In Dennis John Adams (No 2) (CA)(Crim Div) 16 October 1997, the Court of Appeal considered for the third time the admissibility of instruction on the use of Bayes Theorem to combine items of evidence. This was the first time that matter had been properly argued but the Court, including on this occasion the Lord Chief Justice, preferred the uninstructed comments of their brethren to the argument put before them. The Court drew comfort from a favourable comment on one of the two previous cases in the Cases and Comments section of Crim LR but did not consider any of the unfavourable comment.

In Adams, the prosecution evidence was treated as consisting only of a DNA analysis for which a very high likelihood ratio was given. Other items of evidence, such as the failure of the victim to pick out the accused on an ID parade, tended to favour the accused. The defence expert witness, Professor P Donnelly, gave the jury instruction on how to use Bayes Theorem to combine the defence evidence with the prosecution’s DNA evidence so as to reduce the level of probability of guilt below that required for conviction. After conviction, the central issue on appeal became the admissibility of this instruction. Detailed analysis of the Court of Appeal’s judgment in Adams (No 1) can be found in our article in the January and February 1997 issues of The Criminal Lawyer. We discuss the second case in the series, Doheny, in the May 1997 issue of the same journal. Adams was subsequently reconvicted and appealed again to the Court of Appeal.

All three judgments assume that Bayes Theorem is a technical matter appropriate for expert evidence. The Court in Adams (No 2) also agreed that Bayes Theorem could not be faulted "so long as it is applied to appropriate subject matter". The judgment assumes a clear distinction between scientific evidence, to which the Bayesian method can be applied and non-scientific evidence to which it cannot.

But there is no such clear distinction. While DNA evidence may be highly "statistical", other scientific evidence, such as fibre analysis, is less so. Nor is there any reason in principle why less quantitative evidence such as eye-witness identification should not be subjected to statistical analysis.

There cannot be two different ways of thinking about evidence. There can be only one logical approach which is not limited to particular "subject matter". Bayesian reasoning is merely the formalisation of that generally applicable logical approach.

That being the case,we doubt that such reasoning is properly a matter for expert evidence. There is clearly no objection to counsel suggesting to jurors how they should combine and compare items of evidence in informal terms so it is hard to see why there should be any objection to counsel doing the same in formal terms. In fact, the logical rectitude of this form of reasoning should be a matter of general education and knowledge and therefore of judicial notice.

Whether formal instruction of the jury would be a wise approach is another matter but the Court’s fears that instruction on the use of Bayes Theorem would confuse and distract the jury do not seem to have been borne out. The jury convicted Adams and the Court of Appeal was content that there had been no miscarriage of justice.

The Court sets down guidelines on how juries should approach cases involving DNA evidence. The Court suggests that "perhaps" the jury will apply the DNA evidence to the population of the UK and consider how many people are not excluded by the DNA evidence of being the source of the mark. They would then consider whether the defendant was one of those people and go on ask themselves whether the defendant was the perpetrator by considering the other evidence in the case.

Two points arise from this suggestion:

First, this approach is not improper but it is unnecessarily prescriptive. Logically, the order in which the evidence is considered does not matter and it seems unusual for the Court to prescribe one. It is even more unusual for the Court to suggest, as it does, that, if this particular item of evidence is not accepted, that will be the end of the case. The Court’s advice seems even less sensible when applied to evidence producing lower likelihood ratios, such as ordinary blood grouping. It makes little sense to say that there could be half a million men in the UK who match the mark or could have been the father and then to consider the remaining evidence in the case. It seems more sensible to use the other evidence in the case to narrow down the group of suspects and then use the scientific evidence to single out the perpetrator. There is no reason in principle why this should not be done with evidence such as DNA which produces very high likelihood ratios.

Second, the Court, yet again, provides no guidance at all as to how the final stage, the assessment of the non-scientific evidence and its combination with the scientific evidence, is to be done. As we wrote in The Criminal Lawyer, any such advice would either have to comply with the requirements of Bayesian reasoning, or be illogical.

Nor did the Court give guidance as to how scientific evidence is to be presented. This presumably means that we are thrown back on the two previous judgments and the concept invented in them: the "random occurrence ratio". This was interpreted in a newspaper article by one of the counsel involved as ruling out evidence in the form of a likelihood ratio. We are told that, at Crown Court level, confusion is now reigning, different judges making different rulings on whether likelihood ratios may be used or not.

