UK judge rules against probability theory? R v T

Actually, the judge was a bit more considered than my title suggests. In my defence the Guardian says:

“Bayes’ theorem is a mathematical equation used in court cases to analyse statistical evidence. But a judge has ruled it can no longer be used. Will it result in more miscarriages of justice?”

The case involved Nike trainers and appears to be the same as that in a recent appeal  judgment, although it doesn’t actually involve Bayes’ rule. It just involves the likelihood ratio, not any priors. An expert witness had said:

“… there is at this stage a moderate degree of scientific evidence to support the view that the [appellant’s shoes] had made the footwear marks.”

The appeal hinged around the question of whether this was a reasonable representation of a reasonable inference.

According to Keynes, Knight and Ellsberg, probabilities are grounded on either logic, statistics or estimates. Prior probabilities are – by definition – never grounded on statistics and in practical applications rarely grounded on logic, and hence must be estimates. Estimates are always open to challenge, and might reasonably be discounted, particularly where one wants to be ‘beyond reasonable doubt’.

Likelihood ratios are typically more objective and hence more reliable. In this case they might have been based on good quality relevant statistics, in which case the judge supposed that it might be reasonable to state that there was a moderate degree of scientific evidence. But this was not the case. Expert estimates had supplied what the available database had lacked, so introducing additional uncertainty. This might have been reasonable, but the estimate appears not to have been based on relevant experience.

My deduction from this is that where there is doubt about the proper figures to use, that doubt should be acknowledged and the defendant given the benefit of it. As the judge says:

“… it is difficult to see how an opinion … arrived at through the application of a formula could be described as ‘logical’ or ‘balanced’ or ‘robust’, when the data are as uncertain as we have set out and could produce such different results.”

This case would seem to have wider implications:

“… we do not consider that the word ‘scientific’ should be used, as … it is likely to give an impression … of a degree of  precision and objectivity that is not present given the current state of this area of expertise.”

My experience is that such estimates are often used by scientists, and the result confounded with ‘science’. I have sometimes heard this practice justified on the grounds that some ‘measure’ of probability is needed and that if an estimate is needed it is best that it should be given by an independent scientist or analyst than by an advocate or, say, politician. Maybe so, but perhaps we should indicate when this has happened, and the impact it has on the result. (It might be better to follow the advice of Keynes.)

Royal Statistical Society

The guidance for forensic scientists is:

“There is a long history and ample recent experience of misunderstandings relating to statistical information and probabilities which have contributed towards serious miscarriages of justice. … forensic scientists and expert witnesses, whose evidence is typically the immediate source of statistics and probabilities presented in court, may also lack familiarity with relevant terminology, concepts and methods.”

“Guide No 1 is designed as a general introduction to the role of probability and statistics in criminal proceedings, a kind of vade mecum for the perplexed forensic traveller; or possibly, ‘Everything you ever wanted to know about probability in criminal litigation but were too afraid to ask’. It explains basic terminology and concepts, illustrates various forensic applications of probability, and draws attention to common reasoning errors (‘traps for the unwary’).”

The guide is clearly much needed. It states:

“The best measure of uncertainty is probability, which measures uncertainty on a scale from 0 to 1.”

This statement is nowhere supported by any evidence whatsoever. No consideration is given to alternatives, such as those of Keynes, or to the legal concept of “beyond reasonable doubt.”

“The type of probability that arises in criminal proceedings is overwhelmingly of the subjective variety, …

There is no consideration of Boole and Keynes’ more logical notion, or any reason to take notice of the subjective opinions of others.

“Whether objective expressions of chance or subjective measures of belief, probabilistic calculations of (un)certainty obey the axiomatic laws of probability, …

But how do we determine whether those axioms are appropriate to the situation at hand? The reader is not told whether the term axiom is to be interpreted in its mathematical or lay sense: as something to be proved, or as something that may be assumed without further thought. The first example given is:

“Consider an unbiased coin, with an equal probability of producing a ‘head’ or a ‘tail’ on each coin-toss. …”

Probability here is mathematical. Considering the probability of an untested coin of unknown provenance would be more subjective. It is the handling of the subjective component that is at issue, an issue that the example does not help to address. More realistically:

“Assessing the adequacy of an inference is never a purely statistical matter in the final analysis, because the adequacy of an inference is relative to its purpose and what is at stake in any particular context in relying on it.”

“… an expert report might contain statements resembling the following:
* “Footwear with the pattern and size of the sole of the defendant’s shoe occurred in approximately 2% of burglaries.” …
It is vital for judges, lawyers and forensic scientists to be able to identify and evaluate the assumptions which lie behind these kinds of statistics.”

This is good advice, which the appeal judge took. However, while I have not read and understood every detail of the guidance, it seems to me that the judge’s understanding went beyond the guidance, including its ‘traps for the unwary’.

