American Philosophical Quarterly Volume 49, Number 2, April 2012

EVIDENTIARY fAllACIES AND EMPIRICAl DATA Michael Byron

T

he Prosecutor’s fallacy is a well-known hazard in the assessment of probabilistic evidence that can lead to faulty inferences.1 It is perhaps best known via its role in the assessment of DNA match evidence in courts of law. A prosecutor, call him Burger, presents DNA evidence in court that links a defendant, Crumb, to a crime. The conditional probability of a DNA match given that Crumb is not guilty, or p(M | ~G),2 is very low: according to Burger, one chance in tens of millions. Burger goes on to argue that this very low probability entails another low probability. He asserts that it is very improbable that Crumb is not guilty given the match, and so p(~G | M) is also very low. As this latter probability is precisely what the jury is called upon to assess, Burger’s assertion is likely to lead the jury into convicting Crumb. The defense attorney, Mason, knows his probability theory and calls attention to Burger’s fallacious inference. from the mere fact that p(M | ~G) is low, nothing in fact follows about p(~G | M). As tempting as it might be to infer that these conditional probabilities are approximately equal, to do so would be a mistake. After boring the jury with Bayes’s Theorem and some arithmetic, Mason convinces them that they have reasonable doubt, and Crumb gets off. All of this is ordinary probability theory.

Recently, however, objections have surfaced to the use of the Prosecutor’s fallacy in a real-world context.3 According to one view, Mason and his ilk “try to educate the wider public about how to deal with these probability issues but in the end they just manage to muddy the waters even more with their logically flawed analysis.”4 The flaw turns out not to be logical, but empirical: “The basic mistake in their analysis consists in disregarding a crucial aspect of the situation, which should have been taken into account and which indeed significantly lowers the estimated value of [p(~G | M)], and consequently increases the probability of the defendant’s guilt.”5 The “crucial aspect” in the present case would be the fact that Crumb’s data is in the police database. The argument is that, once this fact is taken into account, the probability of guilt given the available information is high, not low: in one example drawn from the literature,6 the posterior probability of guilt ends up being 0.99, which is very high indeed. Notice that the complaint here is not against the Prosecutor’s fallacy itself, which is simply a reflection on the probability calculus. The objection is that the posterior probability of guilt goes up once more empirical evidence is taken into account, for instance, that the suspect’s data appears in a police database (D). That is, the probability of guilt given both

©2012 by the Board of Trustees of the University of Illinois

APQ 49_2 text.indd 175

2/10/12 3:18 PM

176 / AMERICAN PHIloSoPHICAl QUARTERlY a DNA match and the fact that the suspect appears in a police database, p(G | M&D), is significantly higher than p(G | M). once this additional piece of evidence is factored in, the probabilities might seem to change. The fact that a suspect’s data is in the police database appears to make it more likely that the suspect is guilty. Individuals who have committed crimes are more likely than others to commit crimes, and so p(G | D) would be relatively high. Discounting segments of the population who, based on age, sex, and so forth, almost certainly could not have committed the crime, the posterior probability of the suspect’s guilt seems nearly certain. And, once again, the complaint here is not with the Prosecutor’s fallacy per se: the contention is that, as ordinarily formulated, the probability equations omit crucial empirical facts. once one inserts the missing empirical information, the prosecutor’s reasoning to a conclusion of “guilty” becomes cogent. Such critiques of the importance of probabilistic fallacies are fatally undermined by the fact that they go beyond the fallacy, which is simply that of conflating two conditional probabilities (p(M | ~G) with p(~G | M) in the Prosecutor’s fallacy). As soon as one extends the critique by bringing in extraneous factors such as empirical data, one loses sight of the key issue raised by the fallacy. Suppose, for example, one decides that the discussion of the Prosecutor’s fallacy should include generic empirical features of DNA matching, then one must consider the empirical fact that the “one in 20 million” (or whatever) numbers that prosecutors often use regarding DNA matches ignores an important fact. These probabilities stem from the idea that DNA is unique to individuals, and so the probability of a perfect match between two different individuals is vanishingly small in principle. Yet in practice one must contend with laboratory error. In the lab setting, samples become contaminated or mixed, technicians mistakenly expand the criteria for a match, sequencers

