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

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The limits of pragmatism

This is a personal attempt to identify and articulate a fruitful form of pragmatism, as distinct from what seems to me the many dangerous forms. My starting point is Wikipedia and my notion that the differences it notes can sometimes matter.

Doubt, like belief, requires justification. Genuine doubt irritates and inhibits, in the sense that belief is that upon which one is prepared to act.[2] It arises from confrontation with some specific recalcitrant matter of fact (which Dewey called a “situation”), which unsettles our belief in some specific proposition. Inquiry is then the rationally self-controlled process of attempting to return to a settled state of belief about the matter. Note that anti-skepticism is a reaction to modern academic skepticism in the wake of Descartes. The pragmatist insistence that all knowledge is tentative is quite congenial to the older skeptical tradition

My own contribution to things scientific has been on some very specific issues, but which I attempt to generalise:

  • It is sometimes seems much too late to wait to act on doubt for something that pragmatic folk recognize as a ‘specific recalcitrant matter of fact’. I would rather say (with the skeptics) that we should always be in some doubt, but that our actions require justification, and should only invest in relation to that justification. Requiring ‘facts’ seems to high a hurdle to act at all.
  • Psychologically, people do seek ‘settled states of belief’, but I would rather say (with the skeptics) that the degree of settledness ought to be only in so far as is justified. Relatively settled belief but not fundamentalist dogma!
  • It is often supposed that ‘facts’ and ‘beliefs’ should concern the ‘state’ of some supposed ‘real world’. There is some evidence that it is ‘better’ in some sense to think of the world as one in which certain processes are appropriate. In this case, as in category theory, the apparent state arises as a consequence of sufficient constraints on the processes. This can make an important difference when one considers uncertainties, but in ‘small worlds’ there are no such uncertainties.

It seems to me that the notion of ‘small worlds’ is helpful. A small world would be one which could be conceived of or ‘mentally modelled’. Pragmatists (of differing varieties) seem to believe that often we can conceive of a small world representation of the actual world, and act on that representation ‘as if’ the world were really small. So far, I find this plausible, even if not my own habit of thinking. The contentious point, I think, is that in every situation we should do our best to from a small world representation and then act as if it were true unless and until we are confronted with some ‘specific recalcitrant matter of fact’. This can be too late.

But let us take the notion of  a ‘small world’ as far as we can. It is accepted that the small world might be violated. If it could be violated as a consequence of something that we might inadvertently do then it hardly seems a ‘pragmatic’ notion in terms of ordinary usage, and might reasonably said to be dangerous in so far as it lulls us into a false sense of security.

One common interpretation of ‘pragmatism’ seems to be that we may as well act on our beliefs as there seems no alternative. But I shall refute this by presenting one. Another interpretation is that there is no ‘practical’ alternative’. That is to say, whatever we do could not affect the potential violation of the small world. But if this is the case it seems to me that there must be some insulation between ourselves and the small world. Thus the small world is actually embedded in some larger closed world. But do we just suppose that we are so insulated, or do we have some specific closed world in mind?

It seems to me that doubt is more justified the less our belief in insulation is justified. Even when we have specific insulation in mind, we surely need to keep an open mind and monitor the situation for any changes, or any reduction in justification for our belief.

From this, it seems to me that (as in my own work) what matters is not having some small world belief, but in taking a view on the insulations between what you seek to change and what you seek to rely on as unchanging. And from these identifying not only a single credible world in which to anchor one’s justifications for action, but in seeking out credible possible small worlds in the hope that at least one may remain credible as things proceed.

Dave Marsay

See also my earlier thoughts on pragmatism, from a different starting point.

Which pragmatism as a guide to life?

Much debate on practical matters ends up in distracting metaphysics. If only we could all agree on what was ‘pragmatic’. My blog is mostly negative, in so far as it rubbishes various suggestions, but ‘the best is trhe enemy of the good’, and we do need to do something.

Unfortunately, different ‘schools’ start from a huge variety of different places, so it is difficult to compare and contrast approaches. But it is about time I had a go. (In part inspired by a recent public engagement talk on mathematics).

