Likelihoods, Paternity Probabilities, and the Presumption of Innocence in People v. Gonis

In People v. Gonis, 1/ Illinois prosecuted Kenneth Gonis for sexual penetration with his daughter, T.G., when she was 16 years old. T.G. had two children. The first, J.G., was born when she was 17 years old; A.G. was born two years later. To investigate the sexual assault charge, the Illinois State Police Joliet laboratory conducted DNA tests of Gonis, T.G, and the two children. The lab sent the results to the Northeastern Illinois Regional Crime Laboratory for interpretation. That laboratory’s DNA technical leader, Kenneth Pfoser, “entered the DNA profiles into a computer containing a statistical calculator.” He learned that

• “at least 99.9999% of the North American Caucasian/White men would be excluded as being the biological father of [J.G. and A.G.]”;
• the “paternity index” with respect to J.G. was about 195,000,000 and with respect to A.G., it was 26,000,000; and
• “the probability that defendant was the biological father of J.G. and A.G. was 99.9999%.”

In a bench trial before Judge Lance Peterson, the court admitted these findings and convicted Gonis. On appeal, Gonis argued that the trial court erred in denying a pretrial motion to exclude the DNA test results. In an opinion written by Justice Daniel Schmidt, Illinois’ intermediate court of appeals described the motion as asserting only that

[T]he tests were inconsistent with the presumption of innocence because a statistical formula used in the testing assumed a prior probability of paternity. Specifically, the motion alleged:

Assuming that the Northeastern Illinois Regional Crime Laboratory tested the DNA sample using widely accepted practices in the scientific community, said testing was conducted using a statistical mathematical formula. These formulae, as their basis, include a component to determine paternity which by its nature ‘assumes’ that sexual intercourse has in fact taken place.

In other words,

The motion alleged that to allow such paternity test results would violate the presumption of innocence because “the state would be allowed to introduce statistical evidence presuming sexual intercourse, in order to prove an act of sexual intercourse.”

The argument is fallacious for three reasons. First, the probability pertains to the chance that the child was conceived by the mother and the accused man. Conception—the fertilization of an ovum—can occur without penetration.  In the hearing on the motion to exclude, the technical leader referred to artificial insemination, but as insemination and hence pregnancy can occur without penetration by natural mechanisms as well.

Second, even if conception were not merely improbable, but impossible without penetration, it would not follow that a probability of paternity presumes penetration. After all, a probability is not a certainty. To say that an electron has a probability Ψ*ΨdV of being located in a small volume dV is not to presume that the electron is actually located there. To say that the probability of an extended trade war between the U.S. and China is 0.5 (or some other number less than 1) does not presume that this event will occur. That the paternity probability for the defendant is 0.5 (or 0.99999, or any other other number less than 1) also does not presume that the defendant truly is the source of the fertilizing spermatazoon.

Finally, the evidentiary aspect of the presumption of innocence merely directs the judge or jury not to use the fact of the indictment as evidence of guilt. The probabilities in question do not change depending on whether or not a man is indicted.

The opinion in Gonis seems to rely on the activity-level possibility of artificial insemination to reject the defendant's presumption-of-innocence objection. It also comes close to recognizing the second rejoinder, for it states that "Logically, since Bayes's Theorem allowed for the possibility that defendant may not be the father of T.G.'s children, it did not assume that defendant necessarily had sexual intercourse with T.G."

But the court thought that the details of Bayes' Theorem rather than the very definition of probability made the computation compatible with the presumption of innocence. The opinion states that

Pfoser testified that Bayes's Theorem was a likelihood ratio based on two competing hypotheses: (1) defendant was the father, or (2) a random, unrelated individual was the father. Pfoser stated that Bayes's Theorem took “the assumed probability that the person in question is the father of the child” and divided it “by the probability that some unrelated person within the same race group in the general population is the father of the child.” Thus, Pfoser's testimony indicated that Bayes's Theorem posited that either defendant or an individual other than defendant could have been the father of T.G.'s children. Logically, since Bayes's Theorem allowed for the possibility that defendant may not be the father of T.G.'s children, it did not assume that defendant necessarily had sexual intercourse with T.G.

Although a likelihood ratio appears in Bayes' rule, that is not all there is to it, and the description of how the rule works is garbled. The probability that the defendant is the father is not obtained by dividing a probability that he is the father by the probability that an unrelated man is the father. If the expert knew the probability that an unrelated man is the father (and no other alternatives to the defendant's paternity were worthy of consideration), Bayes' rule would be surperfluous. The probability not assigned to the random man is the defendant's probability, so if we have the random-man probability, all we need to do is to subtract it from 1. What remains is the defendant's probability.

