Probability, anti-resilience, and the weight of expectation

Law, Probability and Risk

Volumn 11, Issue 2-3, 2012, Pages 135-158

  Hamer, David ^a  

^a UNIVERSITY OF SYDNEY   (Australia)

Author keywords

Evidence; Inference; Probability; Resilience; Standard of proof; Weight

Indexed keywords

EID: 84866414503     PISSN: 14708396     EISSN: 1470840X     Source Type: Journal    
DOI: 10.1093/lpr/mgs004     Document Type: Article

References (85)

1. At one point in time the expression 'probabilistic certainty'may have appeared an oxymoron, like 'relatively unique'. Many medieval philosophers viewed probability as the absence of certainty; the two were 'opposed categories' like true and false: Lorraine Daston, Classical Probability in the Enlightenment (Princeton University Press, Princeton, However, Bernoulli and his successors put the two on the same scale. '[S]eventeenth century mathematicians [came] to view the relationship of probability to certainty as that of parts to the whole1988) 37.
2. Laurence H Tribe, 'Trial byMathematics: Precision and Ritual in the Legal Process' (1971) 84 Harvard Law Review 1340-41, 1346-50. Another often discussed hypothetical is the Gatecrasher Case: LJ Cohen, The Probable and the Provable (Clarendon Press, Oxford 1977) 75.
3. Eg Grogan v Garner 498 US 279 (1991).
4. Eg Davies v Taylor AC 207, 219 (Lord Simon).1974
5. But see James Brook 'The Use of Statistical Evidence of Identification in Civil Litigation:Well-worn Hypotheticals, Real Cases, and Controversy' (1985) St Louis University Law Journal 293-299.
6. JM Keynes, A Treatise on Probability (MacMillan and Co, London 1921) 71
7. LJ Cohen, '12 Questions about Keynes's Concept of Weight' (1985) 37 British Journal of Philosophy of Science 263
8. Barbara Davidson and Robert Pargetter, 'Weight' (1986) 49 Philosophical Studies 219
9. Dale Nance, 'The Weights of Evidence' [2008] Episteme 267-268
10. Alex Stein, Foundations of Evidence Law (OUP, Oxford 2005) 47-48.
11. Richard Friedman, 'Assessing Evidence' (1996) 94 Michigan Law Review 1810-1819
12. Hock Lai Ho, A Philosophy of Evidence Law - Justice in the Search for Truth (OUP, Oxford 2008) 166.
13. Eg David Kaye, 'The Paradox of the Gatecrasher and Other Stories' [1979] Ariz State LJ 101.
14. David Hamer, 'The Civil Standard of Proof Uncertainty: Probability, Belief and Justice' (1994) 16 Sydney Law Review 506-516.
15. [2011] UKSC 10 (9 March 2011).
16. As the Court noted, this question has been considered elsewhere, eg Sienkiewicz (n 11) [85] (Lord Phillips), discussing Merrell Dow Pharmaceuticals Inc v Havner 953 SW 2d 706 (1997).
17. In addition to this theoretical concern, the court raised practical concerns about the significance and reliability of the statistics, the availability and reliability of evidence of the plaintiff's situation, and whether the statistics cover the plaintiff's situation: eg [98] (Lord Phillips).
18. Sienkiewicz (n 11) [154], [163] (Lord Rodger); [170] (Lady Hale); [190] (Lord Mance); [205] (Lord Kerr).
19. Eg JJ Thomson, 'Liability and Individualized Evidence' (1986) 49 Lawand Contemporary Problems 199
20. D TWasserman, 'The Morality of Statistical Proof and the Risk of Mistaken Liability' (1991) 13 Cardozo Law Review 935
21. Amit Pundik, 'Statistical Evidence and Individual Litigants: A Reconsideration of Wasserman's Argument from Autonomy' (2008) International Journal of Evidence and Proof 303.
22. Above (n 11) [163] (emphasis added).
23. Above (n 11) [95] (Lord Phillips); [171] (Lady Hale); [217] (Lord Dyson); drawing on earlier discussions in Herskovits v Group Health Cooperative of Puget Sound 664 P 2d 474 491 (Brachtenbach J) and Hotson v East Berkshire Area Health Authority [1987] AC 750, 789 (Lord Mackay).1983
24. Above (n 11) [95]-[96] (Lord Phillips).
