The probabilistic analysis of testimony

Journal of Statistical Planninp and Inference 20 (1988) 327-354 North-Holland

327

emetics und Stat&tics, A~rthwestern University, Evanston, IL 60208, U.S.A. ay 1987; revised manuscript received I2 April 1988 Recommended by E.J. Wegman

4bstract: T5e probabilistic analysis of testimony is surveyed. The coverage is not comprehensive; at -ent%a is focused on several problems of particular interest or complexity. The theory often

contains implicit assumptions, and some attempt is made to clarify the role these play. The theory originally arose in an attempt to understand the logic of belief. It was not empirically grounded, however, and later died out in the I9th century when its conclusions became largely self-evident, and its non-empirical background suspect. AMS Subject Ckus~ficotion: Primary 6043; Secondary 01-01. Key words andphrases: Testimony; evidence; Condorcet; Laplace; Hume; Campbell-Price paradox; Eddington.

considerable controversy, and discussions of it were a co

fered from a nearly universal fail ingly frequentistenvironmentof

0378-3758/88/$3.50 0 1988, Elsevier Science

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memoirs’. evertheless, mixed amongst the chaff are a number of problems and paradoxes of surprising interest, and it is around the paper is centered.

though the mathematical discussion of the testimony of a single witness does not mark the earliest use of probability to analyze testimony, we wiZ depart from a strictly chronological treatment of the subject by first considering this simple case, and only later pass on to the more complex questions attending successive and concurrent testimony_ 1.1. Condorcet’s formula et us begin with a problem discussed by Condorcet: The prior probability of an event is p, and the probability that a witness speaks the truth is t. What is the probability that the event actually occurred, given that the witness states it did? Condorcet (1786) gives as his answer Condorcet’s formula: pt/(pt + (1 -p)(l

-

t)).

Condorcet’s formula is a special case of the general Bayesian solution (see below, Section 1.3). It makes the apparently harmless assumption that the witness is as iikely to say the event occurred, if it did, as to say the event did not occur, if it did not. s Todhunter (1865, p. 400) remarks, Condorcet gives this formula “with very little and its application “is not free from difficulty.” Let us illustrate it with a simple example discussed by Laplace in his Essaiphilosophique (1814, Chapter 11): An urn contains 999 black balls and one white ball. One ball is drawn and a witness announces that the ball drawn is white. Suppose further that experience has made known that this witness deceives one time in ten.

Thus, p= l/l and t = 9110, so that, applying Condorcet’s formula, it follows that the probability is

i.e., approximately 1 in 1 to the substantitil a prior mains quite small.’

relative reliability of the e its posterior pro

1 The phenomenonis today a familiar one. For example, if a diagnostic screening test is employed to detect a rare condition, such as tuberculosis, the probability that a person has the condition, given a positive test outcome, may be r (1977, pp. 25-2 e te ositives, although ..noCS.l \ u*clua~p

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hat was Condorcet’s a famous controversy in the 18th century: th lt was ccmnody argued, for example, 1705, that the divine o the occurrence of mira enciary support for miracles and rection. It was against this background that ume argued that it was in the nature of a evidence could suffice to make it probable, due to its inherent improbability: ,Xotestimn:dnyis sufficient to establish a miracle, unless the testimony be of such a kind, that A lralsehood would be more miraculous, than the fact, which it endeavours to establish; and even in that case there is a mutual destruction of arguments, and the superior only gives us an assurance suitable to that degree of force, which remains, after deducting the inferior. [Hume, 1748, pp. 115-M]

argument was purely qualitative, and Condorcet’s formula was an attempt to quantify the phenomenon; the section of his paper which introduces the formula is entitled “‘Surla probabilite des faits extraordinaires.” “has received more critical attention than anything ume’s essay else Hume wrote on religion” (Gaskin, 1978, p. KS), and “m&as, bcyon most provocative publication of his lifetime” (Flew, 1986, p. 79)? many who rose to counter ume’s argument was the who had a few years earlier edited ayes’s manuscripts for public later’s death.3 In the last of his Dissertpations(1768), ‘ ‘The Wu&s

d and a critical analysis of Hume’s argument, 2 For further information on the 18th centur of Religion (1978, Chapter 7). The definitive see J.H.C. Gaslcin’s excellent book, Hume’s account of the deist controversy still remains Sir Leslie Stephen’s Hi&toryofEnglish Thought in the Eighteenth Century (1876),although it has now been superceded in part by R.M. Burns’s The Great Debate

on Miracks (1981). There is an extensive philosophical literature on miracles; two useful introductions with further references are Anthony Flew’s entry on miracles in the Encyclopedia of Philosophy, and Swinburne (1970). For recent probabilistic analyses of Hume’s argument, see Sobel (1987), Owen (1987), (1988). 3 A dissenting preacher of Welsh extraction, Richard Price (1723- 1791) enters the history of probabihty and statistics at several junctures: in addition to his role in publishing Rayes’s papers and his dispute with Hume, Price partici 53-M), and wrote exten 381409). A colorful pers and Priestly (who preached his funeral oration); his correspondents Job

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Christianity, the Nature of Historical Evidence, and Miracles”, Price mounted a two-pronged attack on Hume’s argument. First, appealing to Bayes’s papers, argued that the evidence for a law of nature could at most establish a probability in its tavor, adding that it is “very inaccurate to speak on this subject prevously to calculation”. Second, Price argued that the apriori improbability of an outcome could not alone cast doubt on the veracity of an otherwise credible witness: if a number is drawn in =1lottery, any outcome will be improbable and yet one would not thereby call it into question. This later argument which had been made earlier by George Campbell (1762, pp. 19037), who considered the case of a ferry which safely makes a crossing 1000 times ix both directions, and which is then reported to have sunk.4 1.3. The Campbell-Price paradox Thus an apparent paradox involving the ‘Hume-Condorcet’ formula arises. As Todhunter (1865, p. 400) notes: “Suppose . . . a trustworthy witness asserts that one ticket of a lottery of 10000 tickets was drawn, and that the number of the ticket drawn was 297. Here if we put p = l/l we obtain such a very small value of the truth of the witness’s statement that we lose our confidence in the formula.” For example, if t = 999/1000, Condorcet’s formula states that the probability that the ticket is actually numbered 297 is 999/10* (999/10~)+(1/1000)(9999/100009