APQ 49_2 text.indd 176

become miscalibrated, and various other errors occur. It is a commonplace of scientific practice that no result can be more precise than the rate of error. According to research, a good laboratory error rate would be 1 percent; the best lab might hope for an error rate of 0.1 percent or 1 error in 1,000 cases.7 But then the probability of a DNA match can be no smaller than 1 in 1,000, and it would never be appropriate to use the miniscule numbers that prosecutors typically use.8 Incorporating lab error rates into the probability equations dwarfs the significance of empirical data about DNA profiles in databases. Even if one agrees that ex-cons are more likely than others to commit crimes, the significance of a DNA match falls sharply once one takes account of lab error. Using an example from the literature and an absurdly optimistic error rate of 0.01 percent, the posterior probability of guilt, given a match and the fact that Crumb is in the database, p(G | M&D), is only 0.09. Even if the lab’s experimental error were a miraculous 0.001 percent, p(G | M&D) rises to just 0.5. Evidence from such an unbelievably clean and reliable DNA lab would raise the posterior probability of guilt to just 50–50. A defense attorney would not have to be Perry Mason to succeed in arguing reasonable doubt when presenting those odds to a jury. Apart from deciding which empirical data to include in the formulas of the Prosecutor’s fallacy, something is suspect about the idea that one needs to include more empirical data. The problem is that statements M and D are not independent; indeed, the fact that DNA evidence from the crime scene matches Crumb’s entails that his data was in the database. The value of p(G | M) represents the likelihood that a suspect is guilty, given that evidence from the scene matched the suspect’s. If a match is given, then it is also given that the suspect’s data was in the database, since that is what the evidence from the crime scene matched.9 And so adding D

2/10/12 3:18 PM

EVIDENTIARY fAllACIES AND EMPIRICAl DATA /

to the condition adds no new information and cannot in principle modify the relevant probability formulas. The same issues occur with regard to the Interrogator’s fallacy,10 which is the mistake of assuming that confessions always make guilt more likely. This fallacy, which also concerns the use of probability calculations in legal situations, raises similar issues when applied to criminals. The probability calculus dictates that a confession increases the posterior probability of guilt just when the probability of confessing given that the person is guilty is higher than the probability of confessing given that the person is not guilty. The biconditional goes both ways: so if G is the statement that Crumb is guilty and C is the statement that Crumb confessed, the claim is that p(G | C) > p(~G | C) just in case p(C | G) > p(C | ~G). That says, roughly, that a confession is probative of guilt just in cases where guilty people are more likely than the innocent to confess. Under what circumstances would one find it less likely that a guilty person confessed than that an innocent one did? The standard example is that of a hardened criminal who knows how to resist interrogation. Especially under circumstances of harsh or aggressive interrogation, an innocent person without training or fanatical devotion to a cause who happens to be a suspect might end up confessing. Given that scenario, one might expect the likelihood of confession to be reversed, so that p(C | G)  p(C | ~G), and so the innocent person is more likely to confess p(G | C)  p(~G | C). An objection in the literature draws a parallel to the argument about the Prosecutor’s fallacy: the fact of the interrogation can by itself raise the posterior probability of guilt, even when one might expect that an innocent person is more likely to confess.11 This fact could be relevant because it is possible that the police are more likely to interrogate criminals than innocent people. When one takes account of

