Probability as a guide to life

Probability is the very guide to life.’

Cicero may have been right, but ‘probability’ means something quite different nowadays to what it did millennia ago. So what kind of probability is a suitable guide to life, and when?

Suppose that we are told that ‘P(X) = p’. Often there is some implied real or virtual population, P, a proportion ‘p’ of which has the property ‘X’. To interpret such a probability statement we need to know what the relevant population is. Such statements are then normally reliable. More controversial are conditional probabilities, such as ‘P(X|Y) = p’. If you satisfy Y, does P(X)=p ‘for you’?

Suppose that:

1. All the properties of interest (such as X and Y) can be expressed as union of some disjoint basis, B.
2. For all such basis properties, B, P(X|B) is known.
3. That the conditional probabilities of interest are derived from the basis properties in the usual way. (E..g. P(X|B1ÈB2) = P(B1).P(X|B1)+P(B2).P(X|B2)/P(B1ÈB2).)

The conditional probabilities constructed in this way are meaningful, but if we are interested in some other set, Z, the conditional probability P(X|Z) could take a range of values. But then we need to reconsider decision making. Instead of maximising a probability (or utility), the following heuristics that may apply:

• If the range makes significant difference, try to get more precise data. This may be by taking more samples, or by refining the properties considered.
• Consider the best outcome for the worst-case probabilities.
• If the above is not acceptable, make some reasonable assumptions until there is an acceptable result possible.

For example, suppose that some urn, each contain a mix of balls, some of which are white. We can choose an urn and then pick a ball at random. We want white balls. What should we do. The conventional rule consists of assessing the proportion of white balls in each, and picking an urn with the most. This is uncontroversial if our assessments are reliable. But suppose we are faced with an urn with an unknown mix? Conventionally our assessment should not depend on whether we want to obtain or avoid a white ball. But if we want white balls the worst-case proportion is no white balls, and we avoid this urn, whereas if we want to avoid white balls the worst-case proportion is all white balls, and we again avoid this urn.

If our assessments are not biased then we would expect to do better with the conventional rule most of the time and in the long-run. For example, if the non-white balls are black, and urns are equally likely to be filled with black as white balls, then assessing that an urn with unknown contents has half white balls is justified. But in other cases we just don’t know, and choosing this urn we could do consistently badly. There is a difference between an urn whose contents are unknown, but for which you have good grounds for estimating proportion, and an urn where you have no grounds for assessing proportion.

If precise probabilities are to be the very guide to life, it had better be a dull life. For more interesting lives imprecise probabilities can be used to reduce the possibilities. It is often informative to identify worst-case options, but one can be left with genuine choices. Conventional rationality is the only way to reduce living to a formula: but is it such a good idea?

Dave Marsay

Uncertainty is not just probability

I have just had published my paper, based on the discussion paper referred to in a previous post. In Facebook it is described as:

An understanding of Keynesian uncertainties can be relevant to many contemporary challenges. Keynes was arguably the first person to put probability theory on a sound mathematical footing. …

So it is not just for economists. I could be tempted to discuss the wider implications.

Comments are welcome here, at the publisher’s web site or on Facebook. I’m told that it is also discussed on Google+, Twitter and LinkedIn, but I couldn’t find it – maybe I’ll try again later.

Dave Marsay

Instrumental Probabilities

Reflecting on my recent contribution to the economics ejournal special issue on uncertainty (comments invited), I realised that from a purely mathematical point of view, the current mainstream mathematical view, as expressed by Dawid, could be seen as a very much more accessible version of Keynes’. But there is a difference in expression that can be crucial.

In Keynes’ view ‘probability’ is a very general term, so that it always legitimate to ask about the probability of something. The challenge is to determine the probability, and in particular whether it is just a number. In some usages, as in Kolmogorov, the term probability is reserved for those cases where certain axioms hold. In such cases the answer to a request for a probability might be to say that there isn’t one. This seems safe even if it conflicts with the questioner’s presuppositions about the universality of probabilities. In the instrumentalist view of Dawid, however, suggests that probabilistic methods are tools that can always be used. Thus the probability may exist even if it does not have the significance that one might think and, in particular, it is not appropriate to use it for ‘rational decision making’.