’

or approximately l/100. Obviously a similar reasoning would be valid whatever the number announced. Thus, although the reliability of the witness is high, whatever number is announced apparently has a very small posterior probability of being true. Condoret attempted to deal with this difficulty be modifying the value of p, which he terms ‘Ye probabilite prop&‘. As Todhunt er (1865, pp. 400-406) notes, Condorcet first proposesbne method which he later abandons for another, but he is inconsistent in his application of the latter and, in any case, both seem arbitrary and neither defensible. It was left rather to Laplace to resolve the paradox. As Laplace realized, the resolution of the Campbell-Price paradox lies in recognizing that the problem, as posed, is incompletely specified. f the number 297 4 The argument appears to have originated with Joseph Butler (Stephen_, 1962, pp. 338-339), but Campbell was the first to direct it against Hume. A Scottish divine, George Campbell (1719- 1796) was Principal, later Professor of Divinity at Marischal College. Commonly thought the most able of Hume’s critics, Campbell sent a copy of his Look to Hume for comments before its publication, and the two exchanged letters, Hume departing from this usual custom of not corresponding with critics. Ironically, oswell visited Hume on 7 Jul oted that e had ss, m a copy of Campbell’s ret ric well’s Jou entry rch 1777).

is announced, this can come about in one of two ways: ( and announced (wi robability t); (2) a number other than

7 is drawn (with pro-

the witness. Since tickets are drawn at a=FiI’=n

lX=n]

b=P[X=m]

[Y=nJX=m] m#n

so that 0 1 b s 0.001)(0.99W) = 0. [X=n 1 Y=n]rl, ding on tlt;emissinp ;rarameters P r further discussi Diaconis ar:? %edman, 1981.) As Laplace notes in discussion in th philosophique,if the parameters are taken to be uniform in n, then the posterior probability reduces to t, i.e., the reliability of the witness. If, however, the witness has some interest in seeing that the particular number announced was drawn, then the distribution will not be uniform and the posterior probability will be less than f? Thus the key question is: what is the probability that Mis announced, given that another number m has occured? This highlights a shortcoming of much of the literature on testimony. It is often assumed that one is in a ‘black-white’ situation, where assertions or answers are dichotomous. In the case of a diagnostic test, this may well be realistic. If, however, the possible answers are diverse in nature, then both the analysis and the possible outcomes becomes much more complex; and in particular, the whole notion of ‘reliability’ becomes much more diffuse? The correct general formula for the posterior probability may be readily calculatayes’s theorem. It is pt/(pt+(l

-p)(l

-t”)),

where t is the probability that the witness asserts. the event in question, given that it did occur, and t* is the probability that the witness asserts the event in question, given that it did not. As ill correctly notes, 5 There is an amusing postscript to the ‘Todhunter paradox’. Todhunter refers to both Laplace and De Morgan (1851) and thus was clearly familiar with the resolution of the paradox discussed above. point was that Condorcet’s explanation was inadequate and that the formula can be easily misused: its application “is not without difficulty.” Some recent critics of the theory of subjective probability have misread this passage in Todhunter as pointing to a major difficulty for the theory. For example, L. Jonathan Cohen, citing Todhunter, asks (rhetorically): “A witness of 99.9% reliability asserts that the tickets was, say, 297: ought we really to reject number of the single ticket drawn in a lottery of 1 1981, p. 329). that proposition just because of the size of the lo claim otherwise. Zt was The answer, of course, is no - and no serious dispel such confusions that LapYacediscussed the ) notes that this distinctio he result was to make perfect, and in some respects erroneous”.

ic itio first art of the subject obscure and im-

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TO know whether a coincidence does or does not require mqre evidence to render it credible than an ordinary event, we must refer, in every instance, to first principles, and estimate afresh what is the probability that the given testimony would have been delivered in that instance, supposing the fact which it asserts not to be t.rue. [ 25,

WI

1.4. Tire criticisms of Venn; witness credibility and veracity The AchilIes heeI of the Condorcet-Laplace formulation lies in its notion of testimonial ‘reliability’. At this stage it is important to discuss an ambiguity often present in the testimoniaI literature: the meaning of ‘retiability’. The problem was noted by Keynes: The manner in which the resultant probability is affected depends upon the precise meaning we attach to ‘degree of reliability’ or ‘coefficient of credibility’. If a witness’s credibility is represented by X, do we mean that if a is the true answer, the probability of his giving it is X, or do we mean that if he answers a the probability of a’s being true is X? These two things are not equivalent. [Keynes, 1921, p. 1831 Let us agree to adopt the foIlswing, admittedly

Credibility: P[Assertior?/I is true Veracity: P[Witness asserts A

1

1

arbitrary terminology:

Witness asserts A].

Assertion A is true].

The first of these, the witness’s credibility, is a single-case, epistemic conditionaI probability: it is our posterior probability that a specific statement A is true, given the testimony of the witness. The second, the witness’s veracity, is also a single-case probability, but in certain cases may be identified with a frequency. It is the latter definition that is usually, but not always, intended in the testim~tiai !iterat~e.’ In Nicholas Bernoulli’s thesis of 1709, he states: Since the credibility of witnesses ought to be diligently examined before they are admitted to give testimony, it would not be far from the subject to give here a rule by whose help it is possible to measure anyone’s trustworthiness and to reckon how great the probability is that he speaks the truth or not. This is the rule: Divide the number of times in which he had been found to speak the truth by the sum of those and the number of those times on which he was found to lie, and you will have the degree of trustworthiness. micholas Bernoulli, 1709, Chapter 91 That is, if we view the witness’s statement about A to be embedded in a universe of assertions about different events A 1,AZ, A3, . . . , and if the fraction of true asser’ A similar ambiguity arises in the literature on diagnostic screening tests. For example, Fleiss (1973) defines the ‘false-positive rate’ to be the fraction of persons testing positive who in fact have the disease, while Wilson and Jungner (1968) define this rate in the opposite sense, i.e., the fraction of those having the disease who test pmiiive. The confusion of the conditional probabilities P[A 1l?] and P[B 1A] is, of course, a very common and much more general phenomenon, pervasive throughout the literature and by no means limited to the uninitiated; its distinguished victims include

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tions in this universe is X, then - assuming that is in some sense a ‘random’ member of this universe - we may identify the veracity with the frequency x.s The assumption that such a ‘truth-frequency’ was a meaningful and relevant parameter came under heavy attack during the 19th century, most notably by Venn in his Logic of Chance (11866).Venn had champione a frequentist view of probability, and criticized the testimonial literature on the unds that such a frequency could not be meaningfully determined. In order to apply frequent% probabilities to single-case events, su events must be embedded in a class of events for which the frequency is known. oreover, such a class must be the smallest possible. Venn derided the possibility of doing so in the testimonial case. The problems of when such an identification is valid are not unique to the testimony problem; they arise whenever there is an inferential passage from a classfrequency to r, single case. When the specific outcome or sample is a random sample from c !r iger, prespecified populaticll :hen there is, in principle, no difficulty; i.e., when one begins with the population and generates the sample. ut when one goes in the opposite direction - when one begins with the sample and the populations is, in effect, a hypothetical construct - then problems arise for two reasons. First, the hypothetical population need not be unique. This problem may sometimes be eliminated by a ‘principle of total evidence’: each item of information about the outcome narrows the possible universe or reference class, and the appropriate universe is taken to be the smallest containing the outcome, i.e., that which takes all available information into account. (A natural problem which then often arises is that the frequency for a trait of interest may be known for various larger universes, but not for the smallest.) ,dnce creates Unfortunately, in solving the first problem, the principle of tota! a second: the smallest reference class may become a singleton, the outcome itself. indeed, it is clear that this must essentially aiways be the case: unless there are artificial limitations on the information available, it will always specify a unique individual. There is a way out, however: past a certain point information may become irrelevant, in the sense &at the frequency for all subuniverses from some point on may become constant, or at least vary within negligibie iimits. It is precisely this problem of the appropriate reference class that Venn points to. Taking, as he did, an exclusively frequentist view of probability, he was only willing to assign a witness a veracity to the extent that the asser identified as a member of a population of such state quency. Similar criticisms were also voiced by

