We have seen here recently how appellate judges on the same court can differ sharply over what conclusions can be drawn from the evidence in a case. Is there a right way to arrive at conclusions from facts?
In trials juries are given little guidance on how to reason, other than being told that the drawing of inferences is a process of using logic and common sense, something people do all the time in their daily lives. It is assumed that people have an innate ability to reach proper conclusions. This assumption must be correct. We tend to be right more often than we are wrong, but without examining why.
Judges have to give reasons for their decisions. The process of articulating reasons imposes a discipline on judicial thinking, and until that process is completed a judge may not know what conclusion is going to be reached. An echo of this is the instruction to jurors to keep an open mind while evidence is being given.
Bayesian analysis is useful in revealing or guarding against errors of logic when inferences are drawn from facts. Dr Fisher has used this in his report. This does not mean that Ian Binnie was wrong to not use it in his. Most people have no idea what Bayes' Theorem is and they infer correct conclusions without using it. The interesting question is whether Dr Fisher has revealed any error of logic that was sufficient to make Ian Binnie's conclusions wrong.
My assessment is that the only candidate for being an error of this significance is Dr Fisher's claim that Ian Binnie failed to consider the evidence cumulatively as opposed to by taking each item at a time. Ian Binnie has denied, in an email to the Minister of Justice that has been published, that he made this error.
It would be astonishing if Ian Binnie had made this mistake. Judges habitually stand back after evaluating the probative value of particular facts and look at the overall picture. That is done to enable a conclusion to be drawn from the combination of the probative values of the facts. This process is what is done instinctively when people exercise their judgment.
There are all kinds of influences, revealed by psychologists, which can cause people to make mistakes. A Bayesian approach to inference drawing can counteract those, but its best application requires extensive statistical information, far beyond what is usually available in court cases.
Nearly all judicial decisions are made only on the balance of probabilities. This standard recognises that we can seldom be certain we are right, and that in the interests of finality a decision on the balance of probability is good enough. Hugely important decisions are regularly made in the courts on that basis.
It is significant that Dr Fisher has not endeavoured to decide whether Ian Binnie's conclusions were right or wrong, but that he correctly restricted his report to Ian Binnie's method. Plainly, if Dr Fisher were to go further in a subsequent report, he would apply the Bayesian approach. But that should lead to the same conclusions that Ian Binnie reached unless radically inappropriate assessments of likelihood were made by Ian Binnie over critical facts to such an extent as to influence the result of considering the combined probative values of all the relevant facts.
I was surprised when reading Ian Binnie's report at how he treated the evidence of the luminol footprints. Depending on the length of those footprints, they could have removed the case from being an exercise in assessing probabilities and made this a case of direct evidence of innocence. That was recognised in the Privy Council hearing, as both sides agree. However Ian Binnie has been generous to the prosecution by recognising some doubt over the accuracy of the measurement made by the police scientist of the footprints on the carpet at the scene.
Whether the measurement could really have been susceptible to error to an extent sufficient to cast doubt on who left it there is a matter of judgment for those who have looked at the evidence that was given on this point.
At the measured 280mm the footprints were exact matches for prints that would be left by Robin Bain's foot.
If all the other evidence in the case proved guilt to a probability of 0.95, the footprint evidence would reduce that to 0.80; if all the other evidence in the case proved guilt to a probability of 0.99, the footprint evidence would reduce that to 0.96. So when I say there is "no real possibility that he is guilty" I do acknowledge that 0.96 may be proof beyond reasonable doubt for some people. The estimates of probabilities are used as follows. The first question is, what is the probability of getting these footprints, on the assumption that David is guilty? Experiments showed that it is most unlikely that David's foot could have left prints of that size, so this probability might be assessed as, say, 0.25, which seems rather generous to the prosecution. The second question is, what is the probability of getting these footprints, on the assumption that David is innocent? Because they fit well with the size of Robin's feet, this probability may be close to 1. The third question is, what is the ratio of these probabilities? This ratio, the likelihood ratio, reflects the probative value of the footprint evidence, and with the first as the numerator it is the probative value of the evidence for the prosecution case. It is, on these assessed figures, approximately 0.25. That is the assumption I make when I say that the footprint evidence reduces the probability of David being guilty. The assumption may be far too generous to the prosecution, because the probability that David's foot could have left a print of the size discovered could well be much less that 0.25: in none of the tests did his foot leave a print of that size. There is a possibility that a stretched sock may move under the sole of the foot so that its heel is closer to the toes, and so produce a shorter blood print than the foot wearing it, and I assume that the people who carried out the tests were alert to this.
More relevantly to a civil standard of proof, for a probability of David's guilt of 0.49, thus qualifying him for compensation, and for all the other evidence in the case suggesting a probability of guilt of 0.95, the probability of his foot leaving a print of the size found would need to be 0.05. That is, out of every 100 footprints David made, 5 would be of the size found at the scene. Given that there is no scientific evidence that he ever left the size of footprints found, it should seem reasonable to allow that he might do so no more than 5 times in a hundred. In science, measurements are routinely considered acceptable if they are in the plus or minus 5% range, although obviously greater accuracy is preferred. To disqualify himself for compensation he would need to leave footprints of the size found at least 6 times in every hundred footprints.
[Update: on 2 August 2016 a second and final report was published. At  its author, the Hon Ian Callinan QC (formerly of the High Court of Australia), states: "... the question is not whether the case could or could not accommodate the presence of Mr Robin Bain's footprints, but the reliability and probative value of the evidence of the footprints themselves in the light of all the evidence." If a criticism of this is to be made it would be that it could mean that the question of what the measurements were is determined by all the evidence in the case. Indeed, Mr Callinan appears to consider evidence other than that of footprint measurements as part of his evaluation of the evidence of their length, and says at  that the footprints are "inconclusive". In my opinion it is necessary first to determine what the evidence is, before turning to the question of its probative value. Mr Callinan does not demonstrate why the footprint evidence does not have the importance that it was acknowledged to have at the Privy Council hearing. He neglects to get to grips with what the evidence is: the objective evidence is the experimental results obtained by Drs Walsh and "Sandilands" [Binnie sic, Sandiford], see Binnie at -. The range of prints obtained from a 300mm foot would be 288-310 (Walsh) or 300-315 (Sandiford). And the range for a 270mm foot would be 258-280 (Walsh). None of the results gave a print shorter than the foot by 15mm which is what would be needed on all the assumptions favourable to the Crown and allowing a range of error for the measurement at the scene of plus or minus 5mm. The conclusion should have been that David was innocent, certainly on the balance of probabilities. It is easy to find support in the circumstantial evidence for any conclusion, but the footprints were direct evidence of who the killer was.]