Saturday, July 2, 2016

State Supreme Court Approves of Variable-Prior-Odds Presentation of the Bayesian Probability of Paternity

In 2002, the Supreme Court of Kentucky wrote a disappointing opinion on the admissibility of the "probability of paternity" in a case of rape and sexual abuse of a minor. In 2016, the court corrected its mistakes in a similar case. The more recent opinion is a welcome addition to the jurisprudence on probability evidence in criminal cases.


In the first case, Butcher v. Commonwealth, 96 S.W.3d 3 (Ky. 2002), Larry Butcher lived with a woman and her young daughter in Johnson County. After the woman had twin girls fathered by Larry, he "began a pattern of sexual abuse with H.B., who was then ten years old. . . . H.B. became pregnant in April 1987, at age fourteen. H.B. gave birth to a baby girl on January 19, 1988." Larry was convicted of eleven counts of first-degree rape and other offenses.

The evidence against him included "a DNA paternity test indicating a 99.74 percent likelihood that [he] was the father of H.B.'s child." The court explained that a "Mr. DeGuglielmo testified that tests performed on the eight genetic markers previously discussed yielded a paternity index of 388/1, meaning Appellant was 388 times more likely to be the father of the child than a randomly selected male of the same race."

The meaning the court ascribed to the paternity index is clearly wrong. The paternity index of 388 meant that the genotypes were 388 times more probable if Larry were the father than if a randomly selected man was. The probability that Larry was the father instead of a different man is something else. To mechanically equate the two is to transpose conditional probabilities. As is now widely recognized, the "probability of paternity" is a posterior probability that depends on both the likelihood ratio and the prior probability of paternity. If and only if the prior probability is one-half are the prior and the posteriors exactly equal.

Having initially conflated the paternity index with the posterior probability, the Butcher court went on to explain that the "final part of the paternity test translates the paternity index into a percentage that is more understandable . . . using Bayes' Theorem, a formula that takes into account actual events and circumstances, as opposed to random sequences of events." It thought that "[i]n paternity tests generally, this formula combines the paternity index and another value representing the prior probability that an event occurred, including such factors as access to the mother, fertility, and date of conception."

Of course, paternity test reports in both civil and criminal cases do not generally consider these factors. Rather, they "typically insert a standard prior probability of .5 regardless of any other factors, which indicates a fifty percent chance that the alleged father actually had sexual intercourse with the mother." (This is not quite right either, since not all intercourse leads to children.)

In any event, Butcher concluded that "there was a 99.74 percent likelihood that Appellant was the father of H.B.'s child. Although Mr. DeGuglielmo did not testify that he had used a prior probability value of .5 in reaching his expert conclusions, the mathematical results of the test suggest that he did."

In response to a garbled argument from Butcher that the use of a prior probability of one-half offended the presumption of innocence in a criminal case, the court relied on a 1998 Texas Court of Appeals opinion’s strange conclusion that "[t]he use of a prior probability of .5 is a neutral assumption. The statistic merely reflects the application of a scientifically accepted mathematical theorem which in turn is an expression of the expert's opinion testimony."

This is strange because any value from 0 to 1 for the prior probability would be part of "the application of a scientifically accepted mathematical theorem"—namely, Bayes’ rule. The expert supposedly has no opinion about the prior probability and is using the midpoint of 0.5 to express ignorance about what the value actually is. Yet, to assign a probability of 0.5 to the defendant is to credit the fact that he—as opposed to all other men in the population—has been charged with a crime. Letting an accusation of criminality create a substantial probability of guilt does seem to offend the presumption of innocence.


This year’s case, Ivey v. Commonwealth, No. 2014-SC-000345-MR, 2016 WL 671153 (Ky. Feb. 18, 2016), is a major leap forward. Alfred Ivey, Jr., lived in Hardin County with his half-sister and her daughter from a previous relationship. The daughter became pregnant at the age of 13. Seven years later, she revealed that Ivey was the father and had repeatedly raped her before then. A paternity test confirmed that Ivey was the father, and a jury convicted him of two rapes of a minor.

