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.
The scientific LR of 20 to 1 in favour of a match is used to calculate the numerator of the LR at trial: the probability of getting the evidence of a match on the assumption that the defendant is guilty. Often a scientific LR for DNA is in the millions, resulting in a trial LR numerator of approximately 1. But here the numerator will be much less: a ratio of 20 to 1 is a probability in favour of 20/21 = 0.95.
We can compare what the probability of guilt would have been if the scientific LR had been in the millions and the trial LR numerator approximately 1, with what the probability of guilt would be under the revised scientific LR and corresponding new trial LR, for priors (that is, the ratio of probability of guilt to probability of innocence, based on all the other evidence in the case, for this single-issue case) of various levels. (For multi-issue cases the priors will be assessed separately for each issue.)
Fortunately this comes down to a simple relationship: the new probability of guilt is 0.95 times the old probability of guilt. The assumption is that with an LR in the millions, the probability of guilt is 1, whereas with a reduced LR of 20 the probability of guilt is 0.95, the priors being the same in each alternative.
This means that the scientific match evidence isn't as probative as it was when the scientific LR was in the millions, but the reduction is fairly small - although it could make the difference between proof beyond reasonable doubt and acquittal:
- If the old probability of guilt, on all the evidence at trial, had been 0.95, the reduced scientific LR decreases the final new probability of guilt to 0.90.
- If the old probability of guilt was higher, say 0.98, the reduced scientific LR reduces that to 0.93.
- And, if the old probability of guilt was 0.90, the reduced scientific LR decreases that to 0.86.
- 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.
Which perspective was appropriate in this case? The Court was anxious to remove what it called the CSI effect (the jury might wrongly assume that a DNA match was conclusive evidence of guilt), and this is supported by the first perspective. The danger was that the second, and more relevant, perspective would be overlooked.