I have often come across seemingly sensible people who use ‘sophisticated mathematics’ in strange ways. I think perhaps they take an instrumentalist view of mathematics as a whole, and not just probability theory. This instrumentalist mathematics reminds me of Keynes’ ‘pseudo-mathematics’. But the key difference is that mathematicians, such as Dawid, know that the usage is only instrumentalist and that there are other questions to be asked. The problem is not the instrumentalist view as such, but the dogma (of at last some) that it is heretical to question widely used instruments.

The financial crises of 2007/8 were partly attributed by Lord Turner to the use of ‘sophisticated mathematics’. From Keynes’ perspective it was the use of pseudo-mathematics. My view is that if it is all you have then even pseudo-mathematics can be quite informative, and hence worthwhile. One just has to remember that it is not ‘proper’ mathematics. In Dawid’s terminology  the problem seems to be that the instrumental use of mathematics without any obvious concern for its empirical validity. Indeed, since his notion of validity concerns limiting frequencies, one might say that the problem was the use of an instrument that was stunningly inappropriate to the question at issue.

It has long seemed  to me that a similar issue arises with many miscarriages of justice, intelligence blunders and significant policy mis-steps. In Keynes’ terms people are relying on a theory that simply does not apply. In Dawid’s terms one can put it blunter: Decision-takers were relying on the fact that something had a very high probability when they ought to have been paying more attention to the evidence in the actual situation, which showed that the probability was – in Dawid’s terms – empirically invalid. It could even be that the thing with a high instrumental probability was very unlikely, all things considered.

Artificial Intelligence?

The subject of ‘Artificial Intelligence’ (AI) has long provided ample scope for long and inconclusive debates. Wikipedia seems to have settled on a view, that we may take as straw-man:

Every aspect of learning or any other feature of intelligence can be so precisely described that a machine can be made to simulate it. [Dartmouth Conference, 1956] The appropriately programmed computer with the right inputs and outputs would thereby have a mind in exactly the same sense human beings have minds. [John Searle’s straw-man hypothesis]

Readers of my blog will realise that I agree with Searle that his hypothesis is wrong, but for different reasons. It seems to me that mainstream AI (mAI) is about being able to take instruction. This is a part of learning, but by no means all. Thus – I claim – mAI is about a sub-set of intelligence. In many organisational settings it may be that sub-set which the organisation values. It may even be that an AI that ‘thought for itself’ would be a danger. For example, in old discussions about whether or not some type of AI could ever act as a G.P. (General Practitioner – first line doctor) the underlying issue has been whether G.P.s ‘should’ think for themselves, or just apply their trained responses. My own experience is that sometimes G.P.s doubt the applicability of what they have been taught, and that sometimes this is ‘a good thing’. In effect, we sometimes want to train people, or otherwise arrange for them to react in predictable ways, as if they were machines. mAI can create better machines, and thus has many key roles to play. But between mAI and ‘superhuman intelligence’  there seems to be an important gap: the kind of intelligence that makes us human. Can machines display such intelligence? (Can people, in organisations that treat them like machines?)

One successful mainstream approach to AI is to work with probabilities, such a P(A|B) (‘the probability of A given B’), making extensive use of Bayes’ rule, and such an approach is sometimes thought to be ‘logical’, ‘mathematical, ‘statistical’ and ‘scientific’. But, mathematically, we can generalise the approach by taking account of some context, C, using Jack Good’s notation P(A|B:C) (‘the probability of A given B, in the context C’). AI that is explicitly or implicitly statistical is more successful when it operates within a definite fixed context, C, for which the appropriate probabilities are (at least approximately) well-defined and stable. For example, training within an organisation will typically seek to enable staff (or machines) to characterise their job sufficiently well for it to become routine. In practice ‘AI’-based machines often show a little intelligence beyond that described above: they will monitor the situation and ‘raise an exception’ when the situation is too far outside what it ‘expects’. But this just points to the need for a superior intelligence to resolve the situation. Here I present some thoughts.