Ivey objected that the probability of paternity—a whopping 99.9999%—was inadmissible under Daubert v. Merrell Dow Pharmaceuticals, Inc., 509 U.S. 579 (1993). At a pretrial hearing on his motion to exclude,
Ivey proffered an academic article, Ernest P. Chiodo et al., An Error in Statistical Logic in the Application of Genetic Paternity Testing, J. Mod. Applied Stat. Methods, Winter 2002, Vol. 1, Issue 1, at 126, but did not offer any additional proof, such as expert testimony. His argument at that hearing was based solely on the article, which criticizes the use of the 50% prior probability to calculate the probability of paternity in some circumstances because it "highlights a serious error in statistical methodology known as the ‘principle of indifference.’"
The trial court admitted testimony about the parentage test anyway, and the state supreme court revisited its opinion in Butcher. This time around, it recognized that
The paternity index "does not measure the chance of defendant's paternity compared to that of a randomly selected man." D.H. Kaye, Plemel as a Primer on Proving Paternity, 24 Willamette L.Rev. 867, 877 (1988). . . . Technically speaking, the paternity index is the ratio of "the probability of the alleged father transmitting the alleles and the probability of selecting these alleles at random from the gene pool." D.H. Kaye, The Probability of an Ultimate Issue: The Strange Cases of Paternity Testing, 75 Iowa L.Rev. 75, 89 (1989–1990). It has been described as a "likelihood ratio," that is, "a ratio of two conditional probabilities," but "is not itself a probability." 1 McCormick on Evidence § 211 n. 17 (7th ed. 2013) "Unlike probabilities, which are bounded by 0 and 1, the likelihood ratio can exceed one. The larger the ratio, the more probative the evidence."
As such, the court acknowledged that its understanding of the paternity index in Butcher was "not, strictly speaking, completely accurate." Likewise, Ivey is more careful in describing the nature of a prior probability.

Nevertheless, the court seemed uncertain whether "the use of the 50% prior probability is a logical error." It wrote that "the ‘error,’ to the extent it is one, appears to matter only under unrealistic conditions." It noted that the likelihood ratio of 11,900,000 in the case swamped any plausible prior probability:
[I]n this case, even with very low assumed prior probabilities, the probability of paternity exceeded 99.99%. The Commonwealth's expert explained generally to the jury how the prior-probability notion affected the probability of paternity, albeit without getting into the details of Bayes' theorem, and postulated several variations less than the assumed standard 50%. For example, the expert explained that if the prior probability were only 1%, or .01, then the probability of paternity dropped only to 99.9991%. The extreme example of a low prior probability, .1%, or .001, used in the Chiodo article also results in very high probability of paternity for Ivey. Though the expert did not testify to this number, it can easily be calculated with the formula laid out elsewhere in this opinion, with the resulting probability of paternity being 99.9916%.
After seemingly sidestepping the question of whether the 0.5 prior probability is "a logical error," however, the court agreed with the commentary correcting the claims of "neutrality" for this number: "Those critical of using a 50% prior probability have a point. [T]hat prior probability is fairly high (almost satisfying the civil burden of proof by itself). Despite our assertions in the past, it is not a truly neutral number. It, at best, gives us a ‘useful working hypothesis.’" That said, "a 50% prior probability is far from neutral."

Yet, the court was unfazed by the fact that the prosecution’s expert testified that "she tried to be neutral, though ‘ideally’ the probability of paternity would take into account all the other evidence. . . . She testified that 50% was the only neutral number that would not weigh the results one way or the other."

What, then, can be done to enable a laboratory to explain the import of its genetic findings? Not being in a position to evaluate the nongenetic evidence in a case, the laboratory has no basis for estimating a case-specific prior probability. Neither does it have a usable statistic for the prevalence of paternity on the part of defendants in criminal conduct that results in pregnancies. Consequently, it cannot apply Bayes’ rule to give a single posterior probability to the jury.

