957), 179. , ContributiohS to Mathematical Statistics, New York, ----=-~--:John Wiley and Sons, 1950.

___~=' The Design of Experiments, Edinburgh and London, Oliver and Boyd, 1935. __..".,-__' liThe Fiducial Argument in Statistical Inference," Ann. .. Eugenics, 6(1935), 391-398 . ___-:::-_, "Inverse Probability," Proc. Camb. Phil. Soc., 26(1930), 528-535· , "The Logic of Inductive Inference," JRSS (A), 98(1935), ---=3~9--;:8=2 .

"A Note on Fiducial Inference," AMS, 10(1939), 383-388. , "On Some Extensions of Bayesian Inference Proposed by ----:M~r-.--=Lindley," JRSS (B), 22(1960), 299-301. Critical comments on some remarks by D. V. Lindley in a review of Fisher's Statistical Methods and Scientific Inference. , "On the Mathematical Foundations of Theoretical Statistics," ---=Ph~i~l-. Trans. Roy. A. Soc., 222(1922), 309-368.

__-,-__, "Probability, Likelihood., and Quantity of Information in the Logic of Uncertain Inference," Proc. of the Royal Soc. (A), 146(1934) . Smoking and mng Cancer, Edinburgh, Oliver and Boyd, 1958.

, "Some Examples of Bayes' Method of the Experimental --=De-'t"'-e-rmination of Probabilities A Priori," JRSS (B), 24(1962), 118-124. , Statistical Methods for Research Workers, Edinburgh and --~--London, Oliver and Boyd, 1925. __...".-~-,..' "Statistical Methods and Scientific Induction," JRSS (B), • 17(1955), 69-78. Denounces the Neyman-Pearson-WaJd approach to statistical inference in terms of decisions--choices between hypotheses. Fisher argued that pure scientists never definitely accept an hypothesis. , Statistical Methods and Scientific Inference, Edinburgh, --"'="Ol""-i"""v-er and Boyd, 1959 (2nd ed.). See Review in JASA, 52(1957), 322-330 of the 1st edition. Also check D. V. Lindley's review that Fisher later attacked. 12 Fisher, Ronald A., "Theory of Statistical Estimation," Proc. of .e The Camb. Phil. Soc., 22(1923-25)' __-."."...".--..,..' "The Underworld of Probability," Sankhya, J8(1957), 201-210. __-=-__, "TWQ New Properties of Mathematical Likelihood," Proc. Roy. Soc. (A), 144(1934), 285-307.

Fraser, D. A. S~, "Fiducial Inference for Location and Scale Parameters," Biometrika, 51(1964), 17-24. ,"The Fiducial Method and Invariance," Biometrika, ----r4"'?:'8'T':(1:-="961), 261-280. '

Fraser, D. A. S., "On the Sufficiency and Likelihood Principles," JASA, 58(1963), 641-647. Discussion of likelihood and sufficiency principles proposed by Burnbaum.

, "On the Consistency of the Fiducial Method," JRSS (B), --~2~4(~1~962), 425-434. ----

__..,....",....."...,..,. "On the Definition of Fiducial Probability,~' Bull. ISI, 40(1964), 2, 842-856. "On Fiducial Inference," AMS, 32(1961), 661-676. Frechet, Maurice, "Emile Borel, Initiator of the Theory of Psychological Games and its Application," Econometrica, 21(1953), 95-96. Frechet, Maurice, and J. von Neumann, "Cormnentary on the Borel Notes," Econometrica, 21(1953), 118-127· Freund, John E., "Puzzle or Paradox?;' Amer. Stat., 19(1965), 4, 29 and 44. Friedman, Milton, "Choice, Chance, and Personal Distribution of Income," ... Journal of Political Economy, 61(1953), 277-290· Friedman, Milton, and L. J. Savage, "The Utility Analysis of Choices Involving Risk," Journal of Political Economy, 56(1948), 279-304. , "The Expected-Utility Hypothesis and the Measurability ---of--U""'tility," Journal of Political Economy, 60(1952), 463-474.

Fry, T. C., "A Mathematical Theory of Rational Inference (A Non mathematical Discussion of Bayes' Theorem)," Scripta Mathematica, 2(1934), 205-221. ' Geisser, S. and J. Cornfield, "Posterior Distributions for Multivariate Normal Parameters," JRSS (B), 25(1963), 368-376. 13 Good, I. J., "The Appropriate Hathematical Tools for Describing and Measuring Uncertainty," Uncertainty and Business Decisions, Carter, Meredith and Schackle, eds., Liverpool, University Press, 1954.

, "A Compromise Between Credibility and Subjective ---=---:--::Probability," International Congress of Mathematicians, Abstracts of Short Communications, Stockholm, 1962, 160.

__....",...-,,-_' "A Derivation of the Probabilistic Explication of Infonnation," JRSS (B), 28(1966), 578-581.

, Discussion of "Consistency in Statistical Inference and ---=---Decision," JRSS (B), 23(1961), 28-29·

--~,..",..,,-' Discussion of "On the Foundations of Statistical Inference," JASA, 57(1962), 312.

__~__, The Estimation of Probabilities: An Essa on Modern Bayesian Methods, Cambridge, M.I.T. Press, 19 5. Concerned with methods for the estimation of probabilities from "effectively small" samples and with some implications of these methods. Text reqUires a modern Bayesian approach. A fairly lengthy bibliography follows.

, "How Rational Should a Manager Be? ," Management Science, ---~8~(1~9~62), 383-393.

"Kinds of Probability," Science, 129(1959), 443-446. -----, "The Paradox of Confinnation," Brit. J. Phil. Sc., 1960. __-=-.."...."._, Probability and the Weighing of Evidence, New York, Hafner Publishing Co., 1950. Good's view of probability as credence or degree of belief in a proposition is shared (to a certain extent) by Savage, deFinetti, Koopman, Ramsey, and others. See Savage's review in JASA, 46(1951), 383-384. "Rational Decision," JRSS (B), 14(1952), 107-ll4.

