Messing with LCN DNA

The delights of Bayesian probability reasoning are sufficient to draw from me another case comment!

Our Court of Appeal, in Manoharan v R [2015] NZCA 237 (11 June 2015), has said that a likelihood ratio of 20 (meaning 20 to 1, or in context, 20 times more likely under the prosecutor’s hypothesis than under the defendant’s) is “not strong” [52].

This was an LCN DNA analysis and obviously compared to the usual DNA results of likelihood ratios in the many millions, 20 is not high. But in the context of a case it can be, as can be seen from the results of applying Bayes’ Theorem (this was a single-issue case involving independent items of evidence).

I should emphasise that the single-issue nature of this case (“who did it?”, an actus reus issue) simplifies the use of Bayes’ theorem. Where several elements of the offence charged are in issue, the theorem must be applied to each separately, and only evidence relevant to that issue is used in each application. This prevents, for example, a huge scientific LR on the issue of identity from swamping the issue of intention. For each issue the “probability of guilt” means the probability that that issue is proved to the standard required for guilt to be established.

The likelihood ratio used by the scientists in declaring a match is not necessarily the same as the likelihood ratio for the match evidence used by the fact-finder in the trial. This is because the defence may have an innocent explanation for the match which brings the denominator of the likelihood ratio (the probability of getting the match on the assumption that the defendant is innocent) close to the value of the numerator. Then, the probative value of the test result for the prosecutor would be very reduced or even extinguished.


A perspective on the size of a LR of 20 is obtained from considering a case where the evidence is not given, compared to one where it is.

  • If, without the evidence, the probability of guilt is 0.90, then with the evidence that probability increases to 0.99.
  • If, without the evidence, the probability of guilt is 0.60, then with the evidence that probability increases to 0.96.
  • And, if, without the evidence, the probability of guilt is 0.30, then with the evidence that probability increases to 0.89.


Perhaps the Court under-emphasised the strength of the DNA evidence in this case. It approved defence counsel’s submission that the jury should have been told that in New Zealand it was likely that there are 200,000 people who would have the same profile. The fallacy is to include people who had no opportunity to commit the crime.

Update: on 28 October 2015 the Supreme Court refused leave to appeal: Manoharan v R [2015] NZSC 156. The Court could see no basis to indicate that the Court of Appeal had been wrong when it had held that there had been no miscarriage of justice.

Another update: Bernard Robertson has commented on this case in”Likelihood ratios in evidence” [2016] New Zealand Law Journal 22. He points out that the Court at [39] correctly describes the witness’s evidence as stating a likelihood ratio, but earlier in the judgment, at [17], it had transposed the conditional when referring to the same evidence. Mr Robertson also points out that the “20 times more likely” assertion should have been clarified because juries will not know what to do with a likelihood ratio, and that in this case source level propositions (where the sample at the scene came from) were mixed with activity level propositions (how it got there).


Search seminar

Notes for the talk on the law of search that I gave at the Auckland District Court on 1 July are available here. They are complementary to the more detailed paper that has been distributed to lawyers, and now is available here.