Source: https://www.llrx.com/2013/12/calculating-justice-mathematics-and-criminal-law/
Timestamp: 2019-04-19 16:43:56+00:00

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So it is that the relationship of law and math is a mile wide and a mile deep as is the bibliography of these disciplines. Thus, this article is only an attempt to collect recent, historical and otherwise notable materials regarding the role and risks of mathematics in criminal cases.
“The principal issue in the appellant’s trial was whether he was one of the perpetrators of the robbery. Detective Simmons’ testimony, attempting to establish that a large percentage of those arrested by him for robbery were ultimately proven guilty, undertook to collaterally establish the detective’s investigative successes, but had no probative value in tending to establish the proposition in issue — the identity of the appellant as one of the robbers — and was thus patently irrelevant. . . .
Permitting the detective to relate syllogistically — though imperfectly — before the jury, the high probability of the appellant’s guilt, tended to portray the officer as a ‘super-investigator’ and thus clothed his testimony, with a greater weight than that which might have been given to the testimony of the other witnesses. Thus, the jury’s basic function of weighing the conflicting evidence in arriving at a conclusion of guilt ‘beyond a reasonable doubt,’ was subjected to the counterbalancing effect of the detective’s irrelevant and extraneous opinion. Indeed in the absence of any showing of similarity between the investigation which led to the appellant’s arrest and those other investigations which led to the detective’s conviction rate, the premise posited before the jury appears to have been invalid. . . .
“Appellant claims error with the Commonwealth’s reference during closing argument to statistical probabilities concerning the DNA match between appellant and the saliva extracted from the cigarettes found in Larsen’s apartment. . . .
“Dr. Matthews had made no tests on which he could reasonably base his probabilities of one in ten on soil color, one in one hundred on soil texture, or one in one thousand on soil density (which he multiplied together to obtain his one-in-one-million figure), nor did he base his testimony on studies of such tests made by others. He admitted that his figures were predicated on ‘estimates’ and ‘assumptions’. In short, there is no foundation upon which to base his probabilities of one in a million.
‘An expert witness’ view as to probabilities is often helpful in the determination of questions involving matters of science or technical or skilled knowledge. * * * It is necessary, however, that the facts upon which the expert bases his opinion or conclusion permit reasonably accurate conclusions as distinguished from mere guess or conjecture. * * * To admit expert testimony deduced from a scientific principle or discovery, the thing from which the deduction is to be made must be sufficiently established to have gained general acceptance in the particular field in which it belongs.’ 20 Am.Jur, Evidence, S 795, p. 668.
“Our concern is more with the impact of the probability statistic upon the jury than the foundation for the testimony. The statistic was patently irrelevant. Probability theory is necessarily silent on the crucial question before the jury: Of the relatively small percentage of the population with consistent characteristics, which one, if any, committed the crime? (People v. Collins (1968), 68 Cal.2d 319, 330, 438 P.2d 33, 40, 66 Cal. Rptr. 497, 504.) Jurors would be hard pressed to explain how the 1-in-500 chance of an accidental match did not equate with a 1-in-500 chance that defendant was innocent. (See Tribe, Trial by Mathematics: Precision and Ritual in the Legal Process, 84 Harv. L. Rev. 1329, 1355 (1971).) Of course, the statistic means nothing of the sort. Absent a sound basis to limit the number of possible defendants, the defendant here is but one of thousands of people who share these same characteristics. Legion possibilities incapable of quantification, such as the potential for human error or fabrication, or the possibility of a frame-up, must be excluded from the probability calculation. . . .
“Our concern over this evidence is not with the adequacy of its foundation, but rather with its potentially exaggerated impact on the trier of fact. Testimony expressing opinions or conclusions in terms of statistical probabilities can make the uncertain seem all but proven, and suggest, by quantification, satisfaction of the requirement that guilt be established ‘beyond a reasonable doubt.’ See, Tribe, Trial by Mathematics, 84 Harv.L.Rev. 1329. Diligent cross-examination may in some cases minimize statistical manipulation and confine the scope of probability testimony. We are not convinced, however, that such rebuttal would dispel the psychological impact of the suggestion of mathematical precision, and we share the concern for ‘the substantial unfairness to a defendant which may result from ill conceived techniques with which the trier of fact is not technically equipped to cope.’ People v. Collins, 68 Cal.2d 332, 66 Cal.Rptr. 505, 438 P.2d 41.
“Hernandez has one further complaint about the expert. He claims that the expert ran afoul of Jensen when she was allowed to opine that only one percent of the children claiming sexual assault fabricate the account. We reject this argument because of the doctrine of invited response. See State v. Wolff, 171 Wis. 2d 161, 168, 491 N.W.2d 498, 501 (Ct. App. 1992). It was Hernandez himself who brought up the question during cross-examination of the expert, asking whether the expert was aware of fabricated stories concocted by children in the past. The expert answered, “I had a case where there was fabrication, Yes.” On redirect examination, the prosecutor asked the expert whether she was aware of any percentages about how many cases are fabricated. Upon objection, the prosecutor explained that the door had been opened. The trial court overruled the objection and allowed the expert to answer. The reason why this is an invited response is because Hernandez elicited an admission that children may fabricate. The prosecutor was entitled to rehabilitate the expert to explain that fabrication was not common. There was no error.
