Notes Neither indeed would I have put my selfe to the labour of writing any Notes at all, if the booke could as well have wanted them, as I could easilie have found as well, or better to my minde, how to bestow my time. Marcus Aurelius Antoninus. Chapter 1 1. W.S. Gilbert, The Mikado, 1885. 2. Commentators are divided on whether Hazlitt should be regarded as an essayist or a critic; thus Priestley [1960, p. 6] prefers to de­ scribe him as an essayist, while Brett [1977, p. 5] regards him as a biographical critic. 3. The first abstract presentation of this idea may be found in Plato's Republic, the parable reading as follows in the Analysis to Book VII, Steph. 514-517 of Jowett's translation: Imagine human beings living in an underground den which is open towards the light; they have been there from child­ hood, having their necks and legs chained, and can only see into the den. At a distance there is a fire, and between the fire and the prisoners a raised way, and a low wall is built along the way, like the screen over which marionette players show their puppets. Behind the wall appear mov­ ing figures, who hold in their hands various works of art, and among them images of men and animals, wood and stone, and some of the passers-by are talking and others silent ... , they see only the shadows of the images which the fire throws on the wall of the den; to these they give names, and if we add an echo which returns from the wall, the voices of the passengers will seem to proceed from the shadows. Suppose now that you suddenly turn them round and make them look with pain and grief to themselves at 522 Notes: chapter 1 the real images; will they believe them to be real? Will not their eyes be dazzled, and will they not try to get away from the light to something which they are able to behold with­ out blinking? And suppose further, that they are dragged up a steep and rugged ascent into the presence of the sun himself, will not their sight be darkened with the excess of light? Some time will pass before they get the habit of perceiving at all; and at first they will be able to perceive only shadows and reflections in the water; then they will recognize the moon and the stars, and will at length be­ hold the sun in his own proper place as he is .... How will they rejoice in passing from darkness to light! How worth­ less to them will seem the honours and glories of the den! But now imagine further, that they descend into their old habitations;- in that underground dwelling they will not see as well as their fellows, and will not be able to compete with them in the measurement of the shadows on the wall ... Now the cave or den is the world of sight, the fire is the sun, the way upwards is the way to knowledge, and in the world of knowledge the idea of good is last seen and with difficulty, but when seen is inferred to be the author of good and right. [1888, pp. xcviii-xcix] The allegory was rehearsed by Bacon in his Novum Organum of 1620, where we read The Idols [i.e. illusions or false appearances] of the Cave are the Idols of the individual man. For everyone (besides the errors common to human nature in general) has a cave or den of his own, which refracts and discolours the light of nature; owing either to his own proper and peculiar na­ ture; or to his education and conversation with others; or to the reading of books, and the authority of those whom he esteems and admires; or to the differences of impressions, accordingly as they take place in a mind preoccupied and predisposed or in a mind indifferent and settled; or the like. So that the spirit of man (according as it is meted out to different individuals) is in fact a thing variable and full of perturbation, and governed as it were by chance. [Aphorism XLII] This translation of Bacon's original Latin text is by Robert Leslie El­ lis, some of whose statistical work will be discussed in a later chapter. 4. See Romanov [1974], Sabatier [1978] and Talenti [1986]. 5. For a good discussion of moment problems see Shohat & Tamarkin [1943]. Notes: chapter 1 523 6. For further discussion of the role of inverse probability in error mea­ surement see Edgeworth [1911]. 7. For instance Salmon [1966, p. 118] finds that plausibility arguments not only are an essential ingredient in scientific inference, but also "embody considerations relevant to the evaluation of prior probabili­ ties" (Salmon's views on this matter have been discussed, somewhat slightingly, by Weimer [1975]). For a discussion of mathematical mod­ els of uncertainty and indeterminacy see Walley [1991, §1.8]. 