# Bayesian Statistics Ruled Illegal?

Montford formed this opinion from a piece by David Spiegelhalter, who is a named person in statistics, and who writes at the Understanding Uncertainty blog. Spiegelhalter’s “Court of Appeal bans Bayesian probability (and Sherlock Holmes)” concluded the English Court of Appeal “denied that probability can be used as an expression of uncertainty for events that have either happened or not.” Spiegelhalter worried for his students.

I’m not, as Rumpole would say, though readers might dispute, a “Queer Customer”, i.e. a Queen’s Counsel, a sort of superior barrister, but my reading of the case is that, while one judge did make a mistake, Bayesian philosophy has not been ruled illegal.

The Court of Appeal’s decision tells us an electrician named Nulty was working on electrical equipment at a recycling center and, without finishing, went to a canteen for a smoke break. He returned to find the center on fire. Where there’s smoke, there’s fire, and where there’s fire, there’s lawsuits.

The judge of the original case decided that the equipment, left in an imperfect state, was not likely to have caused the fire, but that a discarded cigarette butt was more likely to have done. He also ruled that the cigarette butt belonged to Nulty, and thus Nulty was the cause of the fire. (Nulty was dead by the time the case came to court.)

Where there’s lawsuits, there’s appeals, and where there’s appeals that involve Bayesian philosophy, there’s blog posts.

The Court of Appeals’s decision was to reject the appeal and to conclude the original judge had done a proper job. The original judge used (thank the Lord) non-quantitative Bayesian reasoning, perhaps imperfectly, but Bayes nonetheless. Since it was Bayes, and since the Court of Appeals upheld the trial judge’s decision, Bayes is in no danger.

All parties agreed that the cigarette, electrical equipment, or perhaps a vague something else (intruder, random arsonist, etc.) caused the fire, though the mysterious intruder theory was eventually abandoned. How they arrived at this is not of direct interest to us. The original judge said that, considered by itself, it was unlikely the cigarette caused the fire. And considered by itself, it was unlikely the equipment did. But given a choice between butt and equipment, the butt was much more likely.

That’s Bayesian reasoning all right. (Bayes is forming probabilities assuming certain fixed evidence: when the evidence changes, the probabilities change.)

If our evidence is that we have a 1,000-sided object with just one side labeled “fire” and an 10,000-sided object with just one side labeled “fire”, and if one side on both must show when tossed, then the Bayesian probability the 1,000-sided object shows “fire” is 1/1000, which can be called unlikely. It is also unlikely the 10,000-sided object would show “fire”; indeed, more unlikely.

But if I now add evidence that one and only one of these objects showed “fire”, but I do not tell you which, then you are right to guess it was the 1,000-sided object; it is 10 times likelier.

The appellate judge quoted a case in which Sherlock Holmes appeared: “when you have eliminated the impossible, whatever remains, however improbable must be the truth?” This is Bayesian, too, but in a more complicated, subtle way. Too subtle to do all in one post, so we shall save it for another day.

What rankled Spiegelhalter was when the appellate judge said:

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.

This is wrong (in part). A doctor, citing certain evidence, can say a patient has an X% chance of cancer. But we can form probabilities for any proposition, even those concerning events which happened in the past. For example, ten minutes ago (longer by the time you read this) I flipped a coin. Do you know which side landed? You do not; you are uncertain; you express this uncertainty with probability. Saying “it was a head or it wasn’t” is a tautology, the probability of which is 1, or 100%. Adding tautologies to evidence never changes any probability conclusion.

It’s obvious this judge made his blunder because of his (entirely reasonable) suspicion of quantification; he speaks of the real “danger of pseudo-mathematics”. The discussion of the “balance of probability” test referenced in the appeal came too close to quantification for him, hence his perfunctory statement. However it is clear he prefers (Bayesian) probability done without numbers, which is plain in his opinion, in which he restated that the trial judge’s (Bayesian) reasoning was spot on. (Two brother judges concurred.)

**Update** “Bayes” was nowhere mentioned in the opinion which, if you’re looking for cheering news, is well written, concise, jargon free, and serious. It’s a pleasure to see these judges’ honest struggle to discover the truth.

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*Thanks to Gareth, Mike Gee, Roger Cohen, K.A. Rodgers, and the others who suggested this post.*

I think you mean to say, “Benedict Timothy Carlton Cumberbatch.”