25. Ultimately Sienkiewicz was decided in favour of the plaintiffs by the application of a principle from Fairchild v Glenhaven Funeral Services Ltd 1 AC 32 which, in mesothelioma cases, deems contribution to risk to be causal contribution, even where the risk is insufficient to establish causation on the balance of probabilities. The Sienkiewicz court, however, also expressed misgivings about this departure from orthodoxy: Above (n 11) [186] (Lord Brown); see also at [167] (Lady Hale).2003
26. Historically, the complainant's delay did call into question his or her credibility: Eg R v DD [2000] 2 275 [60
27. R v Johnston (1998) 45 NSWLR 362-367
28. The credibility issue can be clearly separated from the weight/forensic disadvantage issue: David Hamer, 'Trying Delays: Forensic Disadvantage in Child Sexual Assault Trials' [2010] Criminal Law Review 671-672.
29. DavidHamer, 'Delayed Complaint, Lost Evidence and Fair Trial: Epistemic and Non-epistemic Concerns' in Paul Roberts and Jill Hunter (eds), Criminal Evidence and Human Rights, Ch. 9, p. 215 (Hart, Oxford, 2012
30. (1989) 168 CLR 79.
31. In many US jurisdictions: eg Brian L Porto, 'New Hampshire's New Statute of Limitations for Child Sexual Assault: Is It Constitutional and Is It Good Public Policy?' (1991) 26 New England Law Review 141.
32. In Ireland: PL v DPP [2004] 4 IR 494.
33. Eg Joynson [2008] EWCA Crim 3049; R v Littler .[2001] NSWCCA 173
34. Longman (1989) 168 CLR 79, 91 (Brennan, Dawson and Toohey JJ). For the recommended English direction see: Judicial Studies Board, Crown Court Bench Book: Directing the Jury (March 2010), 34.
35. 54 NSWLR 241 [35] (Wood CJ at CL); see also at [118] (Sully J). The direction has recently been toned down by legislation (eg Evidence Act 1995 (NSW) s 165B(4)), but remains strongly pro-defendant.(2002)
36. Richard Friedman 'Assessing Evidence' (1996) 94 Michigan Law Review 1810-1819
37. Stein (n 6) 120; see also at 97-100, 228; Ho (n 8) 278.
38. Turner (unreported, EWCA Crim Div, 27 March 2000) [19].
39. Crampton v R (2000) 206 CLR 161 [45] (Gaudron, Gummow and Callinan JJ); [140] (Hayne J).
40. BWT (n 29) [23] (Wood CJ at CL); see also R v Inston (2009) 103 SASR 265 [112] (Vanstone J).
41. Wood CJ at CL recognised that his views were unorthodox: BWT (n 29) [13]; see also Percival EWCA Crim Div, 97/6746/ X4, June 19, 1998. Other possible explanations such as the presumption of innocence and the defendant's right to present a defence are examined and found wanting in Hamer (n 20) and Hamer (n 21).
42. J Kaplan, 'Decision Theory and the Factfinding Process' (1968) 20 Stanford Law Review 1065
43. David Hamer 'Probabilistic Standards of Proof, Their Complements, and the Errors That are Expected to Flow from Them' (2004) 1 University of New England Law Journal 71
44. Marvin E Frankel, 'The Search for Truth: An Umpireal View' (1975) 123 University of Pennsylvania Law Review 1031, 1033, 1055. As I have previously noted (Hamer (n 38) 71 fn 1) the goal of factual accuracy has also been described as 'foremost' (Adrian S Zuckerman, Principles of Criminal Evidence (OUP, Oxford 1989) 7), 'fundamental' (Vern RWalker, 'Preponderance, Probability and Warranted Factfinding' (1996) 62 Brooklyn Law Review 1075, 1081), 'principal' (Jonathan Koehler and Daniel N Shaviro, 'Veridical Verdicts: Increasing Verdict Accuracy through the use of Overtly Probabilistic Evidence and Methods' (1990) 75 Cornell Law Review 247, 250), 'overriding' (William Twining, Theories of Evidence: BenthamandWigmore (1985) 117), 'necessary' (William Twining, 'Rationality and scepticism in judicial proof: some signposts' (1989) II International Journal for the Semiotics of Law 69, 72), 'primary' (ibid 70) and 'central' (Jack Weinstein, 'Some difficulties in devising rules for determining truth in judicial trials' (1966) 66 Columbia Law Review 223-243).