“At best, we conclude the court’s colloquy with the witness concerning mathematical probabilities was speculative and confusing. The gravamen of the error came during the trial judge’s comment construing the testimony in terms of the probability of making an erroneous identification through the comparison of hair samples. . . .
“In this book you will find the basics of probability theory and statistics. In addition, there are several topics that go somewhat beyond the basics but that ought to be present in an introductory course: simulation, the Poisson process, the law of large numbers, and the central limit theorem. Computers have brought many changes in statistics. In particular, the bootstrap has earned its place. It provides the possibility to derive confidence intervals and perform tests of hypotheses where traditional (normal approximation or large sample) methods are inappropriate. It is a modern useful tool one should learn about, we believe.
“In a recent article in this Review, I [Laurence H. Tribe] undertook to assess the usefulness, limitations, and possible dangers of employing mathematical methods in the legal process, both in the conduct of individual trials, and in the design of procedures for the trial system as a whole. Michael Finkelstein and William Fairley, addressing themselves exclusively to that part of my discussion of the use of mathematical methods in the conduct of trials which criticized their earlier work, reply to several of my criticisms by suggesting that their intentions were far more modest than the methods of mathematical proof I examined. Indeed, if the technique they advocated were as carefully confined as they had evidently intended, some of the problems I discussed would not arise.
“Everyone knows that lawyers are bad at math. Many fields of law, though — from explicitly number-focused practices like tax law and bankruptcy, to the less obviously numerical fields of family law and criminal defense — require interaction with, and sophisticated understandings of, numbers. To the extent that lawyers really are bad at math, why is that case? And what, if anything, should be done about it?
This is the archive of National Public Radio broadcasts by Dr. Keith Devlin, Professor of Mathematics at Stanford University.
1 See Probability and Statistics, Encyclopedia Britannica (last viewed Nov. 26, 2013)(“probability and statistics, the branches of mathematics concerned with the laws governing random events, including the collection, analysis, interpretation, and display of numerical data. Probability has its origin in the study of gambling and insurance in the 17th century, and it is now an indispensable tool of both social and natural sciences. Statistics may be said to have its origin in census counts taken thousands of years ago; as a distinct scientific discipline, however, it was developed in the early 19th century as the study of populations, economies, and moral actions and later in that century as the mathematical tool for analyzing such numbers.”); Bayes’ Theorem, Stanford Encyclopedia of Philosophy, June 28, 2003 (“Bayes’ Theorem is a simple mathematical formula used for calculating conditional probabilities. It figures prominently in subjectivist or Bayesian approaches to epistemology, statistics, and inductive logic.”).
2 See generally McCormick on Evidence (7th ed 2013)(sects. 208 Surveys and Opinions Polls; 209 Correlations and Causes: Statistical Evidence of Discrimination; 210 Identification Evidence, Generally; 211 Paternity Testing); Reference Manual on Scientific Evidence (3rd ed. 2011); Admissibility, in Criminal Case, of Statistical or Mathematical Evidence Offered for Purpose of Showing Probabilities, 36 A.L.R.3d 1194.
3 See, e.g., Dmitri Krioukov, The Proof of Innocence, Annals of Improbable Research, v.18, n.4, p.12 (2012)(“We show that if a car stops at a stop sign, an observer, e.g., a police officer, located at a certain distance perpendicular to the car trajectory, must have an illusion that the car does not stop, if the following three conditions are satisfied: (1) the observer measures not the linear but angular speed of the car; (2) the car decelerates and subsequently accelerates relatively fast; and (3) there is a short-time obstruction of the observer’s view of the car by an external object, e.g., another car, at the moment when both cars are near the stop sign.”); Yves van Gennip et al., Community Detection Using Spectral Clustering on Sparse Geosocial Data, SIAM J. Appl. Math., 73(1), 67 (2013)(“In this article we identify social communities among gang members in the Hollenbeck policing district in Los Angeles, based on sparse observations of a combination of social interactions and geographic locations of the individuals. This information, coming from Los Angeles Police Department (LAPD) Field Interview cards, is used to construct a similarity graph for the individuals. We use spectral clustering to identify clusters in the graph, corresponding to communities in Hollenbeck, and compare these with the LAPD’s knowledge of the individuals’ gang membership. We discuss different ways of encoding the geosocial information using a graph structure and the influence on the resulting clusterings. Finally we analyze the robustness of this technique with respect to noisy and incomplete data, thereby providing suggestions about the relative importance of quantity versus quality of collected data.”).
