Prosecutor's fallacy and conditional probability reasoning: R v Lyttle [2022] NZCA 52

The prosecutor’s fallacy is an error in reasoning that can be very difficult to notice. It came to attention recently in what is now our leading case on costs in criminal cases: R v Lyttle [2022] NZCA 52.

The Court dealt with appeals from three separate trials, and the one relevant here is R v S, the discussion of which begins at [105].

The issue was whether injuries had been caused by assault or by accident.

There was no evidence indicating guilt other than expert medical opinion evidence. The Crown called five experts, the defence two.

Inevitably, the experts’ evidence was all based on studies of similar injuries where the causes were known [165].

Usually, experts give their evidence in the form of a likelihood ratio: for example, the injuries are much more likely to have been caused by assault than by accident. The cases for the prosecution and the defence in the trial are compared in that way.

Although it is not obvious here, it seems that the experts, as summarised from [116], were in effect saying (for the Crown): the injuries here were very much more likely to have been caused by assault than by accident.

In calling attention to the prosecutor’s fallacy, the Court was in effect saying that the probability of seeing the injuries if they were caused by assault is not the same as the probability of assault, given the injuries.

That indeed is so, as is obvious if we consider a simple example of the same error: the probability of seeing that an animal has four legs, given that it is a sheep, is not the same as the probability of  seeing that an animal is a sheep, given that it has four legs.

There are two separate things under examination in these cases: injuries by assault and injuries by accident; sheep and quadrupeds.

The seriousness of this error depends on context. If you are talking about all the sheep and all the quadrupeds in the world, it is a very serious error. But if you mean just those sheep and those chickens in that field, the error will not matter.  If the context tends to exclude innocent explanations it is like the field, but with greater recognition of innocent explanations it becomes more like the world. In a trial the context of the case provides what can be called the prior probabilities of guilt and innocence.

In R v S the context was that of a situation where accidental fall was possible, and where the defendant was of good character, well spoken of as a caring person, and a person of whom months of covert audio surveillance after the event revealed nothing to raise any suspicion.

If we pretend that the combined likelihood ratios of the experts' evidence strongly favoured guilt, say only one chance in a hundred of innocence, that would in mathematical terms be 99 to 1. But without any other evidence of guilt, and to the contrary, every indication of innocence, the prior likelihood of guilt may be 1 in 99.

Combined, the priors and the likelihood ratio on these hypothetical figures would be 1. This is the ratio of the probability, on all the evidence in the case, of guilt to the probability on the same evidence of innocence. This means a one in two probability of guilt, or 0.5, which is not enough to meet the criminal standard of proof. Nor even the civil standard, which requires more than 0.5.

At [166] the Court summarised the approach that the prosecutor had suggested at trial. This seems to have involved the sheep/quadruped fallacy: a low probability of getting the evidence of the injuries if the defendant was innocent does not in context mean a low probability of innocence given the evidence of injuries.

All the evidence in combination had to be considered when assessing the likelihood of guilt.

One lesson of the appeal in R v S is that lawyers need to be trained in reasoning with conditional probabilities. The major hurdle is that people who become lawyers have mostly been taught at an impressionable age that they are no good at maths. People who become lawyers do, however, tend to be good at logical reasoning, so let’s not despair.

Another lesson is that rare events happen, and apparently overwhelming statistical evidence is only about probabilities, not about the actuality in a particular case. This seems to be the point that was made by the witness whose comment is noted at [127]. Indeed, in law facts are only ever established to some degree of probability, expressed in words as on the balance of probabilities or beyond reasonable doubt.

The jury in R v S were unable to agree on a verdict, and the Crown elected not to seek a retrial, as noted at [112], so the charge (manslaughter) was dismissed, which is equivalent to an acquittal.

If you are interested in this sort of thing, there is more on this site: just enter "Bayes" in the search box.

Overruling "final" decisions: Chandler v The State (Trinidad and Tobago) No 2) [2022] UKPC 19

You can’t read Chandler v The State (Trinidad and Tobago) (No 2) [2022] UKPC 19 without thinking of the anticipated overruling by the Supreme Court of the United States of Roe v Wade.

When will a court of final appeal overrule its own earlier decision, particularly on a point of constitutional interpretation?

The Board addressed the jurisprudence on this at [55]-[66]. The main points are:

Strong reasons for departure from the earlier decision are needed [56].

Stare decisis requires the greatest of hesitation before re-opening the issue; the earlier decision must be shown to have been wrong and lacking a sufficient foundation [57].

As the SCOTUS has said, a special reason for departure is needed, beyond a mere belief that the earlier decision was wrong [58].

It is necessary to avoid damage to the rule of law [58].

Something more than simply thinking the earlier decision was wrong is needed: for example it may have hampered the proper development of the law or distorted the law by being in conflict with established legal principles or by having to be distinguished in subsequent cases to such an extent that the law is uncertain [59].

There needs to have been a material change in circumstances since the earlier decision [60].

Where the court interpreted legislation, it will be more appropriate for the legislature to make the change, in contrast to common law which is more appropriately left to judges to change (citing Posner) [61].

The fact that the earlier decision was that of a mere majority does not weaken it, because the law has to have some way of resolving differences between judges [63].

Stare decisis promotes certainty, predictability, planning and the giving of legal advice [64].

As Lon L Fuller said, a system of legal rules has eight “desiderata”, relevantly here - retroactive laws should be avoided, legal rules should be clear, there should be a constancy of law through time, there should be correct administration of law [66].

 I add this: Another thing lawyers can’t help thinking about in this context is the difference between the internal and the external morality of law. If conformity with the requirements of the rule of law is a matter of the internal morality of the law, does such conformity tend to serve the requirements of external morality too? The (sadly, late) Professor of Jurisprudence at Oxford, John Gardner, considered this in his Law as a Leap of Faith (2012, OUP) Chapter 8, where at 220 he concludes, “Fuller is no legalist but he is surely, on this front, a bit of a quietist. He encourages lawyers to think that they are already doing their bit, qua lawyers, to save us from the abyss so long as they are upholding the rule of law. But there is always more, and sometimes more important, work to do.”