witness’s veracit

ould depend on the c

8 Compare Cournot (1843, pp. 410-411), who terms such a truth fre tdmoignage’. Young (1800, p. 84), in contrast, says that ‘the probabi given instance’ is the ratio of ‘the sum of the number of c total number of chances. This is, in effect, a single case propensity.

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result this rendered “&idemment impracticable toute application numerique” (Cournot 1843, pp. 413-414). But Cournot did not dismiss the theory outright, as organ, 1851, p. 12 did Venn and ertrand. (See also the comments of 1.5. The firie stmcture of veracity he terms ‘credibility’ and ‘veracity’ have been used as a convenient shorthan for two different, but closely related conditional probabilities. Pt is apparent, however, that these two probabilities fail to capture anything like the full sense of the two words. For example, a ‘veracity’ - as we have used the term - expresses the probability that a witness will correctly report an event, but such a correct report will depend not only on the truthfulness of the witness, but also on the accuracy of his perception. Indeed, in order for a witness to correctly report an event, he must (1) accurately perceive it; (2) remember it with precision; (3) truthfully state it; and (4) succesfully communicai;e it to others. Any attempt to realistically model human testimony must necessarily take account of every link in this chain. Laplace includes such consideration in part in his analysis (Laplace, 1812, pp. arc somcrimes repeated in later treatments 455-460; 1814, p. lxxxvii), and an, 1844 pp. 470-471). The mathematical (Lacroix, 1833, pp. 259-262; de analysis resembles that of the successive testimony of two or more witnesses (to be discussed below in Section 2), and will not be considered further here. 1.6. The three prisoner paradox The resolution of the Todhunter paradox required a careful analysis of the different possible origins for the information received. There are a number of probability paradoxes centered about the phenomenon that the informational content of a message may depend on the mechanism by which it is transmitted. e is the three prisoner paradox: 6 are three prisoners awaiting execution. A board of pardons meets and commutes the sentence of one of the three, but does not immediately announce risoner A, arguing that he already knows that at least one of e one of the two, who ed, asks the w warden does so, naming cheers reasoning: “ of a pardon were -#;now only C and myself are i ontention for a pardon, and since w;: are both equally Ii ely to receive it, my chance of to 3.” The basic point of the three prisoner paradox is that the releva

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335

doned, and that when he has a c then the posterior probability of execution for n the general ca care nA, ztg, when he has a ch for A, given the jailer names ner paradox is representative of a class of situations where calculariate conditional information is of the second ace: the conditional probability that two cards dealt from a deck are both aces, given that one is an ace, is not the same as the conditional probabi that both are ac:zs, given that one is the ace of spades. The stand probleh? ml&es an implicit assumption about what it means for question to blz ‘given’. As discussed by aber (1976), however, th abilities involved can, and in general will, depend upon how the information stated was actually acquired. Such paradoxes have been recently reviewed by Shafer (1985). Shafer proposes dealing with such difficulties by insisting on the statement of a protocol, “a set of rules that tell, at each step, what can happen next”. Indeed, Shafer goes so far as to claim that the use of the formular ] is questionable whenever a protocol is lacking, and that prot ntegrated into the axiomatic framework of subjective probability ems untenable, given a major use of conditional probabilities is to calculate the probability of compound events whose subevents need not occur in sequential order. Nevertheless, Shafer’s basic point is certainly well taken: when condition-al probabilities are calculated, care is required in enumerating the possible outcomes which might have occurred, end. @lcit statement of a protocol can be a useful discipline towards t one er a formalization of the recess, as Shafer attempts in his p further than a general sensitivity to and awareness of the problems involved seems, however, questionable. There is an amusing twist to th separating his cell from C, an of execution does indeed decrease (to where more than one witness is now turn to an analysis of this proble

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of testimony

witnesses in puraIleLgThe first of these corresponds to the game “Telephone’: one person writes a message down on a piece of paper and then whispers it to a se person, who whispers it to a third, and so on. The point of the game is that the ‘\--sion reported often bears little resemblance to the original if information is relayed through a series of witnesses it is reasonable to expect that the probability of its veracity decreases with the length of the chain. As Locke observed, “Traditional testimonies the farther removed, the less their proof. . . . robability can rise higher than its first original” (Essay Concerning Understanding, ook 4, Chapter 16). 2.1.

oper and the credibility of historical evidence

The probabilistic analysis of testimonial evidence may be dated to 1689, when it first occurs in a little known work by the nglish cleric George ooper, once Chaplin and later to become 11s.The application was s: Hooper’s book wa in part, an attack on the nonscriptural traditions cited in support of the Catholic church. In the course of his argument Hooper asserts that if the credibility of each witness in a testimonial chain of n witnesses is p, then the combined credibility of their testimony is p”. HooperlS book does not give the actual formula, but in a later paper appearing in the Philosophica Transuzctions of the Roya.”Society (Hooper, 1699), mathematical formulas for both the successive and concurrent cases were given. ooper’s book and his later paper appeared anonymously, and it was only quite recently that ooper’s contrib on and his authorship of the 1699 paper ~~8s pointed out (Grie 1982). Indeed, oper’s 1699 paper has frequently been attributed to the Scottish mathematician John Craig, because in the same year Craig published a book, iogiae Christiane thematica, which likewise argued that the cre lity of successive n diminishes with time. owever, as both n (1978, p. 469) and Shafer (1978, p. 347) have observed, m the internal evidence alone that Craig could not have been the author hical Transactions paper; Craig’s book is a much inferior effort. etic, if not entirely convincing attempt to rehabilitate Craig, see Stigler (1986a).) In general, if the credibilities of n successive witnesses are pl, p2, ._.,p,, , then ooper gives as the resulting credibility oper’s rule: -D~P~--~~. _- _ es with credibilities

concur i

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337

ever, at least one class of settings in which this is clearly unreasonable. 2.2.