45. Standards of proof may be set at levels of probability tominimise the expected error costs: Kaplan (n 38); Hamer (n 38). Of course, there may be a range of objections to this explicitly probabilistic approach to proof. I put these to one side in order to focus on the weight issue.
46. Eg IJ Good, Good Thinking: The Foundations of Probability and Its Applications (University of Minnesota Press, Minneapolis, 1983) 178
47. Ronald J Allen and Michael S Pardo, 'The Problematic Value of MathematicalModels of Evidence' (2007) 36 Journal of Legal Studies 107-134.
48. Stein (n 6) 90.
49. ET Jaynes, Probability Theory: The Logic of Science (Fragmentary Edition of March 1996) 404 http://thiqaruni.org/mathpdf9/%2886%29.pdf.
50. Allen and Pardo (42), 134.
51. Stein (6) 122-3
52. Michael S Pardo, 'Reference Classes and Legal Evidence' (2007) 11 International Journal of Evidence and Proof 255-255.
53. David Hamer, '"Chance Would Be a Fine Thing"; Proof of Causation and Quantum in an UnpredictableWorld' (1999) 23 Melbourne University Law Review 557.
54. [2005] 2 AC 176 [14].
55. (1990) 169 CLR 638, 642-3 (Deane, Gaudron and McHugh JJ).
56. Above (n 53) [13]
57. See also Richard Wright, 'Causation, Responsibility, Probability, Naked Statistics and Proof: Pruning the Bramble Bush by Clarifying the Concepts', (1988) 73 Iowa Law Review 1001-1041
58. J Dupre, The Disorder of Things: Metaphysical Foundations of the Disunity of Science (Harvard University Press, Cambridge, Mass 1995) 184-5.
59. Above (n 53) [82].
60. The exception grows still larger considering the potential for humans to interact with, and introduce uncertainty into, what would otherwise be deterministic segments of the world: Dupre (n 59) 190.
61. Stein (6) 48
62. James Logue, 'Resiliency, Robustness and Rationality of Probability Judgments' (1997) 11 International Studies in the Philosophy of Science 21.
63. Ho (n 8) 278.
64. Stein (n 6) 88. See alsoNeil Cohen, 'Confidence in Probability: Burdens of Persuasion in aWorld of ImperfectKnowledge' (1985) 60 New York University L Rev 385. See also Barbara Davidson and Robert Pargetter, 'Weight' 49 Philosophical Studies 219, who define 'weight' in terms of the probability assessment's resistance to further change rather than by reference to the quantity of evidence that has been considered.1986
65. Cohen (n 64); Logue (n 62) 91; Stein (n 6) 48 fn 51.
66. With replacement, unless the number is very large.
67. The 95% confidence interval is most commonly used. Briefly, this can be roughly calculated as p±2 The symbol represents the standard deviation, which can be estimated by?(pq/n), where n is the number of samples, p is the frequency of 'successes' (eg red balls) and q the frequency of 'failures' (eg non-red balls): W Mendenhall, Introduction to Probability and Statistics (Duxbury Press, Mass 1979) 239-40
68. This formula is based upon themathematics of the normal function which can be taken as an approximation of the binomial probability distribution. The normal approximation is better for larger sample sizes, however, Mendenhall suggests that where the frequency is centred, as it is in the present case with p1/40.5, the normal approximation of a binomial distribution of only 10 samples is still 'reasonably good': at 203.
69. See also David Kaye, 'Apples and Oranges: Confidence Coefficients and the Burden of Persuasion' (1987) 73 Cornell Law Review 54
70. Hamer (n 10) 518-20.
71. See also LJ Cohen, An Introduction to the Philosophy of Induction and Probability (Clarendon Press, Oxford 1989) 99-108.
72. Good (n 41) 178; see also B Skyrms, Choice and Chance (Belmont: Wadsworth, 4th ed, 2000) 155.