Suppose you have a method Π that you regard as pragmatic, in the sense that you can always act on it. To justify this, I think (like Popper) that you should have criteria , Γ, which if falsified would lead you to reconsider ∏ . So your pragmatic process is actually

If Γ then ∏ else reconsider.

But this is hardly reasonable if we try to arrange things so that Γ will never appear to be falsified. So an improvement is:

Spend some effort in monitoring Γ. If it is not falsified then ∏.

In practice if one thinks that Γ can be relied on, one may not think it worth spending much effort on checking it, but surely one should at least be open to suggestions that it could be wrong. The proper balance between monitoring Γ and acting on ∏ seems  impossible to establish with any confidence, but ignoring all evidence against Γ seems risky, to say the least.

Some argue that if you have no alternative to ∏  then it is pointless considering Γ. This may be a  reasonable argument when applied to concepts, but not to actions in the real world. Whatever evidence we may have for ∏ it will never uniquely prove it. It may be that it rules out all the alternatives that we have thought of, or which we consider credible or otherwise acceptable, but we should think again. Logically, there are always alternatives.

The above clearly applies to science. No theory is ever regarded as asolute and for ever. Scientists make their careers by identifying alternative theories to explain the experimental results and then devising new experiments to try to falsify the current theory. This process could only ever end when we were all sure that we had performed every possible experiment using every possible means in every possible circumstance, which implies the end of evolution and inventiveness. We aren’t there yet.

My proposal, then, is that very generally (not just in science) we ought to expect any ‘pragmatic’ ∏  to include a specific ‘caveat’, Γ(∏). If it doesn’t, we ought to develop one. This caveat will include its own rules for falsifying, tests, and we ought to regard more severe tests (in some sense) to be better. We then seek to develop alternatives that might be less precise (and hence less ‘useful’) than ∏ but which might survive falsification of ∏.

Much of my blog has some ideas on how tom do this in particular cases: a work in progress. But an example may appeal:

Faced with what looks like a coin being tossed we might act ‘as if’ we believe it to be fair and to correspond to the axioms of mathematical probability theory, but keep an eye out for evidence to the contrary. Perhaps we inspect it and toss it a few times. Perhaps we watch whoever tosses it carefully. We do what we can, but still if someone tosses it and over a very large runs gets an excess of ‘Heads’ that our statistical friends tell us is hugely significant, we may be suspicious and reconsider

In this case we may decline from gambling on coin tosses even if we lack a specific ‘theory of the coin’, but it might be better if we had an alternative theory. Perhaps it is an ingenious fake coin? Perhaps the person tossing it has a cunning technique to bias it? Perhaps the person tossing it is a magician, and is actually faking the results?

This seems to me a like a good approach, surely better than acting ‘pragmatically’ but without such falsifying criteria. Can it be improved upon? (Suggestions please!)

Dave Marsay

What logical term or concept ought to be more widely known?

Various What scientific term or concept ought to be more widely known? Edge, 2017.

INTRODUCTION: SCIENTIA

Science—that is, reliable methods for obtaining knowledge—is an essential part of psychology and the social sciences, especially economics, geography, history, and political science. …

Science is nothing more nor less than the most reliable way of gaining knowledge about anything, whether it be the human spirit, the role of great figures in history, or the structure of DNA.

Contributions

As against others on:

(This is as far as I’ve got.)

Comment

I’ve grouped the contributions according to whether or not I think they give due weight to the notion of uncertainty as expressed in my blog. Interestingly Steven Pinker seems not to give due weight in his article, whereas he is credited by Nicholas G. Carr with some profound insights (in the first of the second batch). So maybe I am not reading them right.

My own suggestion would be Turing’s theory of ‘Morphogenesis’. The particular predictions seem to have been confirmed ‘scientifically’, but it is essentially a logical / mathematical theory. If, as the introduction suggests, science is “reliable methods for obtaining knowledge” then it seems to me that logic and mathematics are more reliable than empirical methods, and deserve some special recognition. Although, I must concede that it may be hard to tell logic from pseudo-logic, and that unless you can do so my distinction is potentially dangerous.