The statistical guidance cites the following guidance from the forensic scientists’ professional body:

Logic: The expert will address the probability of the evidence given the proposition and relevant background information and not the probability of the proposition given the evidence and background information.”

This seems sound, but needs supporting by detailed advice. In particular none of the above guidance explicitly takes account of the notion of ‘beyond reasonable doubt’.

Forensic science view

Science and Justice has an article which opines:

“Our concern is that the judgment will be interpreted as being in opposition to the principles of logical interpretation of evidence. We re-iterate those principles and then discuss several extracts from the judgment that may be potentially harmful to the future of forensic science.”

The full article is behind a pay-wall, but I would like to know what principles it is referring to. It is hard to see how there could be a conflict, unless there are some extra principles not in the RSS guidance.

Criminal law Review

Forensic Science Evidence in Question argues that:

 “The strict ratio of R. v T  is that existing data are legally insufficient to permit footwear mark experts to utilise probabilistic methods involving likelihood ratios when writing reports or testifying at trial. For the reasons identified in this article, we hope that the Court of Appeal will reconsider this ruling at the earliest opportunity. In the meantime, we are concerned that some of the Court’s more general statements could frustrate the jury’s understanding of forensic science evidence, and even risk miscarriages of justice, if extrapolated to other contexts and forms of expertise. There is no reason in law why errant obiter dicta should be permitted to corrupt best scientific practice.”

In this account it is clear that the substantive issues are about likelihoods rather than probabilities, and that consideration of ‘prior probabilities’ are not relevant here. This is different from the Royal Society’s account, which emphasises subjective probability. However, in considering the likelihood of the evidence conditioned on the suspect’s innocence, it is implicitly assumed that the perpetrator is typical of the UK population as a whole, or of people at UK crime scenes as a whole. But suppose that women are most often murdered by men that they are or have been close to, and that such men are likely to be more similar to each other than people randomly selected from the population as a whole. Then it is reasonable to suppose that the likelihood that the perpetrator is some other male known to the victim will be significantly greater than the likelihood of it being some random man. The use of an inappropriate likelihood introduces a bias.

My advice: do not get involved with people who mostly get involved with people like you, unless you trust them all.

The Appeal

Prof. Jamieson, an expert on the evaluation of evidence whose statements informed the appeal, said:

“It is essential for the population data for these shoes be applicable to the population potentially present at the scene. Regional, time, and cultural differences all affect the frequency of particular footwear in a relevant population. That data was simply not … . If the shoes were more common in such a population then the probative value is lessened. The converse is also true, but we do not know which is the accurate position.”

Thus the professor is arguing that the estimated likelihood could be too high or too low, and that the defence ought to be given the benefit of the doubt. I have argued that using a whole population likelihood is likely to be actually biased against the defence, as I expect such traits as the choice of shoes to be clustered.

Science and Justice

Faigman, Jamieson et al, Response to Aitken et al. on R v T Science and Justice 51 (2011) 213 – 214

This argues against an unthinking application of likelihood ratios, noting:

  • That the defence may reasonable not be able explain the evidence, so that there may be no reliable source for an innocent hypothesis.
  • That assessment of likelihoods will depend on experience, the basis for which should be disclosed and open to challenge.
  • If there is doubt as to how to handle uncertainty, any method ought to be tested in court and not dictated by armchair experts.

On the other hand, when it says “Accepting that probability theory provides a coherent foundation …” it fails to note that coherence is beside the point: is it credible?

Comment

The current situation seems unsatisfactory, with the best available advice both too simplistic and not simple enough. In similar situations I have co-authored a large document which has then been split into two: guidance for practitioners and justification. It may not be possible to give comprehensive guidance for practitioners, in which case one should aim to give ‘safe’ advice, so that practitioners are clear about when they can use their own judgment and when they should seek advice. This inevitably becomes a ‘legal’ document, but that seems unavoidable.

In my view it should not be simply assumed that the appropriate representation of uncertainty is ‘nothing but a number’. Instead one should take Keynes’ concerns seriously in the guidance and explicitly argue for a simpler approach avoiding ‘reasonable doubt’, where appropriate. I would also suggest that any proposed principles ought to be compared with past cases, particularly those which have turned out to be miscarriages of justice. As the appeal judge did, this might usefully consider foreign cases to build up an adequate ‘database’.

My expectation is that this would show that the use of whole-population likelihoods as in R v T is biased against defendants who are in a suspect social group.

More generally, I think that anyguidance ought to apply to my growing uncertainty puzzles, even if it only cautions against a simplistic application of any rule in such cases.

See Also

Blogs: The register, W Briggs and Convicted by statistics (referring to previous miscarriages).

My notes on probability. A relevant puzzle.

Dave Marsay 

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About Dave Marsay
Mathematician with an interest in 'good' reasoning.

One Response to UK judge rules against probability theory? R v T

  1. Pingback: Uncertain Urns Puzzle « djmarsay

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