APQ 49_2 text.indd 177

177

this additional bit of evidence, the posterior probability of guilt in a situation where a hardened criminal is less likely to confess than an innocent suspect could nevertheless rise as high as 0.95. To be clear about what this argument claims: it does not claim that interrogators can prove that a suspect is guilty merely by pointing to the fact of an interrogation. No one would suggest that p(G | I), the probability that Crumb is guilty given that he was interrogated, must be greater than 0.5 (or any arbitrary number). The probative value of interrogations is not intrinsic, but rather depends on other facts about how the police select individuals for interrogation. If the police do their job, guilty people are more likely to be interrogated than the innocent, so the ratio of p(I | G) / p(I | ~G) should be greater than 1. This is the claim that drives the objection to the Interrogator’s fallacy: if it is right, then the fact that Crumb has been interrogated appears to make his guilt more likely. As with the Prosecutor’s fallacy, this critique of the Interrogator’s fallacy is based on an attempt to move beyond the simple logic underpinning the fallacy, specifically the need to ensure that the evidence presented is more likely to come from the guilty than from the innocent. And again, the danger of extending the argument is that it opens the door to arbitrary empirical modifications of the relevant probability calculations. Suppose one agrees that the proper posterior probability is not p(G | C), or the probability of guilt given a confession, but rather p(G | C&I), which includes in the condition the fact that the suspect was interrogated (I). But why stop there? The fact that Crumb confessed entails that he was interrogated. Given standard operating procedure, it also entails that he was “taken downtown” (T). So perhaps one can add yet another condition to the posterior probability and calculate instead p(G | C&I&T). Since it is more likely that a guilty person than an in-

2/10/12 3:18 PM

178 / AMERICAN PHIloSoPHICAl QUARTERlY nocent person is “taken downtown”—perhaps also 9 times more likely, to use the objection’s multiplier—the new formula would add another factor of 9 to the calculation. And so one should conclude that p(G | C&I&T) is 162 times more likely than p(~G | C&I&T), yielding a conditional probability of guilt of 162/163, or 0.99. In today’s high-tech interrogations and given typical procedures, the fact that Crumb confessed also entails that he was videotaped (V), that he was offered a plea bargain (P), and indefinitely many other propositions. By adding arbitrarily many conditions to the probability calculation, one can achieve arbitrarily high estimates of the posterior probability of guilt. Is this all grist for the mill? After all, the objection to the Interrogator’s fallacy offers a counterexample, a possible scenario under which confession makes guilt more likely than not, even when the guilty are less likely than the innocent to confess. Does this not strengthen the objection by providing even more counterexamples? The answer is no, because the entire approach is bogus. The value of p(G | C) is no different from that of p(G | C&I) because given plausible background assumptions, C entails I. The suspect, after all, confessed to the police while being interrogated. The value of p(G | C) represents the likelihood that a suspect is guilty given a confession. If it is given that the suspect confessed, then it is given that the police interrogated the suspect.12 Consequently it would be a mistake to argue that the interrogation provides further or independent evidence of guilt apart from the confession. Adding I to the condition of p(G | C) adds no new information and cannot in principle modify the relevant probability formulas. All of the probative value of the fact of an interrogation is already included in the fact that Crumb confessed, and one does not want to

APQ 49_2 text.indd 178

make the mistake of counting that information twice. It is no surprise that the posterior probabilities of guilt derivable based on this mistake are strikingly high (above 0.95). The objections to the Prosecutor’s and Interrogator’s fallacies are the same in the end: discussions of these fallacies allegedly omit to consider relevant empirical facts, and once one applies the relevant probability formulas in the real world and take these facts into account, the lessons of the fallacies seem to vanish. Readers might have been suspicious of this strategy from the outset: after all, what impact could empirical data have on a logical fallacy? The only way for that to happen would be if there were a logical connection between the purportedly new information and the statements in the fallacy. And sure enough, such a connection exists. In the case of the Prosecutor’s fallacy, the fact of a match entails that the suspect’s data is already in the database. And in the case of the Interrogator’s fallacy, the fact that the suspect confessed entails that the police interrogated the suspect. In neither case does additional empirical information in fact add any probative information not already contained in the posterior probabilities. So on pain of double counting the evidence, one is not entitled to change the probability formulas. The Prosecutor’s and Interrogator’s fallacies remain important contributions to the world of applied probability theory, and attempts to undermine their usefulness by adding empirical data serve only to confuse the lessons they teach. Any proposed critique of these fallacies is fatally undermined by the attempt to go beyond the simple logical errors at their core. To introduce a host of extraneous issues simply fails to draw the sting of these fallacies and cannot bring the resulting inferences into line with “common sense intuition.”