, "Review of a Paper by K. R. Popper," Math. Rev., ---::'1'7"6(7"::1=955), 376.

The Scientist SpeCUlates: An Anthology of Partly-Baked Ideas, New York, Basic Books, 1962. A collection of short articles by leading scientists presenting "half-baked" ideas.

, "Weight of Evidence, Corroboration, Explanatory Power, ---::I=-nf'""'::-"o-rmation and the Utility of Experiments," JRSS (B), , 22(1960),319-331. ---- 14 Good, I. J., ''Which Comes First, Probability or Statistics?," Journal of the Institute of Actuaries, 82-83(1956), 249-255. Goodman, Nelson, Fact, Fiction, and Forecast, Cambridge, Harvard University Press, 1955. Gordon, Charles K., "On the Probability of Past vs. Future Events," Amer. Stat., 18(1964), 2, 20 and 24. Gouraud, Charles, Histoire du Calcul des Probabilites, Paris, Libraire d'Auguste Durand, 1848.

Gridgman,' N.· T. , "The Lady Tasting Tea, and Allied Topics," JASA, 54(1959), 776-783. Grayson, C. J., Jr., Decisions Under Uncertainty: Drilling Decisions by Oil and Gas Operators, Boston, Harvard University Press, 1960. Grundy, P. M., "Fiducial Distributions and Prior Distributions: An Example in Which the Former Cannot be Associated with the Latter," JRSS (B), 18(1956), 217-221. Guttman, Irwin and George C. Tiao, "A Bayesian Approach to Some Best Population Problems," AMS, 35(1964), 825-835· Hacking, Ian, Logic of Statistical Inference, Cambridge, 1965. Attempts to explain the thought underlying statistical inference in fairly simple terms. See the review in JASA, 61(1966), 1233-1235. -----, "Possibility," The Philosophical ReView, 76(1967), 143-168. , "Salmon's Vindication," Philos. of Science, 32(1965), --""'26"'""9---271.

Hailperin, Theodore, "Foundations of Probability in Mathematical Logic," Philosophy of Science, 4(1936-37). Hall, W. J. and M. R. Novick, "A Note on Classical and Bayesian Prediction Intervals for Location, Scale and Regression Models," AMS, 34(1963), 1619· Halmos, Paul R., "The Foundations of Probability," American Mathematical Monthly, 51(1944), 493-510. Harrington, G. M., "Statistics' Logic," Contemporary Psychology, Vol. 6, No.9 (Sept. 1961), 304-305. Hartigan, J., "Invariant Prior Distributions," AMS, 35(1964), 836-845·

Hartley, H. 0., "In Dr. Bayes' Consulting Room," Amer. Stat., ,17(1963), 1, 22-24. 15 Hempel, Carl G. and Paul Oppenheim, "A Definition of 'Degree of Confinnation', " Philosophy of Science, 12(1945), 98-115. Hildreth, Clifford, "Alternative Conditions for Social Orderings," Econometrica, 21(1953), 81-94. , "Bayesian Statisticians and Remote Clients," ------=Ec-o-n-ometrica, 31(1963), 442-448.

Hodges, J. L. and E. L. rehmann, "Testing the Approximate Validity of Statistical Hypotheses," JRSS (B), 16(1954), 261-268. , "The Use of Previous Experience in Reaching Statistical --~~-Decisions," AMS, 23( 1952), 396-407. Hogben, Lance1ot, The Relationshi of Probabi1it , Credibi1it and Error, London, George Allen and UnWin, Ltd., 1957. The author presents the present controversies as a crisis in statistical theory. Viewed from a behaviorist viewpoint.

Hopf, E., "On Causality; Statistics, and Probability," Journal of Mathematics and Physics, 12(1934). Hora, R. B. and R. J. Buehler, "Fiducial Theory and Invariant Estimation," AMS, 37(1966), 643-656. , "Fiducial Theory and Invariant Prediction," AMS, 38(1967), --7""-9-5--.:::'801.

Hume, David, An Enquiry Concerning Human Understanding, London, 1748.

Janossy, L., "Remarks on the Foundations of Probability Calculus," Acta. Phys. Acad. Hungar., 4(1955), 333-349. Jaynes, E. T., "Bayes Again!," St. Louis, Washington University, 1965 (mu1tilithed) .

Jeffrey, R. C., The Logic of Decision, New York, McGraw-Hill, 1965· An approach to decision theory from a Bayesian point of view. Final chapters treat philosophic questions about inference and objectivity within the framework of subjective probability theory.

Jeffreys, Harold, '~n Invariant Fonn for the Prior Probability in Estimation Problems," Proc. Roy. Soc. (A), 186(1946), 453-461. , Scient1ficInference, Cambridge, Cambridge University ----....Pr-e-s-s, 1931. (3rd ed. 1961). Jeffreys' conception of probability is similar to Keynes. This book is a defense of scientific method based on the author's conception of subjective probability. 16

Jeffreys, Harold, TheoEl of Probabilit~, Oxford, Clarendon Press, 1939, (3rd ed. 1961). Here Jeffreys distinguishes between estimation and hypothesis testing. He recommends different priors for each.

Jeffreys, Harold, and Dorothy Wrinch, "On Some Aspects of the Theory of Probability," The Philosophical Magazine, Ser. 6, 38(1919), 715-731.

Johns, M. V., "Non-parametric Empirical Bayes Procedures," AMS, 28(1957), 649-669. Johnson, W. E., Appendix to "Probability: Deductive and Inductive Problems," Mind, 41(1932), 421-423.

--..-,---' "Probability: The Deductive and Inductive Problems," Mind, 41(1932), 409-423. Kaufmann, Felix, "On the Nature of Inductive Inference," Philos. and Phenom. Research, 6(1945-46). ____--, ''Verification, Meaning, and Truth," Philos. and Phenom. Research, 4(1943-44). Kemble, E. C. "Is the Frequency Theory of Probability Adequate for all Scientific Purposes'?," Amer. Journal of Physics, 10(1942), 6-16. -----, "The Probability Concept," Philosophy of Science, 8( 1941) . Kemeny, John G. and Paul Oppenheim, "Degree of Factual Support," Philos. of Science, 19(1952).

