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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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

Can polls be reliable?

Election polls in many countries have seemed unusually unreliable recently. Why? And can they be fixed?

The most basic observation is that if one has a random sample of a population in which x% has some attribute then it is reasonable to estimate that x% of the whole population has that attribute, and that this estimate will tend to be more accurate the larger the sample is. In some polls sample size can be an issue, but not in the main political polls.

A fundamental problem with most polls is that the ‘random’ sample may not be uniformly distributed, with some sub-groups over or under represented. Political polls have some additional issues, that are sometimes blamed:

  • People with certain opinions may be reluctant to express them, or may even mislead.
  • There may be a shift in opinions with time, due to campaigns or events.
  • Different groups may differ in whether they actually vote, for example depending on the weather.

I also think that in the UK the trend to postal voting may have confused things, as postal voters will have missed out on the later stages of campaigns, and on later events. (Which were significant in the UK 2017 general election.)

Pollsters have a lot of experience at compensating for these distortions, and are increasingly using ‘sophisticated mathematical tools’. How is this possible, and is there any residual uncertainty?

Back to mathematics, suppose that we have a science-like situation in which we know which factors (e.g. gender, age, social class ..) are relevant. With a large enough sample we can partition the results by combination of factors, measure the proportions for each combination, and then combine these proportions, weighting by the prevalence of the combinations in the whole population. (More sophisticated approaches are used for smaller samples, but they only reduce the statistical reliability.)

Systematic errors can creep in in two ways:

  1. Instead of using just the poll data, some ‘laws of politics’ (such as the effect of rain) or other heuristics (such as that the swing among postal votes will be similar to that for votes in person) may be wrong.
  2. An important factor is missed. (For example, people with teenage children or grandchildren may vote differently from their peers when student fees are an issue.)

These issues have analogues in the science lab. In the first place one is using the wrong theory to interpret the data, and so the results are corrupted. In the second case one has some unnoticed ‘uncontrolled variable’ that can really confuse things.

A polling method using fixed factors and laws will only be reliable when they reasonably accurately the attributes of interest, and not when ‘the nature of politics’ is changing, as it often does and as it seems to be right now in North America and Europe. (According to game theory one should expect such changes when coalitions change or are under threat, as they are.) To do better, the polling organisation would need to understand the factors that the parties were bringing into play at least as well as the parties themselves, and possibly better. This seems unlikely, at least in the UK.

What can be done?

It seems to me that polls used to be relatively easy to interpret, possibly because they were simpler. Our more sophisticated contemporary methods make more detailed assumptions. To interpret them we would need to know what these assumptions were. We could then ‘aim off’, based on our own judgment. But this would involve pollsters in publishing some details of their methods, which they are naturally loth to do. So what could be done? Maybe we could have some agreed simple methods and publish findings as ‘extrapolations’ to inform debate, rather than predictions. We could then factor in our own assumptions. (For example, our assumptions about students turnout.)

So, I don’t think that we can expect reliable poll findings that are predictions, but possibly we could have useful poll findings that would inform debate and allow us to take our own views. (A bit like any ‘big data’.)

Dave Marsay

 

The search for MH370: uncertainty

There is an interesting podcast about the search for MH370 by a former colleague. I think it illustrates in a relatively accessible form some aspects of uncertainty.

According to the familiar theory, if one has an initial probability distribution over the globe for the location of MH370’s flight recorder, say, then one can update it using Bayes’ rule to get a refined distribution. Conventionally, one should search where there is a higher probability density (all else being equal). But in this case it is fairly obvious that there is no principled way of deriving an initial distribution, and even Bayes’ rule is problematic. Conventionally, one should do the best one can, and search accordingly.

The podcaster (Simon) gives examples of some hypotheses (such as the pilot being well, well-motivated and unhindered throughout) for which the probabilistic approach is more reasonable. One can then split one’s effort over such credible hypotheses, not ruled out by evidence.

A conventional probabilist would note that any ‘rational’ search would be equivalent to some initial probability distribution over hypotheses, and hence some overall distribution. This may be so, but it is clear from Simon’s account that this would hardly be helpful.

I have been involved in similar situations, and have found it easier to explain the issues to non-mathematicians when there is some severe resource constraint, such as time. For example, we are looking for a person. The conventional approach is to maximise our estimated probability of finding them based on our estimated probabilities of them having acted in various ways (e.g., run for it, hunkered down). An alternative is to consider the ways they may ‘reasonably’ be thought to have acted and then to seek to maximize the worst case probability of finding them. Then again, we may have a ranking of ways that they may have acted, and seek to maximize the number of ways for which the probability of our success exceeds some acceptable amount (e.g. 90%). The key point here is that there are many reasonable objectives one might have, for only one of which the conventional assumptions are valid. The relevant mathematics does still apply, though!

Dave Marsay

Mathematical modelling

I had the good fortune to attend a public talk on mathematical modelling, organised by the University of Birmingham (UK). The speaker, Dr Nira Chamberlain CMath FIMA CSci, is a council member of the appropriate institution, and so may reasonably be thought to be speaking for mathematicians generally.

He observed that there were many professional areas that used mathematics as a tool, and that they generally failed to see the need for professional mathematicians as such. He thought that mathematical modelling was one area where – at least for the more important problems – mathematicians ought to be involved. He gave examples of modelling, including one of the financial crisis.