One solution would be to forget Bayes’ rule—to eschew talk of the posterior of paternity and restrict testimony to the likelihood ratio and its components. Citing publications of the Royal Statistical Society, 1/ a concurring justice urged this approach. The paternity tester might testify along these lines: X% of all children would be expected to inherit the genetic characteristics that the child has if the defendant (or an identical twin) were the father. Only Y% would be expected to have these characteristics if an unrelated man were the father. Only Z% of children with (an unknown) brother to the defendant would be expected to inherit the child’s characteristics. Thus, the DNA types in this case strongly support the conclusion that the defendant is the father. (Obviously, this conclusion only follows if X is much greater than Y and Z, as was the case in Ivey.)

Ivey considered, but did not settle on another approach—sometimes called "variable prior odds" 2/ —which involves presenting jurors with a spectrum of prior and posterior probabilities to show them how the posterior probability changes as a function of the prior probability.

The Ivey court observed that
Though this proposed approach has drawn some criticism, see Meyerson & Meyerson, supra 3/, at 789 (stating that “[t]his proposal, though well-meaning, is hopelessly misguided”), it is not without its charm. It avoids the problem of the expert using a 50% prior probability without disclosing what that figure means, and allows the jury to use “Bayes's rule merely ... as a heuristic device, displaying the force of the evidence across a wide range of prior probabilities.” Kaye, DNA Evidence, supra, at 167. And this approach has been used in some courts, especially before DNA testing became as refined as it is today. See State v. Spann, 130 N.J. 484, 617 A.2d 247, 264–65 (1993); Plemel v. Walter, 303 Or. 262, 735 P.2d 1209, 1219 (1987).
But the court was not sufficiently charmed to insist on the use of variable prior odds:
We need not decide today whether such an approach is required. We note it only as one possible solution to the problem of the 50% prior probability. Whether presenting a spectrum is definitively a better method than simply assuming a 50% prior probability is a question better left, in the first instance, to a trial court presented with evidence to evaluate whether it better fits under Daubert and KRE 702. And it is likely that such an examination will be unnecessary, as even in Butcher, we noted that the use of a 50% prior probability could be disregarded by the jury and that it could “be weakened on cross and in argument.” 96 S.W.3d at 8 . . . . And defendants are free to offer their own expert witness critical of the 50% prior probability, as happened in this case.
Inviting a debate before a jury over Laplace's principle of insufficient reason and assertions of "neutrality," however, is less than ideal. Experts should avoid these dubious rationales in favor of
what was used in this case. [T]he Commonwealth's expert testified as to the effect of several lesser prior probabilities, including 1%, on the resulting probability of paternity. In each instance, the probability of paternity remained above 99.99%. Thus, the jury could see the effect of different prior probabilities on the resulting probability of paternity. ... [¶] As Professor Kaye noted in 1989, as genetic testing becomes "more refined ... the controversy over [probability of paternity] will wither." . . . This prophesy has not completely born out, as questions are still being raised more than 25 years later. But as testing improves, we are convinced those questions are becoming of little consequence.
  1. Colin Aitken, Paul Roberts, & Graham Jackson, Communicating and Interpreting Statistical Evidence in the Administration of Criminal Justice: Practitioner Guide No. 1, Fundamentals of Probability and Statistical Evidence in Criminal Proceedings 35–36 (Royal Statistical Society 2010); Roberto Puch–Solis, Paul Robert, Susan Pope & Colin Aitken, Practitioner Guide No. 2, Assessing the Probative Value of DNA Evidence (2012); Graham Jackson, Colin Aitken, & Paul Roberts, Communicating and Interpreting Statistical Evidence in the Administration of Criminal Justice: Practitioner Guide No. 4, Case Assessment and Interpretation of Expert Evidence 37 (2015).
  2. David H. Kaye, David E. Bernstein & Jennifer L. Mnookin. The New Wigmore: A Treatise on Evidence: Expert Evidence §14.3.2(c) (2d ed. 2011).
  3. Michael I. Meyerson & William Meyerson, Significant Statistics: The Unwitting Policy Making of Mathematically Ignorant Judges, 37 Pepp. L.Rev. 771 (2010).

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