, "Extension of the Methods of Inductive Logic," -----Philosophical Studies, 3(1952).

, ''Fair Bets and Inductive Probabilities," Journal of -----=Sym~b-olic Logic, 20(1955), 263-273.

Kempthorne, Oscar, Discussion of "On the Foundations of Statistical Inference," JASA, 57(1962), 319. , "Some Aspects of Experimental Inference," JASA, 61(1966), ------11---3..,..4. Text of the Fisher Memorial Lecture. Kendall, Maurice G., "On the Reconciliation of the Theories of Probability," Biometrika, 36(1949), 101-116. Kendall, M. G. and A. G. Doig, Bibliography of Stat. Literature, 1950-58, New York, Hafner Publishing Co., 1962. 17 Kendall, M. G. and A. G. Doig, Bibliography of Stat Literature, 1940-49, New York, Hafner Publishing Co., 1965. Kerridge:, D., "Bounds for the Frequency of Misleading Bayes Inferences," AMS, 34(1963), 1109-1110. Keynes, John Maynard, A Treatise on Probability, London and NeiV' York, Macmillan and Co., 1921 (2nd ed., 1929). A classic in the history of probability. To Keynes, probability was a relational property linking a proposition with its partial evidence. It is a primitive idea, ultimate and undefined. An excellent bibliography emphasizes the philosophical aspects of probability in the period from Laplace to Keynes. Kneale, William, Probability and Induction, Oxford, Clarendon Press, 1949· Based on lectures delivered to students of philosophy, this work devotes much effort to the problems of definitions of probability. Also, considerable attention is devoted to Keynes' Treatise.

Kolmogoroff, A. N., Foundations Of the Theory of Probability, New York, Chelsea Publishing Co. 1950. Koopman, B. 0., "The Axioms and Algebra of Intuitive Probability," Annals of Mathematics, Series 2, 41(1940), 269-292. , "The Bases of Probability," Bulletin of the American ~---M-a~t~h-ematical Society, 46(1940), 763-774.

__-..,.....,...".._' "Intuitive Probabilities and Sequences," Annals of Mathematics, Series 2, 42(1941), 169-187.

Kraft, C. H., J. W. Pratt, and A. Seidenberg, "Intuitive Probabili~y on Finite Sets," AMS, 30(1959), 408-419. Kullback, Solomon, Information Theory and Statistics, New York, John Wiley and Sons, 1959. An advanced text on the large-sample theory of tests.

Kyburg, Henry E., "Demonstrative Induction," Philos. and Phenom. Research, 21(1960).

--__7:'r::-::'", "The Justification of Induction," Journal of Philosoph~;' 53(1956), 394-400. , "Probability and Decision," Philos. of Science, ------3~3~(1~966), 250-261. 18 KYburg, Henry E., Probability and the Logic of Rational Belief, Middletown, Conn., Wesleyan University Press, 1961. The first part of this book gives an excellent discussion on the major interpretations of the foundations of probability. The author then tries to defend a definition of probability that corresponds to our intuitive concept of "rational belief. tr

"Probability and Rationality," Philosophical Quarterly,

KYburg, Henry E. and Ernest Nagel (eds.), Induction: Some Current Issues, Middletown, Conn., Wesleyan University Press, 1963. A collection of essays, comments and discussions presented at the Wesleyan Conference on Induction in 1961. Kyburg, Henry E. and H. E. Smokler (eds.), Studies in Subjective Probability, New York, John Wiley and sons, 1964. See the review in JASA, 60(1965), 364. Laplace, Pierre Simon, "Memoire sur la probabilite des causes par les evenements," Mem. de l'Acad. R. de ScL Paris, 6(1774), 621-656.

____~~~, A Philosophical Essay on Probabilities, New York, Dover Publications, 1952. This work formed the introduction to Laplace's monumental Theorie Analytique des Probabilities. Laplace presented his own conception of the "meaning" of probality without resorting to higher methematics. _____, Theorie Analytique des Probabilites, Paris, 1812. One 9f the most impressive and far-reaching contributions made to the theory of probability.

Leblanc, Hughes, "On Chances and Estimated Chances of Being True," Revue Philosophique de Louvain, 57(1959), 225-239. __-::-__, "On ReqUirements for Conditional Probability Functions, " Journal of Symbolic Logic, 25(1960), 238-242. , "On So-Called Degrees of Confirmation," British Journal ---=--"':"':for the Philosophy of Science, 10, 1960. , "Probabilities as Truth-Value Estimates," Philos. of ------Sc-i-e-nce, 28(1961), 414-417.

, "The Problem of the Confirmation of Laws," Philosophical ------=St~u~d~ie·s, 12(1961), 81-84.

, Statistical and Inductive Probabilities, Englewood Cliffs, ---~N~e-w~Jersey, Prentice-Hall, 1962. Attempts to settle "a dispute which cannot but be prejudicial to the philosophy of science." The author defines inductive and statistical probabilities such that the former are estimates of the latter. 19 Leblanc, Hughes, "Statistical and Inductive PrObabilities," Induction: Some Current Issues, Kyburg and Nagel,(eds.), 1963.

Truth and Estimated Truth

, "T.lO Probability Concepts," Journ. of Philosophy, 53(1956), --"7.67::":9:-'--;:'688.

Lehman, R. S., "On Confirmation and Rational Betting," J. Symbolic Logic, 2D(1955), 251-262.

L:::li;1arm, E. L., "Some Principles of the Theory of Testing Hypotheses, " I~S, 21(1950), 1-26.

__..__' Testing Statistical Hypotheses, New York, "Tiley, 1959.

Lenz, J. H., "The Frequency Theory of Probability," The Structure of Scientific Thou ht An Introduction to Philoso h of Science, Madden ed. , Boston, Houghton Mifflin Co., 1960.