The main conclusion seemed very reasonable, and in line with the beliefs of most ‘right thinking’ mathematicians. But on reflection, I wonder if my non-mathematician professional colleagues would accept it. In 19th century professional mathematicians were proclaiming it a mathematical fact that the physical world conformed to classical geometry. On this basis, mathematicians do not seem to have any special ability to produce valid models. Indeed, in the run up to the financial crash there were too many professional mathematicians who were advocating some mainstream mathematical models of finance and economies in which the crash was impossible.

In Dr Chamberlain’s own model of the crash, it seems that deregulation and competition led to excessive risk taking, which risks eventually materialised. A colleague who is a professional scientist but not a professional mathematician has advised me that this general model was recognised by the UK at the time of our deregulation, but that it was assumed (as Greenspan did) that somehow some institution would step in to foreclose this excessive risk taking. To me, the key thing to note is that the risks being taken were systemic and not necessarily recognised by those taking them. To me, the virtue of a model does not just depend on it being correct in some abstract sense, but also that ‘has traction’ with relevant policy and decision makers and takers. Thus, reflecting on the talk, I am left accepting the view of many of my colleagues that some mathematical models are too important to be left to mathematicians.

If we have a thesis and antithesis, then the synthesis that I and my colleagues have long come to is that important mathematical model needs to be a collaborative endeavour, including mathematicians as having a special role in challenging, interpret and (potentially) developing the model, including developing (as Dr C said) new mathematics where necessary. A modelling team will often need mathematicians ‘on tap’ to apply various methods and theories, and this is common. But what is also needed is a mathematical insight into the appropriateness of these tools and the meaning of the results. This requires people who are more concerned with their mathematical integrity than in satisfying their non-mathematical pay-masters. It seems to me that these are a sub-set of those that are generally regarded as ‘professional’. How do we identify such people?

Dave Marsay 

 

More to Uncertainty than 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.

(Actually, my paper was published jan 2016,but somehow this request for comments got stuck in a limbo somewhere. Better late than never?)

Which rationality?

We often suppose that rationality is ‘a good thing’, but is it always?

Rationality is variously defined as being in accord with reason, logic or ‘the facts’. Here ‘reason’ may mean one’s espoused or actual reasons, or it may mean in accord with some external standards. Thus in its broadest interpretation, it seems that anything that has a reason for being the way that it is may be considered broadly rational. But the notion of rationality derives from ‘reason’, one aspect of which is ‘sound judgement, good sense’. This suggests some external standard.

If we use the term ‘simple’ to denote a situation in which there are definite ‘objective’ standards of soundness and goodness, then rationality in simple situations is behaviour that accords with those standards. Philosophers can argue endlessly about whether any such situations exist, so it seems sensible to define rationality more generally as being relative to some set of standards. The question then being: What standards?

My natural inclination as a mathematician is that those standards should always include the best relevant logics, including mathematics. Yet I have witnessed many occasions on which the use of mathematics has tended to promote disasters, and the advocates of such approaches (apart from those few who think like me) have seemed culpable. I have a great deal of respect and sympathy for the view that mathematics is harmful in complex situations. Yet intellectually it seems quite wrong, and I cannot accept it.

In each case there seems to be some specific failing, which many of my colleagues have attributed to some human factor, such as hubris or the need to keep a job or preserve an institution. But the perpetrators do not seem to me to be much different from the rest of us, and I have long thought that there is some more fundamental common standard that is incompatible with the use of reason. The financial crises of 2007/8/9 are cases where it is hard to believe that most of those pushing the ‘mathematical’ view that turned out to be harmful were either irrational or rationally harmful.

Here I want to suggest a possible explanation.

From a theoretical perspective, there are no given ‘facts’, ‘logics’ or ‘reasons’ that we can rely on.This certainly seems to be true of finance and economics. For example, in economics the models used may be mathematical and in this sense beyond criticism, but the issue of their relevance to a particular situation is never purely logical, and ought to be questioned. Yet it seems that many institutions, including businesses,  rely on having absolute beliefs: questioning them would be wasteful in the short-run. So individual actors tend not only to be rational, but also to be narrowly rational ‘in the short run’, which normally goes with acting ‘as if’ it had narrow facts.

For example, it seems to me to be  a fact that according to the best contemporary scientific theories, the earth is not stationary. It is generally expedient to for me to act ‘as if’ I knew that the earth moved. But unless we can be absolutely sure that the earth moves, the tendency to suppose that it is a fact that the earth moves could be dangerous. (You could try substituting other facts, such as that economies always tend to a healthy equilibrium.)

In a healthy society there would be a competition of ideas,  such that society as a whole could be said to be being more broadly rational, even while its actors were being only narrowly rational. For example, a science would consist of various schools, each of which would be developing its own theories, consistent with the facts, which between them would be exploring and developing the space of all such credible theories. At a practical level, an engineer would appreciate the difference between building a bridge based on a theory that had been tested on similar bridges, and building a novel type of bridge where the existing heuristics could not be relied upon.

I do not think that human society as a whole is healthy in this sense. Why not? In evolutionary theory separate niches, such as islands, promote the development of healthy diversity. Perhaps the rise of global communications and trade, and even the spread of the use of English, is eliminating the niches in which ideas can be explored and so is having a long-run negative impact that needs to be compensated for?

Thus I think we need to distinguish between short and long-run rationalities, and to understand and foster both. It seems to me that most of the time, for most areas of life, short-run rationality is adequate, and it is this that is familiar. But this needs to be accompanied by an understanding of the long-run issues, and an appropriate balance achieved. Perhaps too much (short-run) rationality can be harmful (in the long-run). And not only in economies.

Dave Marsay