Morphogenesis

The second law of thermodynamics, and much common sense rationality,  assumes a situation in which the law of large numbers applies. But Turing adds to the second law’s notion of random dissipation a notion of relative structuring (as in gravity) to show that ‘critical instabilities’ are inevitable. These are inconsistent with the law of large numbers, so the assumptions of the second law of thermodynamics (and much else) cannot be true. The universe cannot be ‘closed’ in its sense.

Implications

If the assumptions of the second law seem to leave no room for free will and hence no reason to believe in our agency and hence no point in any of the contributions to Edge: they are what they are and we do what we do. But Pinker does not go so far: he simply notes that if things inevitably degrade we do not need to beat ourselves up, or look for scape-goats when things go wrong. But this can be true even if the second law does not apply. If we take Turing seriously then a seeming permanent status quo can contain the reasons for its own destruction, so that turning a blind eye and doing nothing can mean sleep-walking to disaster. Where Pinker concludes:

[An] underappreciation of the Second Law lures people into seeing every unsolved social problem as a sign that their country is being driven off a cliff. It’s in the very nature of the universe that life has problems. But it’s better to figure out how to solve them—to apply information and energy to expand our refuge of beneficial order—than to start a conflagration and hope for the best.

This would seem to follow more clearly from the theory of morphogenesis than the second law. Turing’s theory also goes some way to suggesting or even explaining the items in the second batch. So, I commend it.

Dave Marsay

 

 

Heuristics or Algorithms: Confused?

The Editor of the New Scientist (Vol. 3176, 5 May 2018, Letters, p54) opined in response to Adrian Bowyer ‘swish to distinguish between ‘heuristics’ and ‘algorithms’ in AI that:

This distinction is no longer widely made by practitioners of the craft, and we have to follow language as it is used, even when it loses precision.

Sadly, I have to accept that AI folk tend to consistently fail to respect a widely held distinction, but it seems odd that their failure has led to an obligation on the New Scientist – which has a much broader readership than just AI folk. I would agree that in addressing audiences that include significant sectors that fail to make some distinction, we need to be aware of the fact, but if the distinction is relevant – as Bowyer argues, surely we should explain it.

According to the freedictionary:

Heuristic: adj 1. Of or relating to a usually speculative formulation serving as a guide in the investigation or solution of a problem.

Algorithm: n: A finite set of unambiguous instructions that, given some set of initial conditions, can be performed in a prescribed sequence to achieve a certain goal and that has a recognizable set of end conditions.

It even also this quote:

heuristic: of or relating to or using a general formulation that serves to guide investigation  algorithmic – of or relating to or having the characteristics of an algorithm.

But perhaps this is not clear?

AI practitioners routinely apply algorithms as heuristics in the same way that a bridge designer may routinely use a computer program. We might reasonably regard a bridge-designing app as good if it correctly implements best practice in  bridge-building, but this is not to say that a bridge designed using it would necessarily be safe, particularly if it is has significant novelties (as in London’s wobbly bridge).

Thus any app (or other process) has two sides: as an algorithm and as a heuristic. As an algorithm we ask if it meets its concrete goals. As a heuristic we ask if it solves a real-world problem. Thus a process for identifying some kind of undesirable would be regarded as good algorithmically if it conformed to our idea of the undesirables, but may still be poor heuristically. In particular, good AI would seem to depend on someone understand at least the factors involved in the problem. This may not always be the case, no matter how ‘mathematically sophisticated’ the algorithms involved.

Perhaps you could improve on this attempted explanation?

Dave Marsay

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

Why do people hate maths?

New Scientist 3141 ( 2 Sept 2017) has the cover splash ‘Your mathematical mind: Why do our brains speak the language of reality?’. The article (p 31) is titled ‘The origin of mathematics’.

I have made pedantic comments on previous articles on similar topics, to be told that the author’s intentions have been slightly skewed in the editing process. Maybe it has again. But some interesting (to me) points still arise.

Firstly, we are told that brain scans showthat:

a network of brain regions involved in mathematical thought that was activated when mathematicians reflected on problems in algebra, geometry and topology, but not when they were thinking about non-mathsy things. No such distinction was visible in other academics. Crucially, this “maths network” does not overlap with brain regions involved in language.