LeVi, I., "Corroboration and Rules of Acceptance," British Journal for the Philos. of Science, Feb., 1963.

Levinson, Horace C., Science of Chance From Probability to Statistics, New York, Rinehart, 1950.

LeWis, C. I., An Analysis of Knowledge and Valuation, LaSalle, Ill., The Open Court Publishing Co., 1946. Lewy, Casimir, "On the Justification of Induction," Analysis, 7(1939).

Lidstone, G. J., "Note on the General Case of the Bayes-Laplace Formula for Inducti ve or a posteriori Probabilities," Trans. Fac. Actur., 8(1920), 182-192.

Lindley, Dennis V., Discussion of "Consistency in Statistical Inference and Decision," JRSS (B), 23(1961), 29-30.

---=-::-=-:-' Discussion of "On the Foundations of Statistical Inference," JASA, 57(1962), 316.

, "Discussion of Session on Fiducial Probability," Bull. ISI, ---~4~0~(1~964), 2, 919-921.

, "Fiducial Distributions and Bayes r Theorem," JRSS (B), ---~2~0r=(1~958), 102-107.

, Introduction to Probabilit and Statistics from a Ba esian ---=::-::---Viewpoint, Vols. I Probability and II Inference, Cambridge, University Press, 1965. A good reference book for the Bayesian approach to many problems in statistical inference. Attempts to relate classical and Bayesian inference. The author has written two earlier volumes from a non-Bayesian Viewpoint. 20 Lindley, Dennis V. ., frOn a Measure of the Information Provided by an Experiment," .AMS, 27(1956), 986-1005.

___,,---,.,..,-' "Professor Hogben I s Crisis'--A Survey of the Foundations of Statistics," Appr. Stat., 7(1958),186-198.

__..""..--,_' Review of "Stat. Methods and Scientific Inference," by « R. A. Fisher, Heredity, 11(1957), 280-283. ____, "Statistical Inference," JRSS (B), 15(1953), 30-65. , "A Statistical Paradox," Biometrika, 44(1957), 187-192. ------See discussion by M. S. Bartlett (1957). __-=--",-_, "The Use of Prior Probability Distributions in Statistical Inferences and Decisions," Proc. Fourth Berkeley Symposium, 1, 453-468. Linnik, Yu. V., "On the Behrens-Fisher Problem," Bull. ISI, 40(1964), 2, 833-841. Loeve, M., Probability Theory, Princeton, Van Nostrand Co., 1960 (2nd ed.).

Lucas, J. R., "The One Concept of Probability, II Philos. and Phenom. Research, 26(1965), 180-201. luce, Robert Duncan, and Howard Raiffa, Games and Decisions: Intro duction and Critical Survey, New York, John Wiley and Sons, 1957. An elementary and comprehensive text on the theory of games. See the review in JASA, 53(1958), 758. Luce, Robert Duncan and P. Suppes, "Preference, Utility, and Subjective Probability," Handbook of Mathematical Ps cholo , luce, Bush and Galanter (eds. , New York, Wiley, 19 5. Machol, R. E. and P. Gray (editors), Recent Developments in Information and Decision Processes, New York, Macmillan Co., 1962. Proceedings of a 1961 symposium. Of special interest are articles by H. Raiffa and L. J. Savage on Bayesian decision theory and statistics..

Mallows, C. L., "The Information in an Experiment, II JRSS (B), 21(1959), 67-72. -- Margenan, H., "The Role of Definitions in Physical Science with Remarks on the Frequency Definition of Probability, II American Journal of Physics, 10(1942). Messick, S. and A. H. Brayfield (editors), Decision and Choice: Contributions of Sidney Siegel, New York, McGraw-Hill, 1964. Michalos, Alex C., "Postulates of Rational Preference," Philos. of Science, 34(1967), 18-22. 21 Michalos, Alex C., "Two Theorems of Degree of Confirmation," Ratio, Dec., 1965.

Milnor, J., "Games Against Nature," Decision Processes, Thrall, Coombs, and David (editors), New York, John Wiley and Sons, 1954. Mises, Richard von, "Fundamentalsatze der Wahrscheinlichkeitsrechnung, " .. Mathematische Zeitschrift, 5(1919), 1-97. , "Grundlagen der Wahrscheinlichkietsrechnung, " Mathematische -----=Ze~i~t-schrift, 5(1919), 52-99. Insisted that probability measure the relative frequency with which the members of a reference set belong to another set. Greatly stimulated the modern discussion' on the foundation of inductive (including statistical) inference. ____-.--;, "On the Foundations of Probability and Statistics," AMS, 12(1941), 191.

, Probability, Statistics and Truth, London, William Hodge, ---a-n""d~Co., 1939, and New York, Macmillan, 1957 (2nd edition).

---=---:-'Providence, R.

Molina, Edward C., "Bayes' Theorem," AMS, 2(1931),23-27. , "Bayes' Theorem, an Expository Presentation," Bell Tel. ------=Sy-s~t-em Tech. Publ. Monograph (B), 1931, 557.

Moore, Asher, "The Principle of Induction," Journal of Philosophy, 49(1952), 741-747. Mosteller, F. and D. L. Wallace, Inference and Disputed Authorship: The Federalist, Reading,Mass., Addison-Wesley Publishing Co., Inc., 1964. An example of the use of Bayesian (and other) methods of discrimination--in this case to determine the authorship of twelve of The Federalist papers.

Munroe, M. E., Theory of Probability, New York, McGraw-Hill Co., 1951. Nagel, Ernest, "A Frequency Theory of Probability," Journal of Philosophy, 30(1933), 533-554. , "Is the LaPlacean Theory of Probability Tenable?," ------P=h-i~l-os. and Phenom. Research, 6(1945-46).

"The Meaning of Probability," JASA, 31(1936), 10-30. 22 Nagel, Ernest, "Principles of the Theory of Probability, ,, International Encyclopedia of Unified SCience, Vol. 1, No. 6 Chicago, University of Chicago Press, 1939. __-=-=-~_, "Probability and Non;..Demonstrative Inf'erence," Philos. and Phenom. Research, 5(1945).