The likelihood ratio is the correct way to present such evidence. It has two overwhelming practical advantages. The first is that it is potentially capable of dealing with all situations whereas the frequency (or "random occurrence ratio") will only produce a correct answer where one suspect is compared with one mark. And the second is that evidence expressed in this form can logically be combined with other evidence. Using frequencies requires an intermediate step in reasoning. Significance tests and probabilities of paternity cannot logically be combined with other evidence at all.

Britain and New Zealand have led the world in the application of likelihood ratios to scientific evidence. In its second report on DNA evidence, the US National Research Council has grasped their advantages compared to earlier methods and European jurisdictions are also adopting their use. There is no longer any major controversy in the forensic scientific world between these methods of presentation and the previous statistical methods. The Court of Appeal has attempted to lay down guidelines of general application after hearing argument in a case concerning the comparison of single suspect with a single sample using outdated technology. By so doing, the Court has unwittingly ignored the advances of the last few years and caused needless confusion.

The three rulings do not forbid using likelihood ratios to present scientific evidence. But what they apparently do is to prevent a witness from explaining what the likelihood ratio means. The witness may give a figure, whether it be 100 or 1 million, but is not allowed to explain its logical use. In any case, the explanation the Court has attempted will be of use only where a single suspect is compared with a single mark. It will be of no use where there are multiple marks at the scene of a crime or other non-standard cases such as where a blood stain, believed to be from someone who has disappeared, is compared with the blood of relatives to determine identity.

Further worries arise when considering other forms of evidence such as psychological evidence in child sexual abuse cases. The clearest way of exposing the flaws in many of the "syndromes" presented as evidence that a child has been abused is analysis using likelihood ratios. The law reports are filled with instances in which psychologists have committed the "prosecutor’s fallacy" and thus misled the court. It is not clear if the defence will now be allowed to explain what is logically wrong with such evidence.

So far as DNA is concerned, this discussion is becoming academic. When the sample quality is good, modern methods produce likelihood ratios approaching, and even exceeding, the population of the world. As a result the FBI has recently adopted a rule that it will give a positive identification (like fingerprint evidence) when the likelihood ratio exceeds 260 billion. (Presumably, creating odds of at least 1000 to 1 against there being a repetition of the profile in the USA). Though we recognise the administrative efficiency of this step we do not wholeheartedly support it for several reasons.

Analysis of many other forms of scientific evidence has been illuminated by the likelihood ratio approach. Glass, transferred fibres and conventional blood-typing have all been reanalysed with the result that the first two, in particular, have been found to be much more useful evidence than was previously realised. Likewise, other forms of evidence can be effectively debunked. Increased understanding of less tractable forms of evidence such as fingerprints, document examination and identification parades is gained by subjecting them to a standard method of logical analysis common to all forms of evidence. This clarification is being threatened by the Court of Appeal which appears to wish to make particular ad hoc rules for particular forms of evidence.

Likelihood ratios are the logical and efficient way to analyse and present scientific evidence. This approach also provides the yardstick against which instructions to juries on considering any other evidence should be measured. This does not necessarily mean that it is sensible to give the jury formal instruction in logic. The analogy, as we have said before, is with a parent teaching a child to ride a bicycle. Neither parent nor child requires formal instruction in the laws of mechanics. But if the parent’s instructions do not comply with them, the child will fall off.

There is a far simpler ground on which such formal instruction could be ruled out. If juries are to be instructed in how to apply Bayes Theorem to non-scientific evidence in a case where scientific evidence is present, there would be no reason why such instruction should not be given in respect of non-scientific evidence alone, ie. in every criminal case. This would clearly induce "vexation, expense and delay".

• There is no clear distinction between scientific and non-scientific evidence.

• Bayes theorem is merely a formalisation of logic and common sense.

• Bayes theorem is therefore not appropriate matter for evidence but for general education and judicial notice.

• It is not necessarily appropriate to instruct juries in formal logic.

• Instructions given to juries must comply with the requirements of logic.

• The cumulative effect of these three Court of Appeal judgments is to replace coherence and consistency in forensic science with confusion.

Bernard Robertson and Tony Vignaux are the authors of Interpreting Evidence: Evaluating Forensic Science in the Courtroom, published by John Wiley and Son Ltd. (UK), 1995, where these matters will be found more fully discussed.