4 See, e.g., Commonwealth v. Ferreira, 460 Mass. 781, 787-788 (Mass. Sup. Jud. Ct. 2011)(“The prosecutor also erred in equating proof beyond a reasonable doubt with a numerical percentage of the probability of guilt, in this case, ninety-eight per cent. “[T]o attempt to quantify proof beyond a reasonable doubt changes the nature of the legal concept of `beyond a reasonable doubt,’ which seeks `abiding conviction’ or `moral certainty’ rather than statistical probability.” Commonwealth v. Rosa, 422 Mass. 18, 28 (1996). “The idea of reasonable doubt is not susceptible to quantification; it is inherently qualitative.” Commonwealth v. Sullivan, 20 Mass. App. Ct. 802, 806 (1985). See Commonwealth v. Mack, 423 Mass. 288, 291 (1996) (“the concept of reasonable doubt is not a mathematical one”).”). See generally Ken Strutin, Forensic Due Process: Lawyering With Science, N.Y.L.J., March 20, 2012, at 5 (“Summation was no substitute for expert testimony that could be tested by the defense and scrutinized by the court. The use of unvetted statistical statements inflated the credibility of the victim’s identification, reducing proof beyond a reasonable doubt to mathematical bolstering.” (footnote omitted)); D. Kim Rossmo, Criminal Investigative Failures: Avoiding the Pitfalls (Part Two), FBI L. Enforcement Bull., Oct. 2006, at 12 (“Part two presents probability errors and organizational traps that can lead investigations astray. It also offers recommendations and additional strategies that investigators may find helpful.”).
5 See, e.g., Jackson v. Pollion, 2013 U.S. App. LEXIS 21983 (7th Cir. Ill. Oct. 28, 2013)(“The discomfort of the legal profession, including the judiciary, with science and technology is not a new phenomenon. Innumerable are the lawyers who explain that they picked law over a technical field because they have a “math block”—”law students as a group, seem peculiarly averse to math and science.” David L. Faigman, et al., Modern Scientific Evidence: Standards, Statistics, and Research Methods v (2008 student ed.). But it’s increasingly concerning, because of the extraordinary rate of scientific and other technological advances that figure increasingly in litigation.”); Angela Saini, A Formula for Justice, The Guardian, Oct. 2, 2011(“Bayes’ theorem is a mathematical equation used in court cases to analyse statistical evidence. But a judge has ruled it can no longer be used. Will it result in more miscarriages of justice?”). Harold E. Potts, Letter to the Editor: An Application of Mathematics to Law, Nature, Vol. 91, Issue 2269, pp. 187-189 (15 May 1913)(Responding to criticism about the author’s article concerning the role of math in the adjudication of patent claims.).
6 See David McCord, Primer for the Nonmathematically Inclined on Mathematical Evidence in Criminal Cases: People v. Collins and Beyond, 47Wash. & Lee L. Rev. 741 (1990) (“Statistics are clearly “facts” and thus there is no difficulty in characterizing them as “evidence.” Probabilities are more of a way of thinking about the significance of evidence and thus more easily characterizable as “arguments” than as “evidence.” Nonetheless, most courts and commentators have characterized probabilities as “evidence” and thus we will consider probabilities as “evidence” for purposes of this Article as well.” Id. at 758 n. 59.).
7 Id. at 758 (“It consists of use of statistics based on empirical sampled data to eliminate possible culprits, while simultaneously not eliminating the defendant as the culprit. The key characteristic of this category is that it involves no probability calculation (although it provides data upon which the second category probabilities of a random match-can readily be based).”).
8 Id. (“Any empirical statistic can be used to form such a probability.”).
9 Id. at 759 (“In this category, empirical statistics form part of the basis for formation of a subjective probability, but not via use of Bayes’ Theorem (e.g., “Based upon my assessment of the probability of truth of the eyewitness testimony plus the statistics regarding blood factors, I believe that the defendant’s guilt is extremely probable.”)”).
10 Id. (“This category is composed of instances where the prosecutor, in line with the mathematical probabilist tenet that all evidence is inherently probabilistic, probabilizes non-empirically sampled data, and then further relies on the probabilist position that the burden of persuasion is probabilistic by arguing that the subjective probability of guilt based upon the non-empirically sampled data is sufficient to convict, all without using Bayes’ Theorem.”).
11 Id. (“[E.]g., “Based upon my evaluation of the probability of truth of the eyewitness testimony, as combined with the blood factor evidence’s probability via Bayes’ Theorem, I believe that defendant’s guilt is extremely probable.””).
12 See Leila Schneps and Coralie Colmez, Math on Trial: How Numbers Get Used and Abused in the Courtroom (Basic Books 2013).
13 Leila Schneps and Coralie Colmez, Justice Flunks Math, N.Y. Times, Mar. 27, 2013, at A23.

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