_____, "Probability and the Theory of Knowledge," Philosophy of Science, 6(1939). __-=~_, "Reichenbach's Theory of Probability," Journal of Philosophy, 47(1950), 551-555. Sovereign Reason, Glencoe, Illinois, 1954. Neumann, John von, "Zur Theorie der Gesellschoftsspiele, Mathematische Annalen, 100(1928), 295-320. Neumann, John von and Oskar Morgenstern, Theory of Games and Economic Behavior, Princeton, Princeton University Press, 1953 (3rd edition). The first book on the modern theory of games--contains a detailed discussion of utilityT theory.

Neyman, Jerzy, "Current Problems of Mathematical Statistics,"Proceedings of the Seventh International Congress of Mathematicians, Amsterdam, Groningen: E. P. Nordhoff, 1954, 349-370. , "Fiducial Argument and the Theory of Confidence Intervals," ------Bi-o-m-etrika, 32(1941), 128-150.

____--, First Course in Probability and Statistics, New York, Henry Holt and Co., 1950.

, "Fisher's Collected Papers," Scientific Monthly, --~4~2(~1~95l), 406-408.

"Indeterminism in Science and New Demands on Statisticians," JABA, 55(1960), 625-639. ~ Invited address delivered at the annual meeting of the ABA in ,1960. • , "'Inductive Behavior' as a Basic Concept of Philosophy ---o-':f~S-cience," Review ISI, 25(1957), 7-22.

, Lectures and Conf'erences on Mathematical Statistics ----.:.-=and Probability, Washington, Graduate School of the U. S. Dept. of Agriculture, 1952. , "L' estimation statistique, traitee comme un propleme -----classique de probabilitie," Actualites scientifiques et industrielles No. 739, Paris, Herman et Cie., 1938. 23 Neyman, Jerzy, "Note on an Article by Sir Ronald Fisher," JRSS (B), 18(1956), 288-294. Another reply to Fisher's attack on the decision theory approach to statistical inference.

"On the Problem of Confidence Intervals," AMS, 6(1935), 111.

__~~_, "On the Two Different Aspects of the Representative Method: The Method of Stratified Sampling and the Method of Purposive Selection," JRSS (A), 97(1934), 558-625.

__---.,.__, "Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability," Philosophical Transactions of the Royal Society (A), 236(1937),333-380. _____, "Raisonnement inductif ou comportement inductif," Proceedings of the International Stat. Institute, 1947. _____, "Two Breakthroughs in the Theory of Statistical Decision Making," Review ISI, 30(1962), 11-27.

Neyman, Jerzy, and E. S. Pearson, "Contributions to the Theory of Testing Stat. Hypotheses: I. Unbiased Critical Regions of Type A and Type Al," Statistical Research Memoirs, 1(1936), 1-161. __...",,-_..,...' "Contributions to the Theory of Testing Statistical Hypotheses: II. Certain Theorems on Unbiased Critical Regions of Type A; III. Unbiased Tests of Simple Stat. Hypotheses Specifying the Values of More than One UnknOlffi Parameter," Statistical Research Memoirs, 2(1938), 25-57. _____, "On the Use and Interpretation of Certain Test Criteria for Purposes of Statistical Inference," Biometrika" 20(1928), 175 240 and 263-294.

, "The Testing of Statistical Hypotheses in Relation to ---p""r-o""'b-:abilities A Priori," Proc. Camb. Phil. Soc., 29(1933), 492-510. NOVick, Melvin R., "On Bayesian Logical Probability," Princeton, New Jersey, Educational Testing Service, Res. Bul. 1964,

NOVick, Melvin R. and J. E. Grizzle, "A Bayesian Approach to the Analysis of Data from Clinical Trials," JASA, 60(1965), 81-96.

Oliver, W. Donald, "A Re-examination of the Problem of Induction," Journal of Philosophy, 29(1952), 769-780. Pap, Arthur, The A Priori in Physical Theory, New York, Kings Crown Press, 1946.

Pascal, Blaise, Pensees, New York, Harper, 1962. (reprint). 24 Pearson, E. S., "Statistical Concepts in Their Relation to Reality," JRSS (B), 17(1955), 204-207. ---- A reply to Fisher's attack on the decision theory approach to statistical inference. ____-.".., "Bayes' Theorem, Examined in the Light of Experimental Sampling," Biometrika, 17(1925), 388-442. , "Some Thoughts on Statistical Inference," .MIS, 33(1962), --3"""94'----:-403·

Pearson, E. S. and C. J. Clopper, "The Use of Confidence or Fiducial Limits Illustrated in the Case of a Binomial," Biometrika, 26(1934) . Perks, W., "Some Observations on Inverse Probability Including a New Indifference Rule," J. Inst. Actu., 73(1947), 285-334. Pinkham, R. S., "On a Fiducial Example of C. Stein," JRSS (B), 28(1966), 53-54. --- Pitman, E. J. G., "Statistics and Science," JASA, 52(1957), 322. An invited review article on Fisher's Statistical Methods and Scientific Inference. Plackett, R. L., "Current Trends in Statistical Inference," JRSS (A) , 129(1966), 249-267. An article arising out of an address to the-1965 conference of the Royal Statistical Society. A good source for a brief sketch of recent developments.

Poincare, Henri, Science and Hypothesis, New York, Dover Publications, 1952• Polya, G., Mathematics and Plausible Reasoning, Volume I-Induction and Analogy in Mathematics, Princeton, Princeton University Press, 1954. The first of two volumes dealing with nondeductive reasoning in mathematics. This volume supplies the more mathematical aspects of the work.

____~~.".., Mathematics and Plausible Reasoning, Volume II-Patterns of Plausible Inference, Princeton, Princeton University Press, 1954. In the author's words, this volume is "an inductive investigation of induction." , "Preliminary Remarks on a Logic of Plausible Inference," -----D~i~a~l-ectica, 3(1949), 28-35·

Popper, Karl R., "Degree of Confirmation," British Journal for the Philosophy of Science, 5(1954-55). 25 Popper, Karl R., The Logic of Scientific Discovery, New York, Basic Books, 1959.