holy messages Prevosfs

f~nmila

Suppose one is in the special case where the message being transmitted can only be in one of two states: yes/no, black/white, etc. In that case, as first noted by Prevost in 1794, a correct answer will be trans if both A and if knth A nnA R lita +!I be OFaAVU&~& A U&IU Y a w* 11 Thrac 1 L&U” tip C. w fr~nmmwv s.“ybe”*.“J Prevost’s formula:

p2 + (1 -

p)’

Similarly, if there are three witnesses, then the probability becomes p3 + 3p(l -P)~, and so on.12 2.3. Lap/ace’s urn model iscussion of testimony, ball is in one of se the state is reporte has the advantage of restiicting attention lo In 1963 the Swedish jurist Per Olof Ekeliif independently derived ooper’s rules for successive and concurrent testimony, and gave a rule for conflicting testimony that can be viewed Lambert’s; see Shafer (1986) for a brief description of EkeloPs work, and an interesti &IIand sion of subsequent attempts by the Swedish philosophers S&en a Bayesian justification of Ekelofs rules.

discovering it. For furt

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becomes uncontroversial and was helpful in resolving the Campbell-Price paradoxc13 In the case of successive testimony, Laplace shows that if there are r witnesses, each having veracity pi, and n possible states, then the probability of the message being correctly transmitted (in effect, the veracity of the chain) is 1 (n-1) Laplace’sformula: yr = 5; + n

(npl-l)(npz-l)*=e(np,-1) (n- 1)

(assuming that the n - 1 different possible erroneous answers are equiprobable). As Laplace notes, if n =oo, then yr=p1p2.mepr, i.e., the formula reduces to the classical answer due to ooper. If, on the other hand, n = 2, then Laplace’s formula reduces to Laplace’s binaryformula: 3 + 3(2pl - 1)(2p2- l)**=(2p,- l), which in the case of equal veracities becomes 3 + +(2p - 1)‘. 2.4. The Artemas

tin problem

There is an amusing duality applicable to the Hooper-Prevost formulas for successive testimony. The formulae, as usually derived, s;;sume the pi to be veracities, are computed by arguing forward from the first witness to the last, and result in a composite veracity or truth-frequency. They are equally valid, however, if the pi are assumed to be credibilities, arguing backward from the last witness to the first, the result being a posterior conditional probability. Consider, for example, the following problem: “A says that B says that a certain event took place; required the probability that the event did take place, pI and h being A’s and B'srespective probabilities of speaking the truth.” [Lubbock and Drinkwater, 1830, p. 91

What is required is 64theprobability that the event did take place,” i.e., a posterior conditional probability, and not the truth-frequency of the chain. Nevertheless, both Lubbock and Drinkwater, and Todhunter (in his Algebra), give p1p2+ (1 -pl )(l -p2) as the answer. This only makes sense if the pi are interpreted as credibilities, and if one argues backward from the second witness to the first. Note, however, that this is defensible only because the question refers to ‘a certain event’. redibilities will virtually never be constant over classes, except for very special cases such as Laplace’s lottery example. above, in general credibilities l3 Note that this is not the same as merely likening true and false testimonies to drawing balls from an urn, red and black say, with red denoting truth and black falsity. That assumption is only equivalent to assuming the existence of a truth frequency. Laplace’s model, in contrast, describes the possible states of nature (the colors of the balls), their prior probability (the frequency with which they occur in the urn), and the range of possible assertions a witness can ma e (the possible assertions about the color or state of the ball drawn). It is natural to a s, does not exceed n2, the number of observable states. fwt =n2, assume aflame, roupin se states eorre5

S.L. Zabell / Analysis of testimony

will depend on the prior ility or question. This problem was apparently first ba~i~ity, and later appe

hunter

ity

i

339

of t

icul

a

ater’s

eve tise

p1p2 + (l-

PlP2+8(1-P1)(1-P2)$#(f-b)(l-P1)

solution is valid under the appropriate assumptions. The first is ment; (2) the statement had the stated content. Cayley’s solution introduces prior probabilities and the possibility of irrelevant answers. l4 2.5. Eddington 3 problem With this background, let us turn a problem discussed by the English physicist Arthur Stanley Eddington in his 1935 book New Pathways in Science: If A, Ip”,C, D each speak the truth once in three times (independently), and that B denies that C declares that D is a liar, what is the probability that D w the truth?

Eddington’s problem contains a new twist: relevance. It is possi or C or both have made statements which do not refer to the preceding person? (It is of course possible that they say nothing, but we will assume this not to have been the case.)

I4 “B told A that the event happened, or he did not tell A this; the onl B told him that the event happened; and the chances are p1 and 1-pl. told pl that the event did not happen, or he did not tell him at all; the the incorrectness of A’s st ent) are fl and 1 -/I; and the /I(1 -PI), and (1 -fl)(ll -pl c&ions of the first a for the event having happenedare p2 and l-912; on the supp is no information as to the event) the chance is PC,the ante&

e last subsection, Cayley’s solutio of ir

‘s statement that , in the latter case, eithe

S.L. Zabell / Analysis of testimony

340 Table 1

Possible TL combinations

LLTT LTLT -- wlLL1 LTTL TLTL TTLL LLLL

l/81 4/81 4/81 CSi

4/81 4/81 4/81 16/81

Impossible TL combinations LTTT TLTT TTLT LLLT TTTL LLTL LTLL TLLL

2/81 2/81 2/81 8/81 2/81 8/81 8/81 8i81

Only relevant answers possible s begin by assuming this to be the case. If so, then the analysis of Eddington’s problem is essentially the same as Cayley’s, albeit more complicated. One simplierrne (1965, pp. 186-187), is the following pr fication, suggested by 6, Ts D a liar?“; C then tells the answer to imagine a referee X, w “did C assert that da is a liar?“; B in turn communinot X. The referee then asks cates the anwer to A. Finally, the referee asks A, “Did B deny that C asserts that D is a liar?“, and A answers. There are 24= 16 TL combinations, and the given information excludes 8 of these. f these, D tells the truth in 4, and lies in four (Table 1). Thus, the probability that tells the truth is (13/81)/(13/81+28/81)= 13141. The presence of a referee is not necessary, simply that each person reports the person’s statement or its negation. (Note that a statement such as “C lies” is equivalent to Y! denies D tells the truth”.)rs ddington’s solution to this problem, instead, is 25171. I-Ie was led to this answer e allowed the possibility that the witnesses make I;assertionsirrelevant eding person’s statement although, as we shall see, even in this case ddington’s answer is incorrect. Ctxse II: Irrelevant answers possible

n order to simplify the discussion, let us first omit A. In that case there are 8 combinations. Suppose we are told denies that C asserts that

is a liar.”