8. See also §5.14. 9. The original reads as follows: At hic tandem nobis aqua hrerere videtur, cum vix in pau­ cissimis prrestare hoc liceat, nec alibi fere succedat. quam in alere ludis, quos primi inventores ad requitatem ipsis conciliandam data opera sic instituerunt, ut certi notique essent numeri casuum, ad quos sequi debet lucrum aut damnum, & ut casus hi omnes pari facilitate obtingere possent. In creteris enim plerisque vel a naturre operatione vel ab hominum arbitrio pendentibus effectis id neutiquam locum habet. [po 223] 10. Bernoulli's words are Verum enimvero alia hie nobis via suppetit, qua quresitum obtineamus; & quod a priori elicere non datur, saltern apos­ teriori, hoc est, ex eventu in similibus exemplis multoties observato eruere licebit; quandoquidem prresumi debet, tot casibus unumquodque posthac contingere & non contingere posse, quoties id antehac in simili rerum statu contigisse & non contigisse fuerit deprehensum. [po 224] 11. The original runs as follows: Hoc igitur est illud Problema, quod evulgandum hoc loco proposui, post quam jam per vicennium pressi, & cujus tum novitas, tum summa utilitas cum pari conjuncta difficul­ tate omnibus reliquis hujus doctrinre capitibus pondus & pretium superaddere potest. [po 227] Both David [1962, p. 136] and Sung [1966, p. 42] translate "super­ addere" as "exceed", the impression thus being conveyed that the novelty, f3c. of the problem exceed the value of the rest of the work. The translation I have given here, which I believe to be more ap­ propriate, agrees with that given by de Moivre [1756, p. 254] and Haussner [1899, II, p. 92]. 524 Notes: chapter 1 12. This passage runs as follows in the original: Sit igitur numerus casuum fertilium ad numerum steril­ ium vel prrecise vel proxime in ratione r / s, adeoque ad numerum omnium in ratione r/(r + s) seu r/t, quam ra­ tionem terminent limites (r+1)/t & (r-1)/t. Ostendendum est, tot posse capi experimenta, ut datis quotlibet (puta c) vicibus verisimilius evadat, numerum fertilium observa­ tionum intra hos limites quam extra casurum esse, h. e. numerum fertilium ad numerum omnium observationum rationem habiturum nec majorem quam (r + l)/t, nec mi­ norem quam (r - l)/t. [po 236] 13. Compare Sung [1966, p. 4]. 14. See also David [1962, p. 137]. 15. The passage translated here is Unde tandem hoc singulare sequi videtur, quod si eventuum omnium observationes per totam reternitatem continuaren­ tur, (probabilitate ultimo in perfectam certitudinem abe­ unte) omnia in mundo certis rationibus & constanti vicis­ situdinis lege contingere deprehenderentur; adeo ut etiam in maxime casu ali bus atque fortuitis quandam quasi neces­ sitatem, &, ut sic dicam, fatalitatem agnoscere teneamur. [p.239] 16. See Hacking [1975, pp. 149,154]. 17. The presence of this passage was drawn to the attention of the sta­ tistical community by S.M. Stigler in 1983. 18. The presence of the expectation operator here rather than the prob­ abilistic one is of no moment. 19. The distinction was, it would appear, also noted by Good [1959], who wrote of Bayes's attempt at an inversion of Bernoulli's Theorem. Stigler [1986a, p. 100] regards the inversion of the results of Bernoulli and de Moivre as "The chief conceptual step taken in the eighteenth century toward the application of probability to quantitative infer­ ence" . 20. The sketch illustrating the situation detailed in the following quota­ tion may be found in Chapter 4, Figure 4.1. 21. Each of these symptoms may be viewed as a complex of symptoms. 22. In translation: "0 happy he who can determine the causes of events!" Commentators agree in stating that Virgil is following Lucretius here. A similar sentiment was expressed in Matthew Arnold's Memorial Verses of 1850: "And he [i.e. Goethe] was happy, if to know, causes of things ... " . 23. The connexion is also considered in Keynes [1921, pp. 401-402,420- 422]. Notes: chapter 2 525 24. The idea of Pr[Plq], the probability of p on data q, is taken as funda­ mental by Jeffreys [1961, p. 15]. 25. Perks's paper of 1947 is an important one in the annals of inverse probability: not only are exceedingly pertinent comments made on this topic, but a new indifference rule is proposed. 26. See Robert [1994, p. 370]. 27. See O'Hagan [1994, p. 134]. 28. Seidenfeld [1979, p. 19]. 29. The name of this "eminent professor" is not given, although a footnote to this page gives the other lecturers as Dr Venn, Professor Weldon and Sir Robert Ball. Chapter 2 1. This description is from Hacking's introduction to the English trans­ lation of Maistrov [1974, p.