__,.""..--,._, "Probabilistic Independence and Corrobora.tion by , Empirical Tests," British Journal for the Philos2Ph~ of Science, 10(1960) .

Pratt, John W., "Bayesian Interpretation of Standard Inference Statements," JRSS (B), 27(1965), 169-203. A valuable djscussion follows this paper.

, Discussion of '~n the Foundations of Statistical ---=--=--Inference," ~.ASA, 57 (1962), 314. Pratt, J. w., H. Raiffa, and R. Schlaifer, "The Foundations of Decision Under Uncertainty: An Elementary Exposition," JABA, 59(1964), 353-375. Gives rules for d.ecisions under uncertainty as derived from principles of consistent behavior and scaling preferences.

___~~_" Introduction to Statistical Decision TheorY, New York, McGraw-Hill, 1965.

Quenouille, M. H., The Fundamentals of Statistical Reasoning, London, Charles Griffin and Co., 1958.

Raiffa, H. and R. Schlaifer, Applied Statistical Decision Theory, Boston, Harvard University Press, 1961- See review in~, 57(1962), 199. Rajski, C., ''The Bayes Postulate and Entropy," Zastosowania Matematyki, 4(1958), 91-94.

___-:--::--_, "Comparing General Populations on the Basis of Bayes' Rule," Zastosowania Matemati1t--!z 1(1954), 330-341.

Ramsey, Frank P., The Foundations of Mathematics and Other LogicaJ. Essays, London, Kegan Paul, 1931 and Patterson, New Jersey, Littlefield, Adams and Co., 1960. Contains "Truth and Probability" (1926) and "Further Considerations" (1928).

___"':"':" ~:' ''Mr. Keynes on Probability," The Cambridge Maaazine, 11(1922). ,

Rapoport, A., Strategy and Conscience, New York, 1964.

Reichenba.ch, Hans, "Are Phenomenal Reports Jibsolutely Certain?," ~ilosQPhical Review, 61(1952), 147-159.

_____~~, erience and Prediction An Anal sis of the Foundations and the Structure of Knowled e , Chicago, University of Chicago Press, 1957. The author attempts to give a foundation of logicaJ. empiricism. 26 .. Reichenbach, Hans,' "On P!'obability and Induction," Fhilosophy ~ of Science, 5(1938). , "On the Justification of Induction," Journal of ---=:-~- Philosophy, 37(1940), 97-103. , Philosophic Foundations of Quantum Mechanics, Los Angeles, U':iversity of California Press, 1946.

___..,,-~, The Theory of Probability: An Inquiry into the Logical ~Dd Mathematical Foundations of the Calculus of Probability, Berkeley, University of California Press, 1949. A revised translation of the 1931~ German -edition. Defines probability in the "relative frequency" ''lay with considerable emphasis placed on the relationships between probability and induction.

Renyi, A., "On a Np.w Axiomatic Foundation of the Theory of Probability," Proceedings of the Seventh International Congress of Mathematicians, Amsterdam, Groningen: E. P. Nordhoff, 1957, 506-5°7. , "On a New Axiomatic Theory of Probability," Acta --~M~a~t~h-ematica Acadamiae Scientiarum HUngaricae, 6(195~285-335.

Richter, Hans, "Zur Grundlegung der Wahrscheinlichkeitstheorie, I, II, III, rv," Mathematische Annalen, 125(1952), 129-139; 125(1953), 223-224, 335-343; 126(1953), 362-374. Robbins, H. E., "An Empirical Bayes Approach to Statistics," Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1956, 157-163. , "The Empirical Bayes Approach to Statistical Decision ---P=-r-o-:-ib-:-lems," AMS, 35(1964), 1-20.

--...",...~..".., "A New Approach to a Classical Statistical Decision Problem," Induction: Some Current Issues, Kyburg and Nagel (eds.), 1963. , "Some Aspects of the Sequential Design of Experiments,lr ---=-Bu~l~l-. Amer. Math. Soc., 58(1952), 527-535.

Roberts, Harry V., "Probabilistic Prediction," JASA, 60(1965), 50-62. , "Statistical Dogma: One Response to a Challenge," ---Am-:--.~Stat., 20(1966), 4, 25-27. Discusses one aspect of the problem of making meaningful empirical comparisons between alternate methods of drawing conclusions. Se Bross (1961) for an explanation of the title. 27 Robinson, Richard, 'Measurement and Statistics: Toward a Clarification of the Theory of Permissible Statistics," Philos. of Science, 32(1965), 229-243- Hoy, A. D., "Some Notes on Pistimertic Inference," JRSS (B), 22(1960), 338-347. ---- Russell, Bertraand, Human Knowledge, Its Scope and Limits, London,' George Allen and Unwin, 1948, and New York, Simon and Schuster, 1948. Sacks', J., "Generalized Bayes Solutions in Estimation Problems," AMS, 34(1963), 751-768. --- Salmon, Wesley C., "On Vindicating Induction," Induction: Some Current Issues, Kyburg and Nagel (eds.), 1963. , "Regular Rules of Induction," Philosophical Review, --"""65""7"(1-:"956) .

, "The Status of Prior Probabilities in Statistical -----Explanation," Philos. of Science, 32(1965), 137-146. This article is followed by a discussion by H. E. Kyburg and an additional reply by Salmon. Savage, Leonard J., Discussion of "On the Foundations of Statistical Inference," JASA, 57(1962), 307. ______, "Discussion of Session on Fiducial Probability," Bull. ISI, 40(1964), 2, 925-927. ______, The Foundations of Statistics, New York, John Wiley and Sons, 1954. An excellent "first-source" text for anyone interested in the foundations of statistics.

, "The Foundations of Statistics Reconsidered," ---'p""r-o-c-eedin s of the Fourth Berkele S osium on Math Stat. and Probability, Berkeley, University of California Press, 19 1.