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prisoner paradox: suitable care must be taken in describing the sample space an which events are excluded by the given information, indeed, a careful statement of just what is ‘given’. A complete analysis se lines was given by .W. Chapman in the article appeared the same year. If a, b speak the truth, are irrelevant, then and b,, cSare the probabilities that the assertions by Chapman shows that d[3( 1 +P} - b,(l -~)@a- l)+cQa+{1+P(2&ij] +$&(2a- I){(2c--I)@&--1)-l]

1)(2h- l)+ b,c@a- l)] tcsd(2a- 1)(2b-1

where = (2a - 1)(2b - 1)(2c- I). In particular, if a= b ==c= =f, the probability is 13 + 6b, + -3/c, - 9bscs + 123b, + cs- 9b,c,

*’ The Eddington problem fir described as “a matical gazette roblem was discuss New Scientist for

In successive testimony, there is an input event and an intervening process is viewed as a black box, then the math to the cwz of a single witness; the novelty of the theory therefore lies in its ity of the testimonial cha.& in terms of the veracities of its mnto model the v concurrent testimony, in contrast, two or more assertions are dividual links. made, and this has two important consequences for the structure of the theory. some assumption must be made about how the statements are reiated. onally, it is either assumed that the responses are binary assertions or denials T of an event, or it is assumed that the responses are complex statement providing evidence in support of a proposition, whose joint probative weight is to be assessed. Second, unlike the case of successive testimony, a veracity cannot be attached to the process as a whole. Thus, non-Bayesian analyses of concurrent testimony must veracities if they are to avoid probability statements indepenSties. uch of the literature on testimony is closely related to, but distinct from that pertaining to two other subjects - the accuracy of judicial decisions, and the combination of arguments. A few brief remarks about both may be in order. 3.1. The combination of evidence The iiterature on the combination of evidence or arguments originates in 1713 ith the posthumous publicatio ernoulli’s Ars Conjectandi, although earlier; see Hacking (1975, Chapters 10 and IQ, Shafer (1978) ernoulli’s qualitative view of the cumulative force of circumstantial evidence was Oman orators Cicero and Quintilian; the real novelty of his contri to analyze the process mathematically (Garber and Zabel hi’s rules were later criticized einrich Lambert (1728- 1777), of it; see generally

book; in which he briefly criticized Eddington’s solution on more general philosophical grounds, a criticism which he later expanded upon in a separate article in response to comments of T.E. Sterne. Dingle’s criticisms were cranky: he thought the combination of base-rate probabilities and veracities implicit in the problem incorrect (despite the fact that Eddington had made clear the purely hypothetical nature of the example, intended to illustrate the principle of exclusion). Instead, he arguedin part, one

rd one of the two and use the other. Nevertheless,Dingleconcludedhis articlewith a solution ~rcsumablyto show reader he was not inca able of solving it; his solution, although ddington’s, was no

ore accu:ate. is cl

re

3.2. Judicial decision making

members or nonunanimous verdicts.

aplace also discussed the

le

344

S.L. Zabell / Analjsis of testimony

The rule quoted by Keynes is: X and Y are independent witnesses (i.e., there is not collusionbetweenthem). The probahflity that X will speak the truth is x, that Y will speak the truth is y+ x and Y agree in a particular statement. The chance that this statement is true is xv xy+U -x)(1 -Y) l

In the (admittedly very special) case of independent witnesses and dichotomous

answers, this formula is certainly correct. Nevertheless, Keynes objects to it on several grounds. lg Keynes begins by asserting: “‘It may, however, be safely said that the principal conclusions on the subject set out by Condorcet, Laplace, Poisson, Coumot, and Boole, are demonstrably false. The interest of the discussion is chiefly due to the memory of these distinguished failures.” Keynes’s discussion, however, is disorganized, unfair to his predecessors, and occasionally in error. First, and ironically, although Keynes later notes the distinction between credibilities and veracities, his discussion at this point apparently confuses the two, Although x and y are almost always intended in the classical discussions he cites as veracities, he first interprets them as credibilities, and correctly notes that mere causal independence or lack of collusion does not justify equating the probability that X and Y both speak the truth with v, the product of their individual credibilites. Keynes then goes on to note that in the special case of dichotomous answers to a single question, the classical answer is correct if x and y are interpreted as veracities and the prior probability for the event in question is 3. (It should be noted however, that the answer is also correct if the conditional probability is interpreted as a conditional fieqacency: in a sequence of trials the fraction of correct agreements. Obviously, if what is desired is the posterior conditional probability that a specific event has occured, then the prior probability of the event must be supplied before the posterior probability can be computed.) When x and y are veracities, the independence invoked in the derivation of the formula is easier to justify. 3.4. The general rule There are several interesting discussions of the general formula for concurrent testimony (where several witnesses are involved). It was briefly alluded to by ooper, and is mentioned by Condorcet, Young, and Prevost and L’ ost mathematically sophisticated analysis, ost insightful critic

M’Sdiscussionis ma h)(b/q b2h) = (ah; latter are sufficient for the

S.L. Zabell / Analysis of testimony

345

ume’s argument against miracles: abbage argued that no matter how small the prior probability of a miraculous event, if there were a sufficiently large number of independent attesti witnesses, the posterior probability w&d be &se ruskal (1988). As u&al notes, if all witnesses have a common veracity X, then the conditional probability of the event attested, given the prior probabili- . ty of the event is p and n observers agree, is

where y is the probability that a witnzss will incorrectly report that the event occurred when it actually did not. Thus if x>y, then nn is monotone increasing with limit 1 as n-a. The sarz phenomenon had earlier been noted qualitative by Campbell (1776, p_ 125), Young (1800, pp. 117-118), and Whately (1846, pp. 62-63). De (1845, p. 472) viewed the matter in a form reminiscent of Fisher’s so-called ‘logical disjunction’ : In cases where a very large number of different events might have been clnssem, the unanimity of many witnesses (or even of two witnesses) entirely destroys all their veracity, and throws the whole difficulty of deciding upon this question, or was there not, collusion?

3.5. Bunyakovsky’s problem proposed an interesting example The Russian mathematician KY. which illustrates some of the above kovsky, 1846). Six distinct letters are selected from the Cyrillic alphabet and placed in a row. If two witnesses independently testify that the resulting string forms the word what is the probability that this is actually the case, assu in the Cyrillic alphabet, and (2) the veracity of each witness is s? Bunyakovsky’s example has a structure additional to aplace’s lottery example because some six letter s nations, and others will form meaningful words in the unce a ‘sense-wor witness lies, then he will such sense-word uming that there are 5 unyakovsky’s example is e calculation as if t osite event that one of the 5

pt2/{ pt2 c (1 -p)( e

t”S

for t

SL. ZabeIl /&w’ysis

346

/((36)(35)(34)(33)(32)(31))= 0356

of testimony

and t = 0.9, to arrive at the answer

yakovsky’s solution neglects to take into account that the two witnesses not only agree that a sense-word has been drawn, but also which word. arkov notes that in this case the appropriate formula is pt2/(pt2+