If the result of the coin you tossed was heads then there is 100% chance that it came up heads, while if it came up tails there is a 100% chance that it came up tails. What those of us who are not Wm. Briggs are expressing through probability is the limits of our knowledge of the coin toss, not the results of the toss itself. To speak properly we should say “there is a 50% chance that based on the information available that the result of the coin toss was heads and a 50% chance that the result was a tails.” We are not even faced with probabilistic animal cruelty charges ala Schrodinger’s Cat because the result of the toss has had an impact on the universe, just not an effect that provides we who are not Wm. Briggs enough information to determine the results of the coin toss.

The offensive comment in question was directed towards confusing the “preponderance of evidence” (“balance of probability” in UK usage) standard with the magic “more than 50%” formula. It is long established that the “preponderance of the evidence” standard can be meet with less than 50% of the evidence is there are more than 2 alternatives, and the court was restating this. Discounting the known problems with coin flipping (more likely to land same side that was up when flipped, high energy plasma beam intercepting coin in the air and instantly vaporizing it, etc) a completely fair coin toss will not meet the magic “more than 50%”, rejecting Lord Toulson’s argument means that a coin toss has no legal existence.

To muddy the waters further, consider Schrodinger’s Cat. The Cat is into the box and left there for a certain period of time. We are told that there is a 50% chance that the poisoning event has occurred and the cat has died. It either has or has not. We just don’t know. We open the box and find out. Was the cat alive or dead before we opened the box?

Noblesse Oblige,

“Was the cat alive or dead before we opened the box?”

Yes.

NO,

Whether the cat lives or dies is a future event for the external universe (outside the box) until the sealed universe (box) is opened. That’s the point of Schrodinger’s thought experiment, until an event has an effect on the universe it is still in the future – the passage of time occurs when there is an effect on the universe. If M. Brigg’s had performed his coin toss in a sealed universe which is still sealed then it would be wrong to refer to the coin toss in the past tense, since as far as the external universe is concerned, the coin has yet to be tossed – properly speaking we would say that several hours ago (at this point) a coin will be tossed in a sealed universe, and when it is opened several hours will have passed in the previously sealed universe.

Max,

The question is not about an event which must be in the future or the past, but about the state of an entity which can be in any tense.

Answering the question “Was the cat alive or dead before we opened the box?” No implies that the cat was neither dead nor alive. Do you think it was some kind of zombie cat?

Alive vs dead is a binary condition. A living entity must be one or the other, there is no in between. There fore the answer is yes, the cat was either alive or dead.

max,

Much better to think of it as personal information. You make it sound like a single observer sets the probability for the entire universe.

The cat’s satus is only fuzzy to those who have not yet looked inside the box. It becomes crisp only to those who have. The rest are stuck with various probabilities for the cat’s status. It may be other than 50/50 based on what any potential observer knows. For example, one could ask someone who did look but perhaps know that the askee is 95% reliable. The cat’s status then moves to 95/5 (or 5/95).

Note that the observer is yourself and no one else.

and just to add: asking someone is, in practice, no different than consulting sensor apparatus.

DAV,

are you saying Schrodinger was wrong? Yes, any single observer sets the “probability” for the entire universe, but that probability is one, when something is observed to exist it exists – this all comes out of the implications of Einstein’s general relativity, take it up with Einstein if you have a problem. As soon as the box is open it makes changes in the universe and an astute enough observer can determine the state of the cat by the changes made in the universe without direct observation of the cat. Prior to the box being opened even with absolute knowledge of the external universe an observer cannot tell the state of the cat because there is no change in the external universe. Time, in effect, in the sealed universe is frozen as viewed from the external universe until it is opened, when all the time passes at once for an observer in the external universe.

If you ask someone who is 95% reliable it does not change the actual state of the cat one iota, all that changes is your ability to correctly guess the actual state of the cat. After asking the 95% reliable person the proper answer is that based upon the limited information available there is a 95% chance that the cat is 100% dead (alive) and a 5% chance the cat is 100% alive (dead). Think of it like the Monty Hall problem, the prizes behind the doors do not change but the intelligent player’s ability to win the big prize increases as more information becomes available, likewise the state of the cat does not change but the ability to correctly guess does as more information becomes available.