The technical leader used Bayes' rule because he did not know the probability that a random man was the father. Let’s look at his explanation of the computation, as presented by the appellate court. The court starts by recounting that

Pfoser testified that DNA paternity testing had three components. The first component of the test involved an exclusion analysis where Pfoser entered the DNA profiles into a computer containing a statistical calculator. If there were any inconsistencies between the alleged father and the child, the computer would give a result of “0” for paternity index.

Apparently, each child shared at least one allele per locus with the defendant, so the computer program did not report an approximate probability of zero, 2/ and the opinion continued:

The next stage involved the calculation of the paternity index, which was a formula used to determine “the likelihood that the assumed alleged father in question is in fact a father as opposed to a random individual that's unrelated in the general population.”

If "likelihood" has the technical meaning of statistical "support" for a hypothesis, this statement could be literally true. But if "likelihood" means probability, as the court evidently and understandably thought, then the explanation is either meaningless or misleading. There is no probability that the defendant is the father "as opposed to a random individual that's unrelated." There is a probability that the defendant is the father (as opposed to everyone else in the population, given all the evidence in the case). And, the paternity index is not even a probability, let alone that one. It is a ratio of two different probabilities. As the court wrote, "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.' ... (quoting Ivey v. Commonwealth, 486 S.W.3d 846, 851 (Ky. 2016) (quoting D.H. Kaye, The Probability of an Ultimate Issue: The Strange Cases of Paternity Testing, 75 Iowa L. Rev 75, 89 (1989))."

The important aspect of the paternity index is that it is a likelihood ratio that expresses the support that the reported DNA profiles of the mother-child-defendant trio provide (if correctly determined) for the hypothesis that the defendant is the biological father relative to the hypothesis that an unrelated man is the father. The idea is that if the profiles are some number L times more probable under one hypothesis than the other, then they support that hypothesis L times more than they support the alternative. This ratio does not assume that one hypothesis is true and the other false. Rather, it treats both hypotheses as equally worthy of consideration and addresses the probability of the evidence when each one is considered. Thus, the use of the ratio to describe the strength of the evidence for the better supported hypothesis does not conflict with the presumption of innocence. Had the expert simply given the paternity index and spoken of relative support, the defendant's objection would have had even less traction that it did.

But the technical leader did not describe the paternity index in this “likelihoodist” way. Instead, to quote from the opinion,

Pfoser testified that the third component of DNA paternity testing converted the paternity index into a probability of paternity percentage using a statistical, mathematical formula called “Bayes' Theorem.” Pfoser explained:

“Bayes' Theorem is essentially a basis for a likelihood ratio. Like I kind of described before, you're basing it on two conflicting hypotheses or two conflicting assumptions. One is that the individual in question is in fact the father as opposed to a completely random unrelated individual could be the father.”

Pfoser further explained:

“[S]o you're taking two, essentially two, calculations, one calculation is * * * the prior probability or the assumed probability that the person in question is the father of the child and that is divided by the probability that some unrelated person within the same race group in the general population is the father of the child.”

And     "Pfonis testified that the prior probability of paternity was set at 50%.”

As the earlier remarks on Bayes' rule indicate, this explanation of Bayes' rule cries out for corrections at every turn, but now I will just focus on the last sentence because the concept of a prior probability is what triggers worries about the presumption of innocence. The idea of a prior probability is intuitive but not easily mapped onto the legal setting. If I want to infer whether a furry animal that I glimpsed running outside my window is a squirrel (as opposed to a groundhog, a rabbit, a chipmunk, a possum, a cat, a skunk, a bear, or any other furry critter in this neck of the woods), I can start by asking how often various creatures go by. Based on my past observations, I would order the possibilities as squirrel, chipmunk, rabbit, groundhog, cat, and so on. If squirrels account for half of the past sightings, I might select 50% for the probability of a squirrel. This is my prior probability.

Now I think about the details of what I saw in the periphery of my vision. How small was it? What color? Did it seem to have short legs? Was it scurrying or hopping? To the extent that the set of characteristics I was able to discern are more probable for squirrels than for other creatures, I should adjust my probability upwards to arrive at my posterior probability.