(1 -p)(l

- t,2g”},

where p = 1/((36)(35)(34)(33)(32)(3I)), t = 0.9 as assumed, and fl is the probability that a witness testifies that the string is the word SKVA, given that this is not the case and he does not correctly testify. (Strictly speaking, this probability will depend on which string is drawn: if it is a nonsense string, then /I= l/S hen jP= l/49999. Since the difference between the two as their common value.) Thus the probabiliarkov takes p = 115 ty is readily computed to be (0.9)2/((0.9)2+ (0.1,2( [(36)(35)(34)(33)(32)(31) - 1]/(5 = (gl)2(6252)/{(81’)(625)2 + (299126.5+) > 0.99. Thus, when the unlikelihood of the coincidence is taken into account, the very small initial probability that the word drawn is OSKVA becomes a substantial posterior probability. arkov, however, notes that the assumption that all sense-words are equally likely to be chosen seems unrealistic, and suggests /p= l/200 as a more reasonable value. Under this hypothesis, the posterior probability that the string actually selected is is approximately N/35141; i.e., it is again highly improbable. he event is sensitive to the choice of the (unspecified) j? kov does not perform the calculation, it is interesting to note that the value of r which the probability becomes 50% is approximately l/l ore generally, one could compile a table of B values for which the PO? robability is 0.1, 0.2, etc.) Bunyakovsky’s problem thus illustrates both the advantages and limitations of the ayesian analysis: qualitatively it identifies the fi parameter as a key quantity of interest, and quantitatively it sp way in which the posterior probability depends on this parameter. however, turns out to be very sensitive to the actual value of the p final conclusion will require further investigation into the value tion, however, is not a weakness: at this stage 0 further conclu

347

the conclusion tfaat it is unimp

that under the appropriate as

Anbm CUI~W~WIJ YVIU

mnffirtrrnatdw

rrfi+ arrmwmi AIVC usr66wuc

ana m:.lmlmt L wfhm w ma t th,,, llrre;uL UC.

n\n&kil;tw in th pw3JWIUL, 1J We assunqtion that the veracities of the witnesses are independent of the event attested. For exa7nnrl-,if the witnesses to a car accident divide evenly on the color of one of the cars, this suggests that the event in question is one for which witness accuracy will be low, whereas unani ous agreement suggests the event in uestion is one where witness accuracy will be high.) UILIVI

(O,-,_

s?Eious

n any application of mathematics to the real worl , the link between mathematical abstraction and practical reality always involves an element of simplification. In the case of the probabilistic analysis of testimony it may well be argued t link is questionable. tly addresses this q hat is the value of such analyses? robabilite des te very beginning of his section on testi in the Essai JV!I The greater part of our judgements being foun ed on the probability of testimonial evidence, it is especially important to submit it to calculation. This, it is true, is often impossible, because of the difficulty of determining the veracity of witnesses, and because of the large number of circumstances attending the facts to which they attest. in several cases one can solve problems which have a sub originally proposed, and whose solutions m guide and to defend us against the errors a would expose us. n approximation of this to the most specious reasonings.

348

S.L. Zabell / Analysis of testimony

afforded, rather than the ability to calculate the specific numerical probabilities in

question.21 This point often seems to have escaped the critics of the theory. Venn, for cxampie, states at the beginning of his chapter on testimony that ‘“thec%raordinaryingenuity and mathematical ability which have been devoted to these problems, considered as questions in Probability, fails to convince me that they ought to have been so considered” (1888, p. 394), and then proceeds to direct the bulk of his fire against the practical computability of an index of trustworthiness for a witness.22 t is only towards the end of the chapter that Venn grudgingly concedes that one might validly use the theory to analyze juror decision-making or the credibility of extraordinary stories. Indeed, Venn provides an interesting example of the utility of the theory, In the 1st edition of the Logic of Chance, Venn subscribes to the curious but ubiquitous fallacy that izi assessing the credibility of a witness attesting to an event in question, one should disregard the prior probability of the event, and pay heed only to the veracity of the witness. (This in the chapter on the credibility of extraordinary stories.) In the 2nd and 3rd editions of the Logic, Venn recants on this issue, and not only adopts the qualitative Laplacean analysis, but found it convenient to appeal to the Condorcet formula in order to illustrate the general principles involved. He states: Reasonswere given in the last chapteragainstthe proprietyof applying the rules of Probabilitywith any strictness to such examples as these. But although all approach to numerical accuracy is unattainable, we do undoubtedly recognize in ordinary life a distinction between the credibilityof one witness and another; such a rough practicaldistinction will 21 Cournot likewise concludes, after discussing the difficulties associated with specifying a witness’s veracity (see Section 1.4 above), “Au surplus, l’impuissance oh nous sommes d’assigner la loi de la variations d’une categoric & l’autre, rend evidemment impracticable toute application numerique” (Cournot 1843, p. 414). Young makes a similar admission but argues, like Laplace: It may perhaps be objected, that if we cannot determine the actual number of chances, alI consideration of the manner of expressing the probability is nugatory. But it is by no means so; because though we cannot determine the exact degree of credit, which we ought to give to each witness, yet we can determine according to what law our belief ought to vary in the case of concurring witnesses . . . (Young 1800, p. 85). It is perhaps worth noting that the bulk of Laplace’s discussion of testimony in the Essaiphilosophiqw (directed towards a lay audience) is devoted to the case of a single witness. Although mathematically trivial, Laplace clearly regarded this as the most instructive one in terms of its implications for the logic of belief. In contrast, little space in the Essai is given to the mathematically more complex (but intuitively simpler) questions associated with concurrent and successive testimony, although their treatment accounts for more than half of the corresponding chapter in the Thc!orieanalytique (which was directed towards a purely mathematical audience). he one exception that Venn was wiIling to recognize involved a racial stereotype: . . . it may be possible m some kind of opinion u sses of witnesses; to say, for instance,

S.L. ZabeD / Analysis of testimony

349

be quite sufficient for the purposes of this chapter. For convenience, a;.ld to illustrate the theory, the examples are best stated in a numerical form, but it is not intended thereby to imply that any such accuracy is really attainable in practice. [

emphasis quoted earlier from Laplace, the two are not incompati strictures might have had some relevance in regards to orcet, it is a psychological curiosity that he overlooked o the limits and utility of the theory.23 ertrand, however, was a much more acute critic; in his criticism of t on judicial decision making, he felt compelled to address the issue and this very passage. ( ertrand clearly regardedLaplace as a far more for poncrt than either Condorcet or Soisson, declaring him to trois, 1~moire imprudent, et incomparable aux deux autres’ question tne aptness of the analogy, but his criticisms are judicial applications of the theory? Terse as he was, Laplace pointed to the key issue; ertrand to its weakest point. The examples discussed above - approximate, oversimplified, unrealistic as they seem - provide several importantinsights into the nature of probabilisticreasoning. It is all too easy to overlook this contribution nowadays, because often such qualitative insights have become so much taken as a matter of course that it is frequently forgotten that many of them were initially many of them remain unobvious. For example, as have documel;ted, failure to consider base rates is a common lay error in assessing probabilities. Anyone who reads through the 18th century literaturecannot help but thsugh

