Peter Tillers has drawn everyone's attention to an interesting discussion of relevance, "Bayesian Wars Redivivus – An Exchange" in International Commentary on Evidence, Vol 8, Issue 1, Article 1 (2010).
Definitions of relevance can appear to be inappropriately restrictive. In New Zealand we have an example in s 7 of the Evidence Act 2006 (compare rule 403 of the Federal Rules of Evidence 2010). Critically, subsection (3) defines relevant evidence:
"Evidence is relevant in a proceeding if it has a tendency to prove or disprove anything that is of consequence to the determination of the proceeding."
Inevitably, however, there are many facts in a case that are not in dispute but that are relied on to give context to the relevant facts. These contextual matters are often called narrative. Strictly speaking, they do not in themselves tend to prove anything, but they help to explain why other evidence does have the necessary tendency. A generous reading of s 7(3) is needed: it could encompass narrative evidence if "evidence" is understood as including the evidence that explains the "tendency to prove or disprove".
Neutralising the opponent's evidence
Sometimes, each party may rely on the same narrative evidence but will invite a different inference from it. A defence stratagem is to neutralise prosecution evidence by showing that it is consistent with innocence, and this applies to narrative evidence too.
This leads to a paradox: if a narrative fact is not in dispute, but is equally consistent with guilt as with innocence, it is needed by both parties even though its tendency to prove a matter in issue is neutral; it is relevant in ways that cancel each other out. Does neutralisation destroy relevance?
This paradox disappears if s 7(3) means that equal and opposite tendencies are still tendencies, because each must be considered separately.
A problem for Bayesian logic?
The tendency requirement for relevance says nothing about the strength of that tendency. The strength of the tendency of relevant evidence to prove a matter in issue is called the probative value of that evidence. In Bayesian logic probative value is expressed as a likelihood ratio. Essentially this is a way of asking (for prosecution evidence) how much more consistent is the evidence with guilt than it is with innocence.
In some ways it is unfortunate that the adjective Bayesian has attached to this thought process, because it is an ordinary and natural way of addressing the question of the strength of the probative value of evidence. There is not even anything necessarily mathematical about it, as strengths and likelihoods can be assessed without numbers.
Much of the argument in the paper cited above is concerned with how a likelihood ratio is to deal with common reliance on the evidence without rendering it irrelevant. I think there is some crossing of the wires here: probative value is treated in this discussion as if it was relevance. This error is introduced by Ronald Allen at p 10 of the exchange. Roger Park tries to correct it at p 11, but David Kaye thinks it makes a different point (p 11). Ronald Allen emphasises his assertion of a problem that a likelihood ratio of one makes for relevance at p 12, David Kaye discusses the probative values in reply (p 12), and Ronald Allen's rejoinder (p 13) corrects the tendency to think that changing probative value changes relevance, while seeming to suggest that it was David Kaye who said that if both sides rely on the evidence it is not relevant at all, when really it was Ronald Allen himself who suggested that this is implicit in Bayesian reasoning. Ronald Allen thinks it is difficult to determine relevancy until all the evidence has been heard (p 13) – but I think that is because he confuses relevancy with probative value. Samuel Gross seems to agree that relevance cannot be assessed without the other evidence (p 15). Bruce Hay usefully distinguishes between the function of the judge and that of the jury (p 19). Peter Tillers chips in with a defence of the proper use of Bayes' Theorem (pp 20-21). David Kaye mentions what I have here called narrative evidence, at p 24. Ronald Allen comes down hard against Bayesianism (p 25) although he acknowledges it has some use (p 26). David Kaye brings narrative evidence into a Bayesian approach (p 29), and then Peter Tillers brings the discussion to cows (p 30) and common sense (the "stories" approach to probative value).
The reality is that juries are commonly told that they may decide to give particular evidence little or no weight (probative value) notwithstanding that it is (necessarily) relevant evidence. Evidence can be relevant although it has only a slight tendency to prove the matter contended for, and it will be admissible unless excluded by some other rule, and its probative value – assessed in the context of all the relevant evidence in the case – may be similarly slight yet its impact on the result of the case will depend on the priors (that is, how close the other evidence brings proof of the prosecution case to the required standard).
Bayesian reasoning can be useful on the issue of admissibility where it can be shown that the likelihood ratio is close to one (the evidence is nearly as consistent with innocence as it is with guilt) if other circumstances in the case make the evidence in question liable to exclusion because of its illegitimately prejudicial effect (see s 8 of the Evidence Act 2006; rule 403 of the Federal Rules). Evidence that is merely narrative should not have the necessary prejudicial effect to require exclusion, but it is commonplace to encounter exclusion of other relevant evidence because of its prejudicial effect.
Stories or statistics?
Usually people decide what to believe on the basis of what seems, without the need for further inquiry, to be consistent with common sense. They are using experience as the basis for judgment. They would have to concede that other people's experience can be useful in helping them make that judgment, and that that experience may come from statistical studies. The usefulness of scientific studies, the results of which are presented statistically, cannot be denied. They can distinguish factual from fictional stories. The significance of probabilities for logical reasoning must be recognised, and the inescapable influence of conditional probabilities on the correct determination of judicial proceedings must be utilised by fact-finders.