2/10/12 3:18 PM

EVIDENTIARY fAllACIES AND EMPIRICAl DATA /

APPENDIx This appendix includes formulas and calculations to support the argument of the text. first consider the Prosecutor’s fallacy, which points out the mistake of assuming that p(~G | M) ≈ p(M | ~G) (where ≈ means “approximately equals”) and inferring the value of the former from the latter. Why is that wrong? According to Bayes’s Theorem, one knows that: p(~G | M) =

p(~G) × p(M | ~G) p(M)

The only way for p(~G | M) ≈ p(M | ~G) would be for p(~G) ≈ p(M). In that case, p(~G) / p(M) would equal approximately 1, and the remaining conditional probabilities would be approximately equal. ordinarily, the prior probability of a random citizen being not guilty of a given crime p(~G) is very high, and the prior probability of a random citizen’s DNA matching the crime scene DNA p(M) is very low. Hence the prospects for p(~G | M) ≈ p(M | ~G) are dim, and the value of one clearly does not determine the value of the other. Next consider the claim that p(G | M&D) > p(G| M), which purportedly demonstrates the impact of considering the probative value of the fact that someone’s data appears in a police database (D). following Donnelly’s example, Sesardic employs a version of Bayes’s Theorem that gives the following: p(G | M) =

p(G) × p(M | G) p(G) × p(M | G) + p(~G) × p(M | ~G)

The next step is to insert some values: in Donnelly’s example, p(G), the prior probability that an arbitrary citizen is guilty, is 1 in 60 million (the population of the U.K.), and so since p(~G) = 1 – p(G), p(~G) ≈ 1. The probability of a DNA match given that a suspect is guilty p(M | G) is 1 (or nearly so), and according to Sesardic the probability of a match given that a suspect is not guilty p(M | ~G) is 1 in 20 million. After some arithmetic, one finds: p(G | M) =

APQ 49_2 text.indd 179

1 ×1 60m 1 1 × 1 + ≈1 × 20m 60m

= .25

179

Next, Sesardic proposes that a more reasonable prior probability of guilt p(G) is 1 in 10 million. The case he discusses is a sexual assault committed by a man, and so he concludes that it is reasonable to rule out 50 million Britons as not possibly the perpetrator (30 million women plus 20 million boys and elderly men). By adjusting this prior probability, the result is: p(G/M) =

1 ×1 10m 1 1 × 1 + ≈1 × 20m 10m

= 0.66

The primary move of the objection is to argue that adding the fact that the perpetrator’s data is in the database, D, to the condition raises the posterior probability even higher—indeed, to the level of subjective certainty (probability nearly 1). The equation used here is: p(G | M&D) =

p(G | D) × p(M | G&D)

p(G | D) × p(M | G&D) + p(~G | D) × p(M | ~G&D)

where the posterior probability of a match given that a suspect is guilty p(M | G&D) is still 1 and p(M | ~G&D) is still 1 in 20 million. At this point Sesardic assumes that being an ex-con makes one 100 times more likely to commit a crime, and infers that p(G | D) is 1 in 100,000. Based on these assumptions, the new conditional probability of guilt is: 1 ×1 100,000 p(G | M&D) = = 0.99 1 99,999 1 ×1+ × 100,000 20m 100,000

and this result is clearly greater than p(G | M), no matter which prior probability of guilt one uses. The calculation changes dramatically once one takes Koehler’s advice and accounts for laboratory error. The prior probability of a match p(M) can be no smaller than the laboratory error rate, which in a best-case scenario might be 0.1 percent or 1 in 1000. Keeping the equation otherwise the same, one would have: 999 1 × 100,000 1000 p(G | M&D) = = 0.0099 1 99,999 1 ×1+ × 100,000 1000 100,000