tPaere is $ clear

difference:

in

23 Another early critic was John Stuart Mill. Unfortunately, Mill - no mathematician - never really understood probability theory, and the successive editions of his Logic (1st ed. 1843) suffer from recurrent confusions, misconceptions, and reversals of positiori (cf. Keynes 1921, p. 268, n. 1). The most notorious of these concerned the very nature of probabi!+y: in the first edition Mill rejected the epistemic, Laplacian view of probability for a more narrow, frecluentist one, but backed away from this position in later editions (Book 3, Chapter 18, 01); see Strong (1978). In the first edition of the Logic, Mill claimed that Laplace’s resolution of the Campbell-Price paradox was ‘fallacious’, but in later editions recanted, conceding that is “irrefrageble in the case which he supposes, and in all others which that case fairly represents” (Book 3, Chapter 25, $6). Mill’s fallback position, that this case was not “a perfect representative of all cases of coincidence”‘, had never been denied seen as a rejection by Laplace; and Mill’s conclusion (qluoted earlier in Section 1.3)9 altho formula. of ‘Laplace’s doctrine’, merely summarizes the key feature of the ge 24 For two other discussions of the aptness of the analogy, see Abbott (1864). Such criticisms seem to miss the was intended to dispel the confusion involving t traordinary events may be based on the general formula discussed in Section 1.3. the appropriate value for t*; Laplace’s

350

S.L. Zabell / Analysis of testimony Table 2 Chronological table of the probabilistic analysis of testimony 1689 1699 1709 1748 1764 1767 1785 1794 1800 1800 1812 1814 1837 1843 1845 1846 1847 1851 1857 1864 1866 1889 1901

I-Looper Wooper, Craig N. Bernoulli Wume Lambert Price Condorcet Prevost Prevost and L’Huilier Young Laplace Laplace Poisson, Babbage Cournot, Mill De Morgan Bunyakovsky De Morgan De Morgan Boole Abbott Venn Bertrand Peirce

be grateful for the anchor the subjective theory can provide in understanding the issues, and wonder at the confusion then prevalent. e complex cases, such as the Eddington problem, the subjective theory again an anchor on which to base analysis. Complex issues such as the meaning of witness reliability, independence among witners, relevance of answers, the mechanism of information transmittal, and the spectrum of possible erroneous answers, emerge in the co tions these issues raise, n

XL. Zabeh / Analysis of testimony

351

I have ckcusse

“I have not read all these books myself, but I have read more of them than it would be good for lny one “4 read again. There are here enumerated many memoirs. ‘Lc list is too long, and I have not always succesfully resisted the impulse to add to it in the spirit of a collector. . . . The student will find many famous names here recorded. The subject has preserved its mystery, and has thus notice, profound or, more often, casual, of most speculative minds” - John ynes (1921, pp. 432-433), referring to the literature of probability. Abbott, T.K. (1864). On the probability of testimony and arguments. Philosophical 12-25. Babbage, Charles (1837). The Ninth Bridgewater Treatise. London (2nd ed., 1838). Ball, W.W.R. (1911). Mathematical Recreations and Essays, 5th ed. Macmillan, London. Bernoulli, Nicolas (1709). De usu artis conjectandi in jure. Basel. Reprinted in Die We&e von Jakob Bernoulli, Vol. 3 (B.L. van der Waerden, Ed.), Basel, 1975. Boole, George (1854). An Investigation of the Laws of Thought. Walton and Reprinted 1976, Dover Publications, New York. the combinaBoole, George (1857). Or, the application of the theory of probabilities to the questio tion of testimonies or judgment. Transactions of the Royal Society of Edinburgh Bertrand, J. (1889). Cakul des probabilitek Gauthier-Villars, Paris. Page references in the text are to the 2nd ed. of 1907. Brown, William Byron Jr. and Myles Hollander (1977). Statistics: A Biomedical Introduction. John Wiley and Sons, New York. Bunyakovsky, Viktor Yakovlevich (1846). Osnovania Matematicheskoy Teorii Veroyatnostey. St. Petersburg. R.M. Burns (1981). The Great Debate on Miracles: From Joseph Glanvill to David Hume. University Press, Lewisburg. Campbell, George (1762). A Garland Publishing, Inc. 9 Campbell, George (1976). ~,ti?n in probabilities. aIlis (1936). Eddington’s probability problem. han (1981). Can human irrationality be experi

352

S.L. Zabell / Ana4ysi.sof t&imo;iy

Coumot, Antoine Augustin (1838). M&moire sur les applications du calcul des chances a la statistique judiciaire. Journal de mathematiquespures et appliquees 3, 257-334 Comet, Antoine Augustin (3.843). Exposition de la theorie des chanceset des probabilites. Librairie de L. Hachette, Paris. De Moivre, Abraham (1756). The Doctrine of Chances, 3rd ed. Printed for A. Millar, in the Strand, London. De Morgan, Augustus (1845). Theory of probabilities. Encyclopedia Metropolitana, Vol. 2, pp 393-490. B. Fellowes et al., London. [written 1836-37.1 De Morgan, Augustus (1847). Formal Logic: or, the Calculus of Inference, Necmaty and Probable. Taylor and Walton, London. Reprinted 1926, The Open Court Company, London. De Morgan, Augustus (1851). On the symbols of logic. Section VI: Application of the theory of probabilities to some points connected W;th testimony. Transactions of the Cambridge Philosophical Society 9, 116-125. Diaconis, Persi and David Freedman (1981). The persistence of cognitive illusions. The Behavorialand

Brain Sciences4, 333-334. Eddington, Arthur Stanley (1935). Afew Pathways in Science. Macmillan, New York. Eddington, Arthur Stanley (1936). The problem of A, B, C, and D. The Mathematicai Gazette 19,

256-257. Faber, R. (1976). Discussion: re-encountering a ~~u&i-i&&i~~

p&&ili;y.

r”ilosophy

of Science 43,

283-285. Fairley, William and Frederick Mosteller (1974). A conversation about Collins. ?&ivi&‘fy of Chicago

Law Review 41, 242-253. Fleiss, Joseph L. (1981). Statistical Methods for Rates and Proportions, 2nd ed. John Wiley and Sons, New York. Flew, Anthony (1986). David Hume: Philosopher of Moral Science. Blackwell, Oxford. Garber, Daniel and Sandy Zabell(l979). On the emergence of probability. Archivefor H&tory of Eyqct

