..when you have eliminated the impossible, whatever remains, however improbable, must be the truth

(Sherlock Holmes in The Sign of the Four, ch. 6, 1890)

In a recent judgement the English Court of Appeal has not only rejected the Sherlock Holmes doctrine shown above, but also denied that probability can be used as an expression of uncertainty for events that have either happened or not.

The case was a civil dispute about the cause of a fire, and concerned an appeal against a decision in the High Court by Judge Edwards-Stuart. Edwards-Stuart had essentially concluded that the fire had been started by a discarded cigarette, even though this seemed an unlikely event in itself, because the other two explanations were even more implausible. The Court of Appeal rejected this approach although still supported the overall judgement and disallowed the appeal - commentaries on this case have appeared here and here.

But it's the quotations from the judgement that are so interesting:

Sometimes the "balance of probability" standard is expressed mathematically as "50 + % probability", but this can carry with it a danger of pseudo-mathematics, as the argument in this case demonstrated. When judging whether a case for believing that an event was caused in a particular way is stronger that the case for not so believing, the process is not scientific (although it may obviously include evaluation of scientific evidence) and to express the probability of some event having happened in percentage terms is illusory.

The idea that you can assign probabilities to events that have already occurred, but where we are ignorant of the result, forms the basis for the Bayesian view of probability. Put very broadly, the 'classical' view of probability is in terms of genuine unpredictability about future events, popularly known as 'chance' or 'aleatory uncertainty'. The Bayesian interpretation allows probability also to be used to express our uncertainty due to our ignorance, known as 'epistemic uncertainty', and popularly expressed as betting odds. Of course there are all gradations, from pure chance (think radioactive decay) to processes assumed to be pure chance (lottery draws), to future events whose odds depend on a mixture of genuine unpredictability and ignorance of the facts (whether Oscar Pistorius will be convicted of murder), to pure epistemic uncertainty (whether Oscar Pistorius knowingly shot his girlfriend).

The judges went on to say:

The chances of something happening in the future may be expressed in terms of percentage. Epidemiological evidence may enable doctors to say that on average smokers increase their risk of lung cancer by X%. But you cannot properly say that there is a 25 per cent chance that something has happened: Hotson v East Berkshire Health Authority [1987] AC 750. Either it has or it has not.

So according to this judgement, it would apparently not be reasonable in a court to talk about the probability of Kate and William's baby being a girl, since that is already decided as true or false (but see note added below). This seems extraordinary.

Part of the problem may be the judges' use of the word 'chance' to describe epistemic uncertainty about whether something has happened or not - this would be unusual usage now (even though Thomas Bayes used 'chance' in this sense). If they had used the term 'probability' perhaps their quote above would seem more clearly unreasonable.

Anyway, I teach the Bayesian approach to post-graduate students attending my 'Applied Bayesian Statistics' course at Cambridge, and so I must now tell them that the entire philosophy behind their course has been declared illegal in the Court of Appeal. I hope they don't mind.

(Note added 1st March 2013: William Hill are currently offering 1000-1 against Chardonnay as the name of the potential future monarch).