____--,-_, "La probabilita soggetiva nei problemi pratici della statistica," Rome, Induzione e Statistica, 1959. , "Recent Tendencies in the Foundations of Statistics," ------Proceedings of the Eighth International COngress of Mathematicians, Cambridge, Cambridge University Press, 1960, 540-544. , The Subjective Basis of Statistical Practice, Ann Arbor, ----U,...n~i-v-ersity of Michigan, 1961.

, "The Theory of Statistical Decision," JASA, 46(1951), 55 ..67· ------A paper discussing Wald's Statistical Decrsron Functions published just after Wald's death. 28 Scheffe, Henry, The Analysi.s of Variance, New York, John Wiley and Sons, 1959. , "A Note on the Behrens-Fisher Problem," AMS, 15(1944) ---'-4--30"'---"-432.

Schlaifer, Robert, Introduction to Statistics for Business Decisions, New York, McGraw-Hill, 1961. Sets forth the classical Neyman-Pearson theory and attempts to show how this is complimented by recent work in various approaches to statistical inference. Closely related to • the author's Probability and Statistics for Business Decisions. ____--, Probability and Statistics for Business Decisions, New York, McGraw-Hill, 1959. Uses the Bayesian approach for an elementary exposition of decision making under uncertainty. Schmetterer, Leopold, "Zur Bayessche Regel," Statistischen Viertelgshresschrift, 5(1952), 174-178. Schrodinger, E., "The Foundation of the Theory of Probability," Proc. Roy. Irish Acad., 5lA(1947), 51-141. , "The Statistical Law in Nature," Nature, wndon, --~1~53~(~1944), 704-705.

Searle, S. R., "Probability-The Difficulties of Definition," of the Institute of Actuaries Students' Societ , , 204-212" Sellars, W., "Inference and Meaning," Mind, 62(1953). Shimony, Abner, "Coherence and the Axioms of Confirmation," Journal of Symbolic Logic, 20(1955), 1-28. Simon, H. A., "Prediction and Hindsight as Confirmatory Evidence," Phil. Sci., 22(1955), 227-230. Skyrms, Brian, "On Failing to Vindicate Induction," Philos. of Science, 32(1965), 253-268. Smith, Cedric A. B., "Consistency in Statistical Inference and Decision," JRSS (B), 23(1961),1-25· Concerns the testing of the strength of a person's belief by finding at what odds he is prepared to bet on them. , "Personal Probability and Statistical Analysis," JRSS --~(~A)~,- 128(1965), 469-499· 29 Smullyan, Arthur,"The Concept of Empirical Knowledge," Philosophical Ull Review, 65(1956), 362-370.

Sprott, D. A., ''Necessary Restrictions for Distributions A Posteriori, II JRSS (B), 22(1960), 312-318. __--,~~, "An Example of an Ancillary Statistic and the Combination of Two Samples by Bayes' Theorem," AMS, 32(1961), 616-618. , "A Transformation Model for the Investigation of Fiducial ----D="'i:-s-:-t-ributions," Bull. ISI, 40(1964), 2, 856-869. Stein, Charles, "An Example of Wide Discrepancy Between Fiducial and Confidence Intervals," AMS, 30(1959), 877-880. , "A Remark on the Likelihood Principle," JRSS (A), --~1~25~(-1962),565-568. -----

Steinhaus, H., "Probability, Verisimilitude,Credibility," Zastosowania Matematyki, 1(1954), 149-172. _

, "Quality Control by Sampling: A Plea for Bayes' Rule, II --"="Co""":l=l-oquium Mathematicum, 2(1950), 98.

Sterling, T. D., ''What is So Peculiar About Accepting the Null Hypothesis?," Psychol. Rep. 7(1960),363-364. Stone, M., "Generalized Bayes Decision Functions, Admissibility and the Exponential Family," AMS, 38(1967), 818-822. , "Comments on a Posterior Distribution of Geisser and -~C:-or-nf-=ield," ~ (B), 26(1964), 274-276. -----, "The Posterior t Distribution," AMS, 34(1963), 568-573. Suppes, P., "Some Open Problems in the Foundations of Subjective Probability," Information and Decision Processes, Machol (ed.), New York, Macmillan Co., 1960. Thatcher, A. R., "Relationships Between Bayesian and Confidence Limits for Predictions," JRSS (B), 26(1964),176-192. See the discussion following this article.

Thornber, Hodson, "An Autoregressive Model: Bayesian Versus Sampling Theory Analysis," Technical Report 6504, Center for Mathematical Studies in Business and Economics, University of Chicago, 1965. Discusses an experiment which indicates a modest gain by Bayesian over sampling theory methods. Thrall, Robert M., Clyde H. Coombs, and Robert L. Davis (eds.), Decision Processes, New York, John Wiley and Sons, 1954. 30

Tiao, George C. and W, y, Tan,. "Bay-esiJm Analy'sis of' Random-Effect Models in the Analysis of Variance,it Biometrika, 52(1965), 37-53, and 53(1966), 477--495. Tiao, G. C. and A. Zellner, "Bayes Theorem and the Use of Prior Knowledge in Regression Analysis," Biometrika, 51(1964), 219-230.

Tintner, Gerhard, "A Contribution to the Non-Static Theory of Choice, It Quarterly Journal of Economics, 56(1942), 274-306.

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Todhunter, Isaac, A Histo of Probabilit From the Time of Pascal to That of La lace , Cambridge, Macmillan and Co., 18 5, and New York) Chelsea Publishing Co., 1949.

Tukey, John W., "Conclusions vs. Decisions," Technometrics, 2(1960), 423-433· , "Discussion of Session on Fiducial Probability," ------B~u~l~l-. lSI, 40(1964), 2, 921-924.

, "Quick and Dirty Methods in Statistics. Part II. Simple --~-::-- Analyses for Standard Designs,it Proceedings of the Fifth Annual Convention of the American Society for Quality Control, 1951, 189-197.