Sciences21, 33-53. Gaskin, J.C.A. (1978). Hume’s Philosophy of Religion. Barnes and Noble, New York. Gelfand, Alan E., and Herbert Solomon (1973). A study of Poisson’s models for jury verdicts in crirni&i and civil trials. Journal of the American StattkticalAssociation 68, 271-278. Glass, David V. (1978). Numberingthe People: TheEighteenthCenturyPopulation Controversyand the Development of Census and Vital Statistics in Britain. Gordon and Cremonesi, London. Grier, Brown (1982). George Hooper and the early theory of testimony. Unpublished manuscript. Hacking, Ian (1975). The Emergenceof Probability. Cambridge University Press. Hailperin, Theodore (1976). Boole’s Logic and Probability. Studies in Logic and the Foundations of Mathematics, Vol. 85. North-Holland Publishing Company, Amsterdam. Hooper, George (1689). A Fair and MethodicalDiscussionof the First a& Great Controversy, Between the Church of Englandand Churchof Rome Concerningthe InfallibleGuide. London. Reprinted in Hooper (1757). Hooper, George (1699). A calculation of the credibility of human testimony. PhilosophicalTransactions of the Royal Society 21, 359-365. Reprinted in Hooper (1757). [Griginal publication anonymous.] Hooper, George (1757). The Works of the Right Reverand Father in God, George Hooper, D.D., Late ii._f0iid miversity Pi-ess, :g55. Bishop of Bath and ~~elis.Oxford. Reprinted iu tws voirrrrre~~ Hume, David (1748). An Enquiry ConcerningHuman Understanding.2nd ed., edited by L.A. SelbyBigge, 1902; 3rd ed., revised by P. Nidditch, Oxford University PressI 1975. Page references are to this edition. n Maynard (1921). A Treatiseon Probability. Macmillan, London. illiam (1988).Miracles and statistics: the casual assumption of independence. Journalof the Statistical Association, to appear, Francois (18 16). Trait&~I~rn~~~~ir~du talc

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Lambert, Johann Heinrich (1764). Neues Organon, oder Gedanken fber die Erforschung und &q&hnung des Wahren und dessen Unterscheidung Von Irrtum und Schein. Leipzig. Reprinted in Lambert’s Philosophische Schriften, Vol. 1, Olms, Hildesheim, 1965. Laplace, Piere Simon (1812). Theorieanalytiquedesprobabilit& Courtier, Paris (2nd ed., 1814; 3rd ed., 1820). Page references are to Oeuvres completes de Laplace, Vol. 7, Gauthier-Villars, Paris, 1886. Laplace, Pierre Simon (1814). Essai philosphique sur les probabilit&. Courtier, Paris. Page references are to Oeuvres compl&s de Laplace* Vol. 7, Gauthier-Villars, Paris, 1886. Leftwich, H.S. (1936). The A, B, C, D problem. The , 309-310. Lubbock, J. and J. Drinkwater (1830). On Probability. of Useful Knowledge, London. MacFarlane, Alexander (1879). Principles of the Algebra of Logic. David Douglas, Edinburgh. Markov, A.A. (1912). Wahrseheinlichkeitsrechnung. B.G. Teubner, Leipzig and Berlin. Mill, John Stuart (1843). A System of Logic, 2 ~01s. London. Many later editions. O’Beime, T.H. (1965). puzzles and Paradoxes. Oxford University Press. Reprinted by Dover, New York. Owen, David (19871. Hume v@rsmPrice on miracles and prior probabilities: testimony and the Bayesian calculLtion - 1he Philosophical Quarterly 37, 187-202. Pearson, Karl (1978). The H&tory of Statistics in the 17th and 18th Centuries. Edited by E.S. Pearson. Macmillan, New York. Peirce, Charles Sanders (1981). On the logic of drawing history from ancient documents especially from testimonies. In: Historical Perspectiwes on Peirce’s Logic of Science (Carolyn Eisele, Ed.). Mouton, Berlin, 705-800. Poisson, Sim&on Denis (1837). Recherches sur la probabilitt! des jugements en mat&e criminelle et en mat&e civile, pr&!d& des regles g&n&ales du calcul des probubilit&. Bachelier, Paris. Prevost, Pierre and Simon L’Huilier (1800). MCm_nve 1 snr !‘application du calcul des probabilit&s a la valeur du tdmoignage. Memoires de 1‘Acudemie Royal des Sciences et Relies Lettres ci Berlin, volume for 1797, 120-152. Price, Richard (1767). Four Dissertations. London (2nd ed., 1768; 3rd ed., i772; 4th ed., 1778). Shafer, Glenn (1978). Non-additive probabilities in the work of Bernoulli and Lambert. Arc3riue SOP History of Exact Sciences 19, 309-370. Shafer, Glenn (1985). Conditional probability. International Statistical Review 53, 261-277. Shafer, Glenn (1986). The combination of evidence. International Journal of Intelligent Systems 1, 155 179. Sobel, Jordan Howard (1987). On the evidence of testimony for miracles: a Bayesian interpretation of David Hume’s analysis. The Philosophical Quarterly 37, 166- 186. Stephen, Leslie (1876). History of English Thought in the 18th Century. 3rd ed., 1982. Reprinted 1962, Harcourt, Brace, and World, New York. Page references are to this edition. Stigler, Stephen M. (1968). The History of Statistics: The Measurement of Uncertainly Before 1900. vard University Press, Cambridge, MA. Stigler, Stephen M. (1986a). John Craig and the probability of history: from the death of Christ to the birth of Laplace. Journal of the American Statistical Association lity of causes’. In: P&l I hel, and the ‘prob Strong, John V. (1978). John Stuart Mill, John sociation, East Lansing, osophy of Science Vol. 1 (P.D. Asquity and I. Hacking, Eds.), 3i-41; artin’s Press), London. Swinburne, Richard (1970). The Concept of Miracle. Terrot, Bishop Charles (1857). On the possibility of combining t ability. Transactio event, so as to form on The Thought and Th Thomas, D.O. (1977). hunter, Isaac (18

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TToner,J&n (11843).On the measure of the force of testimony in cases of legal evidence. Transactions of the Cambridge Philosophical Society Tversky, Amos and Kahncman, Daniel (1982). Evidential impact of base rates. In: Judgment Under Uncertainty: Heuristics and Biases (D. Kahneman, P. Slavic, and A. Tversky, Eds.), Cambridge University Press, Cambridge, 153- 160. Venn, John (1866). The Logic of Chance. Macmillan, London (2nd ed., 1876; 3rd ed., i888). Reprinted 1962, Chelsea Publishing Company, New York. Whately, Richard (1826). Elements of Logic. London. Whately, Richard (1828). Elements of Rhetoric. London (7th ed., 1846). Reprinted 1963, Southern Illinois ‘University Press, Carbondale, IL. “Woolhouse, W.S.B. (187”). On two questions in probabilities. Mathematicsfrom the Educational Times 21, 77-79. Young, Matthew (1800). On the force of testimony in establishing facts contrary to analogy. Transactions of the Royal Irish Academy 7, 79- 118.