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Rabu, 27 Maret 2013

Are Two Heads Better than One? Mathematics and the Amanda Knox Case

In yesterday’s New York Times, “Leila Schneps, a mathematician and mystery writer, and her daughter Coralie Colmez” wrote an op-ed entitled “Justice Flunks Math.” They gave this failing grade to an Italian judge who declined to order a second DNA test in the notorious murder case against the American student, Amanda Knox and her Italian boyfriend, Raffaele Sollecito.

If only the judge understood probability theory, they suggest, he might have ruled in favor of the prosecution's request for another DNA test. In their words:
Whatever concerns the judge might have had regarding the reliability of DNA tests, he demonstrated a clear mathematical fallacy: assuming that repeating the test could tell us nothing about the reliability of the original results. In fact, doing a test twice and obtaining the same result would tell us something about the likely accuracy of the first result. Getting the same result after a third test would give yet more credence to the original finding.
This claim of a mathematical fallacy in the judge’s reasoning, however, rests on an important assumption — that the test results are statistically independent (at least in substantial part). To illustrate their point, Schneps and Colmez explain:
Imagine, for example, that you toss a coin and it lands on heads 8 or 9 times out of 10. You might suspect that the coin is biased. Now, suppose you then toss it another 10 times and again get 8 or 9 heads. Wouldn’t that add a lot to your conviction that something’s wrong with the coin? It should.
But consider this example instead:
You test the surface of a gold coin and test it to make sure it is gold. The test is 90% certain to indicate gold when the metal is gold, so following the advice of Schneps and Colmez, you repeat the test three times and strike gold each time. When you try to sell the coin, however, a more astute buyer weighs it and finds that it does not have the density of gold. It is, in fact a thickly gold plated, lead coin.
Which example is more apt in the Knox case? A Nature blog explained that "[v]ery small amounts of Knox’s DNA were found on a knife located at the crime scene 46 days after Kercher’s murder," but this is not correct. The knife came from Sollecito's kitchen, and a more informed account in the New York Times states that "the court-appointed experts concluded that ... Ms. Knox's DNA was in fact on the handle" -- hardly a surprise given that she may have used it to cook dinner in her boyfriend's apartment. The only thing that made the knife incriminating was a police laboratory finding of DNA from the victim, Meredith Kercher, on the blade. But there was evidence that the knife could not have produced all the wounds, and the court-appointed experts were skeptical of the finding about the blade as well the police laboratory's analysis of "a bra clasp that belonged to the victim found on the floor at the scene 46 days after her murder" said to show Sollecito’s DNA.

Returning to Nature's account:
Speaking at the request of the defence, two forensic scientists, Stefano Conti and Carla Vecchiotti from Sapienza University in Rome, said police investigators failed to follow international protocols for collecting and handling evidence and conducting tests on small genetic samples known as low-copy-number (LCN) DNA analysis. For example, officers were not wearing protective masks or hair caps at the crime scene. ... In addition, Conti said police often used plastic bags, rather than paper, to wrap evidence, heightening the risk of contamination. ... “There are various circumstances do not adhere to protocols and procedures,” Conti told the court.

Consequently, the independent experts concluded that they could not rule out the possibility that the knife and bra had been contaminated by other sources of Knox’s and Sollecito’s DNA, such as other evidence at the crime lab where forensic testing was taking place.
If these experts’ concern — that the original DNA test was simply detecting traces of Kercher's and Sollecito’s DNA that investigators inadvertently transferred to the knife and bra clasp, respectively — then repeating the tests could well continue to detect that DNA — and prove nothing more than the original tests did. If the bra clasp sample showed a mixture of DNA from the victim and Sollecito (and nothing else), for example, then repeating it over and over would not reinforce the prosecution case in the slightest. It would be no different than retesting the surface of the lead coin with its gold plated contamination. The inability of the DNA evidence to demonstrate a convincing link to the defendants would remain after even an infinity of new tests.

Consequently, it is hard to judge Schneps' and Colmez's suggestion that "[t]he judge’s rejection of the retest — at least based on the notion that a confirming retest could tell us nothing about the likelihood that the DNA was a match — was a serious error, one that scuppered an opportunity to get at the truth of Ms. Kercher’s murder."

The judge’s decision may have been mathematically sound, or it may have been as naive and fallacious as Schneps and Colmez propose. They have a nice theory but it is fair to assume that there were uncontaminated samples for new testing? Without some specification of precisely what made the initial testing problematic and whether those problems could be reduced sufficiently with retesting, it seems precipitous to convict the judge who overturned the guilty verdict of "bad math."

Indeed, Schneps and Colmez seem to believe that the judge ignorantly opposed retesting of the small sample of DNA on the blade despite an improvement in the technology of testing low template DNA. They wrote that
Even though the identification of the DNA sample with Ms. Kercher seemed clear, there was too little genetic material to obtain a fully reliable result — at least back in 2007. By the time Ms. Knox’s appeal was decided in 2011, however, techniques had advanced sufficiently to make a retest of the knife possible, and the prosecution asked the judge to have one done. But he refused.
Yet, the judge clearly was open to new methods. He asked the two university experts to ascertain "whether it is possible, by means of a new technical analysis, to identify the DNA present on items 165b (bra clasp) and 36 (knife)." The Conti-Vecchiotti Report, Assignment. Finding "no evidence of cellular material in the samples analyzed," however, his experts concluded that "no DNA suitable for further laboratory investigations (amplification, electrophoresis) was present either on the swabs [tamponature] (A-B-C-D-E-F-G-H-I) taken from Exhibit 36 (knife) or on those (L-M) taken from Exhibit 165B (hooks of the bra)." Id. Conclusions (1). They based this conclusion on the absence of cellular material and the failure of "quantification of the extracts ... conducted via Real Time PCR [to] reveal the presence of DNA." Id. Conclusions (2).

The prosecution disagreed. It asked for still more testing. But Judge Claudio Pratillo Hellmann had had enough. Rather than prolong the appeal -- and the imprisonment of the defendants -- still longer to await tests that his experts told him would be useless, he and the jurors returned their not-guilty verdict. It remains to be seen why this verdict now has been overturned, but it is hardly obvious that "bad judicial math" is the reason.

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