@max

Instead of putting a cat in the box, go sit yourself in it. And instead of releasing a poisonous gas, we release a gas that will make you fall asleep, in order to get the experiment past the ethical commission.

Now, the changes that somebody else will find you asleep or not after opening the box are the same as the change of finding a dead cat.

Now, if the other person opens the box and finds you awake then you will have been awake the whole time during the experiment. And if he finds you asleep, you will have fallen asleep because the gas had been released. Make the sleep gas a bit slow-acting and you would have observed yourself getting drowsy, falling into sleep. So you have observed the whole situation, until the lid opened, or until you were falling into sleep and later being woken up.

The situation that the universe created some weird wave function never occurred, but as far as the observer outside the box is concerned, nothing has changed. He did not look into the box so he did not know whether you were asleep or not until he looked.

max,

That’s just a bizarre way of looking at things. Until you know, it doesn’t make a bit of difference what others know.

Suppose, for example, some event gets triggered by the status of the cat. Are you claiming that event will not happen until someone looks in the box? Really? Or does the event act as an observer?

I don’t know if Schrodinger was wrong. Mostly because I don’t know if he truly intended what you are claiming. As an hypothesis (the need for an observer), how would you ever prove it? It looks untestable to me.

Perhaps we are talking past each other. There is no such thing as in-between alive or dead but the question really boils down to how much you actually know which is what I think Schrodinger was really trying to illustrate.

M.s van der Wal & DAV:

Yes, but while I am in the box (asleep or awake) nothing happens in the external universe and no time passes (relative to me, an observer in the box) until the box is opened. If we accept that rate of time passing is determined by the motion of an observer relative to the observed event (from the Theory of General Relativity) then, if the event and the observer are isolated, time cannot pass at the point of the event from the observer’s viewpoint and the event cannot occur as far as the observer is concerned. This is exactly what Schrodinger set out to show with his cat, a problem with relativity – if time passes when the motion of an observer relative to an event is infinite then General Relativity is wrong (or at least incomplete).

And yes, it is a bizarre way of looking at things but it is unavoidable consequence of Einstein when subjected to reducio ad absurdium. There have been a few proposed solutions; I’m fond of the alternative universe creation theory, inside the box time passes but two universes are created inside the box – one in which the cat lives and one in which the cat dies, when the universe/box is opened one alternative universe dies and the other is absorbed into the external universe. Because the Schrodinger’s Cat problem requires isolating the observer from the event it is impossible to test the solutions against observations, at this point we are beyond the realm of science and into metaphysics.

@max, et al: The Schroedinger’s Cat example was intended to show that a particular interpretation of quantum mechanics was absurd, not to actually reflect reality.

The Cat is either dead or alive in the box. If the box is sufficiently magical (i.e. its contents are totally and absolutely isolated from the rest of the universe) the life or death of the cat can’t influence the rest of the universe but that doesn’t mean it hasn’t happened.

As an example, assume the cat is on a distant planet — let’s say 100 light years away. I could ask you if you think the cat is alive or dead now, but it’s physically impossible for either of us to know until 100 years have passed. That doesn’t mean that the cat lives or dies 100 years in the future, it simply means that we cannot know until 100 years from now whether the cat is actually alive or dead at this moment.

Same with the box: the fact that we don’t know until we look does not mean that the cat is alive, dead, alive-and-dead, or neither-alive-nor-dead at this moment. It’s a matter of knowledge, and perhaps influence, but not fact.

Wayne:

yes it was directly in response to the ’35 EPR paper opposed to quantum entanglement. The EPR paper argued that quantum entanglement does not exist and that apparent non-relativistic interactions were in fact relativistic interactions through a yet-to-be-discovered process. Schrodinger’s Cat and it’s ilk were used to argue that quantum entanglement does exist and the theory of general relativity is wrong (“incomplete” is perhaps better) and produces absurd results. By extending the EPR anti-quantum entanglement argument (or perhaps Einstein’s later sole argument, been a while since I read them) I was trying to point out the weakness of general relativity re quantum mechanics (QM). I think it has been pretty well established that quantum mechanics is not bound by relativity (actual experiments go back over 3 decades showing quantum entanglement at super-relativistic speeds), but by presenting a description of how reality would work if QM were bound by relativity I was hoping to goad readers into rejecting general relativity.