When we state ‘P(A|B)=p’ we are often not just asserting the probability relationship: it is usually implicit that ‘B’ is the appropriate condition to consider if we are interested in ‘A’. Contemporary mAI usually takes the conditions a given, and computes ‘target’ probabilities from given probabilities. Whilst this requires a kind of intelligence, it seems to me that humans will sometimes also revise the conditions being considered, and this requires a different type of intelligence (not just the ability to apply Bayes’ rule). For example, astronomers who refine the value of relevant parameters are displaying some intelligence and are ‘doing science’, but those first in the field, who determined which parameters are relevant employed a different kind of intelligence and were doing a different kind of science. What we need, at least, is an appropriate way of interpreting and computing ‘probability’ to support this enhanced intelligence.

The notions of Whitehead, Keynes, Russell, Turing and Good seem to me a good start, albeit they need explaining better – hence this blog. Maybe an example is economics. The notion of probability routinely used would be appropriate if we were certain about some fundamental assumptions. But are we? At least we should realise that it is not logical to attempt to justify those assumptions by reasoning using concepts that implicitly rely on them.

Dave Marsay

Who thinks probability is just a number? A plea.

Many people think – perhaps they were taught it – that it is meaningful to talk about the unconditional probability of ‘Heads’ (I.e. P(Heads)) for a real coin, and even that there are logical or mathematical arguments to this effect. I have been collecting and commenting on works which have been – too widely – interpreted in this way, and quoting their authors in contradiction. De Finetti seemed to be the only example of a respected person who seemed to think that he had provided such an argument. But a friendly economist has just forwarded a link to a recent work that debunks this notion, based on wider  reading of his work.

So, am I done? Does anyone have any seeming mathematical sources for the view that ‘probability is just a number’ for me to consider?

I have already covered:

There are some more modern authors who make strong claims about probability, but – unless you know different – they rely on the above, and hence do not need to be addressed separately. I do also opine on a few less well known sources: you can search my blog to check.

Dave Marsay

Coin toss puzzle

This is intended as a counter-example to the view, such as Savage’s, that uncertainty can, in practice, be treated as numeric probability.

You have a coin that you know is fair. A known trickster (me?) shows you what looks like an ordinary coin and offers you a choice of the following bets:

1. You both toss your own coins. You win if they match, otherwise they win.
2. They toss their coin while you call ‘heads’ or ‘tails’.

Do you have any preference between the two bets? Why? And …

In each case, what is the probability that their coin will come up heads?

Dave Marsay

Clarification

In (1) suppose that you can arrange things so that the trickster cannot tell how your coin will land in time to influence their coin, so that the probability of a match is definitely 0.5, with no uncertainty. The situation in (2) can be similar, except that your call replaces the toss of a fair coin.

Other uncertainty puzzles .

Uncertain Urns Puzzle

A familiar probability example, using urns, is adapted to illustrate ‘true’ (non-numeric) uncertainty.

Simple situation

The following is a good teaching example:

Suppose that an urn is known to contain black and white balls that are otherwise identical. A subject claims to be able to predict the colour of a ball that they draw ‘at random’.

They ‘predict’ and draw a black ball. What are the odds that they are really able to predict?

From a Bayesian perspective, the final odds are the initial odds times the likelihood ratio. If there are b black and w balls and we represent the evidence by E and likelihoods by P( E | ), then P( E | Predict ) = 1 and P( E | Luck) = b/(b+w). Thus the rarer the phenomenon predicted, the more a correct prediction tends to support the claim, of reliable prediction.

Common quibbles

There is, however, some subjectivity in the estimated probability that the subject can predict:

• In this case, the initial odds seem somewhat arbitrary, and Bayes’ rule seems not to apply. For example, have you considered that the different colours may result in different temperatures? Such a thought is not ‘evidence’ in the sense of Bayes’ rule, but might change your subjective estimate of the probability prior to their draw.
• If we do not know the proportions of black and white balls for sure then the likelihood is uncertain.

Multiple urns

Here we introduce a different type of uncertainty:

Suppose now that the subject is faced with two urns and selects a ball from one. Given the number of black and white balls in each urn, what is the likelihood, P( E | Luck ), of a correct prediction due to luck?

If you think the question is ambiguous, please disambiguate it however you wish.

Suppose you know the total numbers of black and white balls in the two urns. Is the likelihood estimate P( E | Luck) = b/(b+w) reasonable? Could it be biased? How?