73. The relationship between probabilistic certainty and the risk of error ismediated by the standard of proof. If the standard of proof is 50%, the relationship is direct-the apparent risk of error is the complement of the degree of certainty. With a higher standard of proof, the relationship is less straightforward. Eg, in a criminal case, a fact-finder's certainty may increase from50 to 90%while still not eliminating a reasonable doubt. Certainty has increased, but this has only brought an increase in the apparent risk of amistaken acquittal. But while the asymmetric standard of proof is a complicating factor, it remains true that, at the limits, the risk of error is reduced by increasing certainty. Note also that fresh evidence offers the expectation of making a decision with greater utility: see (n 73).
74. At the end of Section 4, and see proof in Appendix A.
75. Good's analysis (n 41) focused on the expected utility of a decision based on a probability assessment. Expected certainty behaves in essentially the same way as expected utility-both are V-shaped linear functions of probability. Certainty has the advantage of being a purely epistemic measure. '[W]eight.. is undoubtedly an epistemic concept, and attempts to account for it or measure it in terms of non-epistemic criteria must be suspect': James Logue, Projective Probability (OUP, Oxford 1995) 84.
76. This model was implemented on an Excel spreadsheet with programming in Visual Basic. My thanks to Joshua Klose for the programming.
77. Because the numbers in the table were generated randomly, the ' and "marks do not signify whether the evidence is favourable or unfavourable to the conclusion, G.
78. Cell entries in the original table were randomly selected numbers between 0 and 100. Because each number has an equal chance of being selected, as the size of the table increases, the prior probability (the ratio of the sum of half the table divided by the sum of the whole table) tends towards 1/2 as table size increases. (This is related to the anti-resilience effect discussed in Section 10.) The cell entries in this notional frequency table were again randomly generated but, to generate a higher prior probability, the G entries were skewed up and the NG entries skewed down.
79. At least this is true on the probabilistic unit interval. The odds and log-odds measures are a different matter. There is insufficient space here to explore the implications of the different measures.
80. Above (nn 45-46).
81. Eg, Hock Lai Ho, 'The Presumption of Innocence as a Human Right', In Roberts and Hunter, Ch 11, p. 259 (201
82. Significance should not be attached to the fact that the police have laid charges and the prosecution service has brought the matter to trial. Whatever incriminating evidence influenced them, if admissible, can be presented to the court in due course: Bell v Wolfish 441 US 520 (1979), 533.
83. Just as crime rates are relatively low in probabilistic terms. Eg among serious crimes, sexual assault has the one of the highest crime rates. The victimization rate for 2010 for Australia was 79.5 per 100 000 (as against 1 in 100 000 for murder): Australian Bureau of Statistics, 4510.0 Recorded Crime - Victims (Commonwealth of Australia, Canberra 2001) 9. This relatively high crime rate is less than 1/10 of 1%.
84. Above (nn 62-64).
85. The standard deviation provides a measure of the variation of sample data around a mean. The probability calculation for the frequency table is not a simple calculation of amean, but instead involves the ratio of means. The probability-the sumof the first column divided by the sumof both columns-is equal to themean of the first column divided by two times the mean of both columns. The standard deviation of functions of means such as this, like the standard deviation of a simple mean, is inversely proportional to the square root of the sample size. See eg, US Commerce Department, NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook