Bayesian probability theory banned by English court

5 Aug, 2014 at 16:58 | Posted in Statistics & Econometrics | 4 Comments

In a recent judgement the English Court of Appeal has 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 …

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’ …

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 … Either it has or it has not“ …

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.

David Spiegelhalter

David Siegelhalter should of course go on with his course, but maybe he also ought to contemplate the rather common fact that people — including scientists — often find it possible to believe things although they can’t always warrant or justify their beliefs. And — probabilistic nomological machines do not exist “out there” and so is extremely difficult to properly apply to idiosyncratic real world events (such as fires).

As I see it, Bayesian probabilistic reasoning in science reduces questions of rationality to questions of internal consistency (coherence) of beliefs, but — even granted this questionable reductionism — it’s not self-evident that rational agents really have to be probabilistically consistent. There is no strong warrant for believing so. Rather, there are strong evidence for us encountering huge problems if we let probabilistic reasoning become the dominant method for doing research in social sciences on problems that involve risk and uncertainty.

In many  situations one could argue that there is simply not enough of adequate and relevant information to ground beliefs of a probabilistic kind, and that in those situations it is not really possible, in any relevant way, to represent an individual’s beliefs in a single probability measure.

Say you have come to learn (based on own experience and tons of data) that the probability of you becoming the next president in US is 1%. Having moved to Italy (where you have no own experience and no data) you have no information on the event and a fortiori nothing to help you construct any probability estimate on. A Bayesian would, however, argue that you would have to assign probabilities to the mutually exclusive alternative outcomes and that these have to add up to 1 — if you are rational. That is, in this case — and based on symmetry — a rational individual would have to assign probability 1% of becoming the next Italian president and 99% of not so.

That feels intuitively wrong though, and I guess most people would agree. Bayesianism cannot distinguish between symmetry-based probabilities from information and symmetry-based probabilities from an absence of information. In these kinds of situations most of us would rather say that it is simply irrational to be a Bayesian and better instead to admit that we “simply do not know” or that we feel ambiguous and undecided. Arbitrary and ungrounded probability claims are more irrational than being undecided in face of genuine uncertainty, so if there is not sufficient information to ground a probability distribution it is better to acknowledge that simpliciter, rather than pretending to possess a certitude that we simply do not possess.

I think this critique of Bayesianism is in accordance with the views of Keynes’s A Treatise on Probability (1921) and General Theory (1937). According to Keynes we live in a world permeated by unmeasurable uncertainty – not quantifiable stochastic risk – which often forces us to make decisions based on anything but rational expectations. Sometimes we “simply do not know.” Keynes would not have accepted the view of Bayesian economists, according to whom expectations “tend to be distributed, for the same information set, about the prediction of the theory.” Keynes, rather, thinks that we base our expectations on the confidence or “weight” we put on different events and alternatives. To Keynes expectations are a question of weighing probabilities by “degrees of belief”, beliefs that have preciously little to do with the kind of stochastic probabilistic calculations made by the rational agents modeled by probabilistically reasoning Bayesians.