A recent paper, discussing the use of a Bayesian Approach by juries states;

“The likelihood ratio and the prior help to avoid the overlooking of information and to see how information is interrelated. For legal scholars, particularly the interpretation of the prior is not easy to grasp. Whereas the likelihood ratio can be seen as the evidence in a case, the prior has no obvious legal equivalent... We also showed that prior and likelihood ratios cannot meaningfully exist without each other. One simply and logically needs a prior.” [my underline]

It should be noted that there is debate on which method a jury should use to calculate such prior probabilities. These are discussed extensively by Meester and Stevens who conclude;

“These four examples show that a likelihood approach (as suggested in Sullivan) or the presumed prior approach (as suggested in Dahlman and Kolflaath) are both not tenable. We could give many more examples in which similar problems arise. Without going into any detail, we mention that similar problems arise when evaluating mixed DNA profiles. In that situation, there may be many different hypotheses, varying the number of possible contributors but also their identity. These hypotheses will all lead to different likelihood ratios. None of them is unnatural or unacceptable, and the court can pick any of them.”

Authors from the National Institute of Science & Technology published another challenge to the use of the LR in forensic science which states;

“Individuals following Bayesian reasoning may establish their personal degrees of belief regarding the truth of a claim in the form of odds (i.e., ratio of their probability that the claim is true to their probability that the claim is false),taking into account all information currently available to them. Upon encountering new evidence, individuals quantify their “weight of evidence” as a personal likelihood ratio. Following Bayes’ rule, individuals multiply their previous (or prior) odds by their respective likelihood ratios to obtain their updated (or posterior) odds, reflecting their revised degrees of belief regarding the claim in question. Because the likelihood ratio is subjective and personal, we find that the proposed framework in which a forensic expert provides a likelihood ratio for others to use in Bayes’ equation is unsupported by Bayesian decision theory, which applies only to personal decision making and not to the transfer of information from an expert to a separate decision maker, such as a juror”

In an attempt to rebut these challenges, another paper states; “The judge or jury form their personal view which might be visualised as a likelihood ratio, LRDM, although it is certainly not this formal. The jury accepts the scientist’s likelihood ratio LRExpert, rejects it, or modifies it as their own. Hence, the jury has the choice of believing, not believing, or partially believing the expert witness. The jury members are not obliged to use Bayes’ theorem. What is important is that the jury members will have had the benefit of hearing an explanation of the pertinent expert considerations in arriving at a balanced assessment of the probative value of the evidence.”

However, it is clear from other work that the only rational way that a LR can be used to infer the probability of a proposition is indeed by the use of Bayes’ Theorem. The problems with this approach have also been recognised by others;

“A related argument may be raised that even if probability is mathematical in principle, it is difficult to measure in the context of a criminal trial because of an absence of statistics and the difficulty in determining its subjective equivalent as a measure of belief. It should be clarified that the method is not only suitable for those cases where the probability of a particular piece of evidence can be independently scientifically verified, as with a DNA profile. The method can also be used when subjective probabilities are involved. This may be seen as a somewhat unscientific application of a scientific method, but of course where a judge or juror has to consider issues such as the fact that the defendant fled as part of their deliberations, they must assess what that specific piece of information tells them about the likelihood of an outcome. The application of the language of mathematics simply helps the decision-maker to clarify the measure of belief that attaches to a particular intuition, and it would do such decision-makers a disservice to exclude subjective probabilities from the application of Bayes’ theorem.”

In our opinion it is simply asking too much of jurors to undertake the exercise required to rationally use the LR reported for the DNA evidence. The Appeal Court in England and Wales have several times rejected this suggestion:

“Of course, it is a matter for the jury how they set about their task, and it is no part of this court's function to prescribe the course which their deliberations should take. But consideration of this case along the lines indicated would in our judgment reflect a normal course for a properly instructed jury to adopt. It is the sort of task which juries perform every day, carefully and conscientiously, on the evidence, as they are sworn to do. We do not consider that they will be assisted in their task by reference to a very complex approach [Bayes Theorem] which they are unlikely to understand fully and even more unlikely to apply accurately, which we judge to be likely to confuse them and distract them from their consideration of the real questions on which they should seek to reach a unanimous conclusion. We are very clearly of opinion that in cases such as this, lacking special features absent here, expert evidence should not be admitted to induce juries to attach mathematical values to probabilities arising from non-scientific evidence adduced at the trial.”

“We turn therefore to the second of the three submissions which we recited, namely that the Bayesian approach is logically sound and approved by expert opinion. We would not for our part wish to take issue with that statement so long as it is applied to appropriate subject matter by persons competent to apply it. We have no reason to doubt, as is stated by a number of highly authoritative experts, that it is a sound and reliable methodological approach in some circumstances. We have, however, the gravest reservations about its use in jury trials in cases such as this.”

"It has been suggested that it may be appropriate for the statistician to expound to the jury a statistical approach to evaluating the likelihood that the defendant left the crime stain, using a formula which gives a numerical probability weighting to other pieces of evidence which bear on that question. This approach uses what is known as the Bayes Theorem. In the case of Adams (Denis ) [1996] 2 Cr App R 467 this Court deprecated this exercise in these terms at p482: 'To introduce Bayes Theorem, or any similar method, into a criminal trial plunges the jury into inappropriate and unnecessary realms of theory and complexity deflecting them from their proper task.' We would strongly endorse that comment."