2/10/12 3:18 PM

180 / AMERICAN PHIloSoPHICAl QUARTERlY and that dramatically undercuts the idea that a DNA match makes guilt probable, even given the supposedly additional fact that the suspect’s data is in the police database. Next consider the Interrogator’s fallacy. That is the mistake of thinking that a confession always makes guilt more likely: that in every case p(G | C) > p(~G | C). The correct relationship is the following biconditional.

probability of guilt of 0.75, that the police are 9 times more likely to interrogate a guilty person rather than an innocent one, but that this hardened criminal is less likely to confess (probability 0.4) than an innocent person (probability 0.6) under harsh questioning. Arithmetic yields: p(G | C&I) 0.75 0.4 = ×9× p(~G | C&I) 0.25 0.6 2 =3×9× 3 = 18

p(G | C) > p(~G | C) ⇔ p(C | G) > p(C | ~G)

That says, roughly, that a confession is probative of guilt just in cases where guilty people are more likely than the innocent to confess. Sesardic offers a counterexample to the left-toright implication: he argues that it is possible to conceive of a case in which a confession increases the posterior probability of guilt even under circumstances where the innocent are more likely than guilty people to confess. This argument begins with the odds form of Bayes’s Theorem: p(G | C) p(G) p(C | G) = × p(~G | C) p(~G) p(C | ~G)

So the posterior probability of guilt p(G | C) is likely to be higher than the prior probability of guilt just in case the likelihood ratio (the last term of the equation) is greater than 1. And that is consistent with what the biconditional states. The next step of the objection is to incorporate the fact that the suspect is interrogated. The equation is the following:

That makes the posterior probability of guilt 18 times greater than that of innocence, which means that the posterior probability of guilt is 18/19 or 0.95. Consequently the ratio of the posterior probabilities is greater than 1, whereas the likelihood ratio is less than one, contrary to the original biconditional. By parity of reasoning, however, one can add other conditions to the formula for the posterior probability of guilt. Suppose that being “taken downtown” makes one a further 9 times more likely to be guilty than innocent. It would follow that another factor of 9 should appear in the equation just above, which yields a ratio of 162 and a posterior probability of guilt p(G | C&I&T) of 162/163 or 0.99. Whatever probative value lies in the confession, it already includes the probative value of being interrogated, taken downtown, videotaped, offered a plea bargain, and all of the other entailments of the confession itself. To add any of these a second time is a mistake.

p(G | C&I) p(G) p(I | G) p(C | G&I) = × × p(~G | C&I) p(~G) p(I | ~G) p(C | ~G&I)

Kent State University

Plugging in some values for a counterexample, Sesardic supposes that an individual has a prior

NoTES 1. The body of the essay develops the main argument intuitively and aims to minimize the use of equations there. Interested readers may consult the Appendix, which explains the equations behind each step. The term ‘Prosecutor’s fallacy’ was coined by William C. Thompson and Edward l. Schumann in “Interpretation of Statistical Evidence in Criminal Trials: The Prosecutor’s fallacy and the Defense Attorney’s fallacy,” Law and Human Behavior, vol. 11, no. 3 (September 1987), pp. 167–187. for a different treatment of the Prosecutor’s fallacy, see Gerd Gigerenzer, Reckoning with Risk: Learning to Live with Uncertainty (New York: Penguin, 2002).