Bayes’ rule is a prescription for making the adjustment. It instructs us to multiply the prior odds by the likelihood ratio. Then, voilà, the posterior odds emerge. Suppose my likelihood ratio is 3. I think the characteristics I perceived are 3 times as probable when a squirrel zips by than when the average non-squirrel does. 3/ If the prior probability is ½, the prior odds are 1 to 1, and the posterior odds are 3 × 1:1 = 3:1. Odds of 3:1 correspond to a posterior probability of 3/(3+1) = 3/4. Following Bayes’ rule, I moved from a prior probability of ½ to a posterior of 3/4.

The expert in Gonis arrived at his posterior probability of paternity by making up a set of prior odds — he chose 1:1 — for defendant’s paternity and multiplying them by the paternity index. This looks like a Bayesian calculation. 4/ But in the squirrel-sighting case, there was an empirical basis for the prior odds. I know something about the animals in my neighborhood. The DNA technical leader apparently offered no such justification for his choice of the same number. And how could he? His expertise does not extend to the sexual and criminal conduct of the defendant and everyone else in the male population. The judge or jury, not the DNA profiling expert, is supposed to consider the nongenetic evidence in the case and to rely on its general background information in processing the totality of the evidence in the case to reach its best verdict.

In Gonis, trial judge, who was the factfinder in the case, was explicit about why he found the prior probability of ½ to be acceptable:

The court noted that the cases cited by the State explained why “the .5 number presumption that they start off with is actually just a truly neutral number. It assumes the same likelihood that the defendant was not the father of the child as it does that he would be the father of the child.

This rationale is specious. For a Bayesian, starting with a probability of ½ amounts to believing, before learning about the DNA profiles, that the defendant owns half the probability and that the other half is distributed across every else in the population. Maybe the other evidence in the case would justify that belief, but it hardly seems “neutral” toward the defendant. It treats him very differently from every other man in the population. The more “neutral” position might be to assign the same per capita probability to everyone, including the defendant, and then make adjustments according to the specifics of the case.

The appellate court took no stand on whether the trial court’s conception of neutrality was scientifically or legally tenable. Construing the defendant’s objection narrowly, the court did "not reach the issue of whether a 50% prior probability is a neutral number."

A bona fide Bayesian procedure would be to display the posterior probability for many values of the prior probability. This “variable prior odds approach” avoids the need for the expert to tell the judge or jury which prior probability is correct. 5/

That said, the uncontested likelihood ratios in Gonis, as Justice Schmidt observed, would swamp most prior probabilities. Even if we regarded all the men in the Chicago metropolitan area as equally likely, a priori, to have fathered the two children, the posterior odds of paternity still would be substantial. There are fewer than five million men (of all ages) living in the metropolitan area. So the per capita prior odds are 1:5 million. For the likelihood ratios of 195 million and 26 million, the posterior odds would be more than 39:1 for the paternity of J.G. and 5:1 for the paternity of A.G.

NOTES

1. 2018 IL App (3d) 160166, No. 3-16-0166, 2018 WL 6582850 (Ill. App. Ct. Dec. 13, 2018).
2. Particularly at a single locus, an exclusion does not mean that the probability of paternity is strictly zero. Mutations at some of the STR loci are known to occur at nonzero rates.
3. The phrasing about an "average non-squirrel" is imprecise. There are n+1 mutually exclusive hypotheses H₀, H₁, H₂, ..., H_n, about the animal. Each H_j has a prior probability Pr(H_j) and a likelihood Pr(E|H_j). Let H₀ be the squirrel hypothesis. The appropriate factor for the multiplication of the prior odds is the squirrel likelihood Pr(E|H₀) divided by a weighted average of the other likelihoods. The weight for each non-squirrel hypothesis H_j (j = 1, .., n) is my prior probability on that hypothesis renormalized to reflect that it is conditional on ~H₀. In other words, the Bayes factor is Pr(E|H₀) × [1−Pr(H₀)] divided by Pr(H₁) × Pr(E|H₁) + ... + Pr(H_n) × Pr(E|H_n).
4. By limiting attention to an unrelated man as the only possible alternative, the technical leader was ignoring the terms in the denominator of the Bayes factor for possible related men. See supra note 3. As a result, the Bayesian interpretation he provided was not strictly correct.
5. For discussions of such proposals and their reception in court and in the scholarly literature, see David H. Kaye, David E. Bernstein & Jennifer Mnookin, The New Wigmore on Evidence: Expert Evidence ch. 15 (2d ed. 2011) (updated annually).

Last updated: 16 May 2019, 1:20 PM