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Urmson, J. 0., "Two of the Senses of 'Probable I ," Analysis, 8(1947-48). Van Dantzig, D., "Statistical Priesthood I (Savage on Personal Probabilities)," Statistica Neerlandic~, 11(1957), 1-16.

, "Statistical Priesthood II (Sir Ronald on Scientific ---I-nf-e-rence)," Statistica Neer1andica, 11(1957), 185-200.

, "Sur l'analyse logique des relations entre Ie calcuI des -----probabilites et ses applications," Co11oque de calcul des Probabilites, Paris, Herman et Cie, 1951. .. Venn, John, The Logic of Chance, London and New York, 1866. Gives perhaps the earliest explicit formulation of the frequency interpretation of probability.

Verhagen, A. M. W., "The Notion of Induced Probability in Statistical Inference," CSIRO, Division of Mathematical Statistics Technical Paper No. 21, Melbourne, 1966. The author attempts to relate a number of concepts by means of what the author calls "induced probabilities."

Vickers, John M., "Some Remarks on Coherence and Subjective Probability," Philos. of Science, 32(1965), 32-38. 31 Wald, Abraham, On the Principles of statistical Inference, Notre Dame, Indiana, 1942. Gives a brief general review of the problem of statistical inference. A good source for one who wants a quick view of the overall field of statistical inference.

__~~_, Selected Papers in Statistics and Probability, New York, McGraw-Hill, 1955. A memorial to A. Waldo Contains a complete bibliography.

__~_.,-,:' Sequential Analysis, New York, John Wiley and Sons, 1947. See the review by G. A. Barnard in~, 42(1947), 658-669.

__-=-__, statistical Decision Functions, New York, John Wiley and ~M,~~. . See the discussion of this .book by L. J. Savage in JASA, 46(1951), 55-67. ---

Wallace, D. L., "The Behrens-Fisher Problem--A Critical Review and Subjective Approach," Institute of Mathematical Statistics Central Regional Meeting, Ames, Iowa, April 1958.

Wallis, W. Allen, and Milton Friedman, "The Empirical Derivation of Indifference Functions," Studies in Math Economics and Econometrics, O. Lange (ed.), Chicago, University of Chicago Press, 1942, 175-189.

Wasserman, William, "Foundations of Statistics and Gdals of the Basic Required Course," American Stat., 17(1963), 3, 15-18.

Watanabe, Satosi, "Information--Theoretical Aspects of Inductive and Deductive Inference," IBM ;Journal, 4(1960), 200-231-

Weibull, Christer, Some Aspects of Statistical Inference with Applications to Sample Survey Theory, Gothenburg, University of' Gothenburg, ,1960. Contains a brief historical study of statistical inference and some comments on the author's position. .

Whiteley, C. H., "On the Justification of Induction," Analysis, 7(1939-1940). , ''More About Probability," Analysis, 8(1947-1948). • Whitworth, William A., Choice and Chance, 5th Edition Iondon, 1901, Reprinted, New York, 1942, Hafner Publishing Company.

Wilks, S. S., The Theory of Statistical Inference, Ann Arbor, Edwards Bros., Inc., 1937.

, ''Fiducial ·Distributions in Fiducial Inference," AMS, ----,9~(1~9~38:), 272. 32 Will, F. L., IIIs There a Problem of' Inducti.on?, II Journal of Philosophy, 39(1942), 505-513"

''Will the Future Be Like the Past?, II Mind, 55(1946). , 'The Justification of Theories,lI Philosophical Review, ---:6...... 4'7":(1""""95 5), 370 -388,

Vlilliams, Donald C., The Ground. of Indu.ction, Cambridge, Mass., Harvard University' Press, 1947.

) __---,,..-._, "On the DerivatiOn of Probabilities from Frequencies," Philoso and Phenom. Res., 5(1945)<

, "The Problem of Probability," Philos and Phenom. Research, ---:6'?7'('::"":19:T"45 -46) ,

Williams, E. J., "A Comparison of the Direct and Fiducial Arguments in the Estimation of a Parameter," JRSS (B), 25(1963), 95-99. A different approach than Fisher's to an aspect of Fiducial inference.

Wilson, E. B., An Introduction to Scientific Research, New York, McGraw-Hill, 1952.

Wisdom, John Oulton, Foundations of Inference in Natural Science, London, Methuen, 1952.

, Causation and the Foundations of Science, Paris, Herman ---et-C-ie, 1946.

Paradox and Discovery, New York, Philosophical Library, 1965. Wolfowitz, J., "Bayesian Inference and Axioms of Consistent Decision," Econometrica, 30(1962), 470-479.

Wright, Georg H. von, "Carnap' s Theory of Probability," The Philosophical Review, 60(1957), 362-374, , The Logical Problem of Induction, Helsinki, 1941. -----See review in Journal of Philosophy, 55(1958).

"On Probability," Mind, 49(1940).

, "The Meaning of Probability," Journal of Symbolic Logic, --2-=-0"""(1-=-95 5).

A Treatise on Induction and Probability, Patterson, New Jersey, Littlefield, Adams and Co., 1960. An examination of various types of inductive arguments considered with respect to their logical correctness. 33 Yates, F., "An Apparent Inconsistency Arising From Tests of Significance Based on Fiducial Distributions of Unknown Parameters, ,II Proceedings of the Cambridge Philosophical Society, 35(1939), 579-591. , IIBases logiques de la p1anification des experiences," ----A~n-n-.-Inst. H. Poincare, 12(1951), 97-112. .. , IIComments on Mr. Lindley's Review," Heredity, 12(1958), ----=1=33~--=-13·5. t , '~iducial Probability, Recognizable Subsets, and Behren's ---T-:-e-s-t-,II Biometrics, 20(1964), 343-360.

, IISir Ronald Fisher and the Design of Experiment,1I -----~Bi~o-m-etrics, 20(1964), 307-321.

Zellner, Arnold, IIBayesian Inference and Simultaneous Equation Models, II Chicago, (multilithed), University of Chicago, 1965. An experimental test of Bayesian vs. "orthodox" methods.