APQ 49_2 text.indd 180

2/10/12 3:18 PM

EVIDENTIARY fAllACIES AND EMPIRICAl DATA /

181

2. The notation is standard: M is the proposition that the suspect’s DNA matches a sample collected at the crime scene, and G is the proposition that the suspect is guilty. The probability notation, p(M | ~G), for example, refers to the conditional probability of the proposition M given the proposition ~G, that is, of a match given that the suspect is not guilty. 3. Numerous studies across the social sciences have purported to show that people are “bad” at probabilistic reasoning and succumb to many probabilistic fallacies, including the Prosecutor’s fallacy. Readers are familiar with the seminal research of Tversky and Kahneman developing prospect theory as an alternative to expected utility theory and its reliance on probability calculations. (See, for instance, D. Kahneman and A. Tversky, “Prospect Theory: An Analysis of Decision under Risk,” Econometrica, vol. 47 [1979], pp. 263–292.) Nisbett and Ross have done important work showing that heuristics more than Bayes’s Theorem are descriptively adequate to human behavior. (See R. E. Nisbett and l. Ross, Human Inference: Strategies and Shortcomings of Social Judgment [Englewood Cliffs, N.J.: Prentice-Hall, 1980].) Rather than merely point out that human inferences often fail to conform to Bayesian models, such approaches develop alternative models that seek greater descriptive adequacy. A rival approach aims to show how inferences do in fact conform to Bayesian models, once those inferences are properly understood. Some of these approaches defend existing legal practice. Along these lines Saunders and colleagues incorporate multiple DNA tests into the evidential calculation, and Meester and Sjerps consider the impact of additional empirical information, such as the fact that a suspect’s information is located in a database. (Sam C. Saunders, N. Chris Meyer, and Dane W. Wu, “Compounding Evidence from Multiple DNA-Tests,” Mathematics Magazine, vol. 42 [1999], pp. 39–43; and Ronald Meester and Marjan Sjerps, “Why the Effect of Prior odds Should Accompany the likelihood Ratio When Reporting DNA Evidence,” Law, Probability, and Risk, vol. 3 [2004], pp. 51–62.) These considerations might lead one to wonder whether prosecutors are in fact guilty of the Prosecutor’s fallacy, which is the issue under discussion. 4. Carmen de Macedo (Neven Sesardic), “Guilt by Statistical Association: Revisiting the Prosecutor’s fallacy and the Interrogator’s fallacy,” Journal of Philosophy, vol. 105, no. 6 (June 2008), pp. 320–332. Sesardic addresses the fallacies, not my example directly. 5. Ibid., p. 324. 6. Peter Donnelly, “Appealing Statistics,” Significance, vol. 2 (2005), pp. 46–48. 7. laboratory error is a standard tool of scientific analysis and does not introduce any probabilistic puzzles of its own. Any measurement includes a standard “margin of error” that defines the precision of the measurement. for a more elaborate treatment of including error rates into forensic evidence, see William C. Thompson, franco Taroni, and Colin G. G. Aitken, “How the Probability of a false Positive Affects the Value of DNA Evidence,” Journal of Forensic Science, vol. 48 (2003), Paper ID JfS2001171_481. 8. The facts stated in this paragraph are drawn from J. J. Koehler, “one in Millions, Billions, and Trillions: lessons from People v. Collins (1968) for People v. Simpson (1995),” Journal of Legal Education, vol. 47 (1997), pp. 214–223. Koehler concludes: “If an error occurs, say, 1 time in 100, it makes no difference whether the DNA frequency statistic is 1 in 1,000,000 or 1 in 57 billion. The chance that a reported DNA match is erroneous is about 1 in 100 regardless of how small the DNA frequency statistic becomes” (p. 223). Error rates might have improved incrementally in the dozen years since Koehler published his research; to accommodate such improvements, the argument considers error rates two orders of magnitude (or 100 times) better than what Koehler suggests was normal in the 1990s. 9. This ignores the potential complication that DNA and other evidence can be subpoenaed from suspects, that DNA databases have expanded over the years to include samples from noncriminals, and other facts about DNA matching that would further undercut the case against the Prosecutor’s fallacy.

APQ 49_2 text.indd 181

2/10/12 3:18 PM

182 / AMERICAN PHIloSoPHICAl QUARTERlY 10. Robert A. J. Matthews introduced the Interrogator’s fallacy to the literature. See Matthews, “Inference with legal Evidence: Common Sense Is Necessary But Not Sufficient,” Medicine, Science and Law, vol. 44 (2004), pp. 189–192; and Matthews, “The Interrogator’s fallacy,” Bulletin of the Institute of Mathematics and its Applications, vol. 31 (1994), pp. 3–5. 11. Sesardic, “Guilt by Statistical Association,” p. 329. 12. This ignores the possibility of a confession without an interrogation, which does not undermine the Interrogator’s fallacy.

APQ 49_2 text.indd 182

2/10/12 3:18 PM