It seems reasonable to suppose that many people do not develop such a maths capability from experience in ordinary life or non-mathsy subjects, and perhaps don’t really appreciate its significance. Such people would certainly find maths stressful, which may explain their ‘hate’. At least we can say – contradicting the cover splash – that most people lack a mathematical mind, which may explain the difficulties mathematicians have in communicating.

In addition, I have come across a few seemingly sensible people who may seem to hate maths, although I would rather say that they hate ‘pseudo-maths’. For example, it may be true that we have a better grasp on reality if we can think mathematically – as scientists and technologists routinely do – but it seems a huge jump – and misleading – to claim that mathematics is ‘the language of reality’ in any more objective sense. By pseudo-maths I mean something that appears to be maths (at least to the non-mathematician) but which uses ordinary reasoning to make bold claims (such as ‘is the language of reality’).

But there is a more fundamental problem. The article cites Ashby to the effect that ‘effective control’ relies on adequate models. Such models are of course computational and as such we rely on mathematics to reason about them. Thus we might say that mathematics is the language of effective control. If – as some seem to – we make a dichotomy between controllable and not controllable systems then mathematics is the pragmatic language of reality. Here we enter murky waters. For example, if reality is socially constructed then presumably pragmatic social sciences (such as economics) are necessarily concerned with control, as in their models. But one point of my blog is that the kind of maths that applies to control is only a small portion. There is at least the possibility that almost all things of interest to us as humans are better considered using different maths. In this sense it seems to me that some people justifiably hate control and hence related pseudo-maths. It would be interesting to give them a brain scan to see if  their thinking appeared mathematical, or if they had some other characteristic networks of brain regions. Either way, I suspect that many problems would benefit from collaborations between mathematicians and those who hate pseudo-mathematic without necessarily being professional mathematicians. This seems to match my own experience.

Dave Marsay

Mathematical Modelling

Mathematics and modelling in particular is very powerful, and hence can be very risky if you get it wrong, as in mainstream economics. But is modelling inappropriate – as has been claimed – or is it just that it has not been done well enough?

As a mathematician who has dabbled in modelling and economics I thought I’d try my hand at modelling economies. What harm could there be?

My first notion is that actors activity is habitual.

My second is that habits persist until there is a ‘bad’ experience, in which case they are revised. What is taken account of, what counts as ‘bad’ and how habits are replaced or revised are all subject to meta-habits (habits about habits).

In particular, mainstream economists suppose that actors seek to maximise their utilities, and they never revise this approach. But this may be too restrictive.

Myself, I would add that most actors mostly seek to copy others and also tend to discount experiences and lessons identified by previous generations.

With some such ‘axioms’ (suitably formalised) such as those above, one can predict booms and busts leading to new ‘epochs’ characterised by dominant theories and habits. For example, suppose that some actors habitually borrow as much as they can to invest in an asset (such as a house for rent) and the asset class performs well. Then they will continue in their habit, and others who have done less well will increasingly copy them, fuelling an asset price boom. But no asset class is worth an infinite amount, so the boom must end, resulting in disappointment and changes in habit, which may again be copied by those who are losing out on the asset class., giving a bust.  Thus one has an ’emergent behaviour’ that contradicts some of the implicit mainstream assumptions about rationality  (such as ‘ergodicity’), and hence the possibility of meaningful ‘expectations’ and utility functions to be maximized. This is not to say that such things cannot exist, only that if they do exist it must be due to some economic law as yet unidentified, and we need an alternative explanation for booms and busts.

What I take from this is that mathematical models seem possible and may even provide insights.I do not assume that a model that is adequate in the short-run will necessarily continue to be adequate, and my model shows how economic epochs can be self-destructing. To me, the problem in economics is not so much that it uses mathematics and in particular mathematical modelling but that it does so badly. My ‘axioms’ mimic the approach that Einstein took to physics: it replaces an absolutist model by a relativistic one, and shows that it makes a difference. In my model there are no magical ‘expectations’, rather actors may have realistic habits and expectations, based on their experience and interpretation of the media and other sources, which may be ‘correct’ (or at least not falsified) in the short-run, but which cannot provide adequate predictions for the longer run. To survive a change of epochs our actors would need to be at least following some actors who were monitoring and thinking about the overall situation more broadly and deeply than those who focus on short run utility. (Something that currently seems lacking.)

David Marsay