Years ago, Professor Roger Park at the University of California's Hastings College of the Law asked this question about eyewitnesses. As he pointed out,

*something*had to go wrong in these cases. That something could be common or rare. In more technical jargon, how can you infer the prevalence of a factor from a retrospective study?

Another law professor, David Harris of the University of Pittsburgh, tried to do just this. In a recent and generally penetrating book on flaws in the criminal justice system, he argued as follows:

David A. Harris, Failed Forensics: Why Law Enforcement Resists Science 76-77 (2012) (emphasis in original).[T]he basic data are available for all to see. Of the more than 250 exonerations now on record, 25% involved "innocent defendants who made incriminating statements, delivered outright confessions, or plead guilty." ... Recall that DNA evidence--the basis for nearly all the exonerations to date--is available only for a fraction of all criminal cases; experts estimate that police recover testable biological evidence in only 5 to 10 percent of all cases. [I]n these other 90 to 95% of the cases, we have no reason to think that interrogation tactics work any differently, or any better, than in cases in which police recover DNA evidence. Thus, if false statements by suspects occur in 25 percent of the DNA-testable cases, we should expect a similar percentage in the other 90 to 95 percent of the cases. Put another way, if there is no reason to think that the DNA-based exoneration cases differ from others in the system, they provide us with a window into the whole criminal justice system. And that means that the problem we see--the 25 percent of the DNA cases in which false statements occur--represents the tip of the proverbial iceberg, and rational, conservative assumptions would lead us to believe that we should expect to see false convictions and statements by defendants in 25 percent ofallcases.

But surely there is a mistake here. By this reasoning, if every exoneration were a case in which an eyewitness identified the defendant, "rational, conservative assumptions would lead us to believe that we should expect to see false convictions and [eyewitness identifications] in 100% of

*all*cases." That hardly seems rational.

What has gone wrong? First, the assumption that cases in which DNA evidence can be recovered are comparable to the large remainder of criminal cases overlooks the fact that convictions are not clearly comparable to nonconvictions (acquittals or dismissals). For the professed equality to hold, defendants who are not convicted would have to be just as likely to confess and to do so falsely as are those defendants who are falsely convicted and whose cases can be studied. This necessary premise is implausible. Because confessions cause convictions, the incidence of confessing should be less in the nonconvicted group. In addition, if the case against a suspect is weak, the police may resort to more coercive tactics to obtain a confession. For such reasons, we would not expect to find that the proportion of false confessions in cases of false convictions in DNA (or any other batch of) exoneration cases equals the proportion in all criminal cases.

Bayes' rule also has a role here. Let's stipulate that 25% of all false convictions—not just those of DNA exonerations—involved confessions. This statistic is compatible with many possible ratios of false confessions to true ones in all cases. To prove this, I'll run through some algebra, then give a numerical example, but you can skip the box of algebra if you want.

For a numerical example, consider a set of 1000 cases in which 230 defendants (23%) confess. Suppose that 1 in 10, or 23, of these confessions are false, and that 90% of these false-confessions cases terminate in convictions. The result is about 21 false convictions. Now consider the other 770 cases with no confessions. If, say, 80% of these cases end in convictions of which 10% are false, that will add another 62 false confessions. Upon examining the 83 false-conviction cases, one would find confessions present in 21/83 = 25% of them. Yet, only 2.3% of all the cases (10% of all the confessions) were false confessions.

Let P(FC) be the unknown prevalence of false confessions in all cases. Let P(TC) be the prevalence of true confessions, and P(NC) be the prevalence of the remainder of cases in which defendant does not confess. Under each of these conditions, there is some probability that a verdict of guilty (V) will be attained. (For simplicity, I am going to restrict the analysis to cases without guilty pleas. Discovering that some innocent defendants plead guilty would not be much of a surprise.) For example, the probability of a (false) conviction (FV) given a false confession is P(FV|FC).

Bayes rule then states that

P(FC|FV) = P(FC)P(FV|FC) / [P(FC)P(FV|FC) + P(TC)P(FV|TC) + P(NC)P(FV|NC)].

Because aguiltyverdict cannot follow a true confession, P(FV|TC) = 0, and we have

P(FC|FV) = P(FC)P(FV|FC) / [P(FC)P(FV|FC) + P(NC)P(FV|NC)].

We observe P(FC|FV) = 0.25 and want to infer P(FC). Solving for P(FC) yields

P(FC) = kzw/(1-k)y,

wherek= P(FC|FV), y = P(FV|FC), z = P(NC), and w = P(FV|NC).

Thus the prevalence of false convictions depends not just on the observed valuek= P(FC|FV) = 0.25. It varies with the other conditional probabilities (w and y) and the prevalence of cases without confessions (z).

The lesson of this exercise is simply that the proportion of confessions within a set of cases of false convictions cannot produce a meaningful estimate of their prevalence. Of course, this lesson is not confined to confessions. It applies to any factor that appears (or does not appear) among the known cases of false convictions. If there were no fingerprint identifications in the DNA exoneration cases, would that demonstrate that latent fingerprint identification is highly accurate?

Judge Richard Posner thought so. In

*United States v. Herrera*, 704 F.3d 480 (7th Cir. 2013), he wrote that "[o]f the first 194 prisoners in the United States exonerated by DNA evidence, none had been convicted on the basis of erroneous fingerprint matches, whereas 75 percent had been convicted on the basis of mistaken eyewitness identification." Id. at 487. I will discuss this assertion and its implications in a later posting.

**Related Postings**

Looking Backwards: How Safe Are Fingerprint Identifications?, July 28, 2014

False Confessions, True Confessions, and the Q Factor, July 2, 2014

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