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 

 

The limits of (atomistic) mathematics

Lars Syll draws attention to a recent seminar on ‘Confronting economics’ by Tony Lawson, as part of the Bloomsbury Confrontations at UCLU.

If you replace his every use of the term ‘mathematics’ by something like ‘atomistic mathematics’ then I would regard this talk as not only very important, but true. Tony approving quotes Whitehead on challenging implicit assumptions. Is his implicit assumption that mathematics is ‘atomistic’? What about Whitehead’s own mathematics, or that of Russell, Keynes and Turing? He (Tony) seems to suppose that mathematics can’t deal with emergent properities. So What is Whitehead’s work on Process, Keynes’ work on uncertainty, Russell’s work on knowledge or Turing’s work on morphogenesis all about?

Dave Marsay

 

Are more intelligent people more biased?

It has been claimed that:

U.S. intelligence agents may be more prone to irrational inconsistencies in decision making compared to college students and post-college adults … .

This is scary, if unsurprising to many. Perhaps more surprisingly:

Participants who had graduated college seemed to occupy a middle ground between college students and the intelligence agents, suggesting that people with more “advanced” reasoning skills are also more likely to show reasoning biases.

It seems as if there is some serious  mis-education in the US. But what is it?

The above conclusions are based on responses to the following two questions:

1. The U.S. is preparing for the outbreak of an unusual disease, which is expected to kill 600 people. Do you: (a) Save 200 people for sure, or (b) choose the option with 1/3 probability that 600 will be saved and a 2/3 probability no one will be saved?

2. In the same scenario, do you (a) pick the option where 400 will surely die, or instead (b) a 2/3 probability that all 600 will die and a 1/3 probability no one dies?

You might like to think about your answers to the above, before reading on.

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The paper claims that:

Notably, the different scenarios resulted in the same potential outcomes — the first option in both scenarios, for example, has a net result of saving 200 people and losing 400.

Is this what you thought? You might like to re-read the questions and reconsider your answer, before reading on.

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The questions may appear to contain statements of fact, that we are entitled to treat as ‘given’. But in real-life situations we should treat such questions as utterances, and use the appropriate logics. This may give the same result as taking them at face value – or it may not.

It is (sadly) probably true that if this were a UK school examination question then the appropriate logic would be (1) to treat the statements ‘at face value’ (2) assume that if 200 people will be saved ‘for sure’ then exactly 200 people will be saved, no more. On the other hand, this is just the kind of question that I ask mathematics graduates to check that they have an adequate understanding of the issues before advising decision-takers. In the questions as set, the (b) options are the same, but (1a) is preferable to (2a), unless one is in the very rare situation of knowing exactly how many will die. With this interpretation, the more education and the more experience, the better the decisions – even in the US 😉

It would be interesting to repeat the experiment with less ambiguous wording. Meanwhile, I hope that intelligence agents are not being re-educated. Or have I missed something?

Also

Kahneman’s Thinking, fast and slow has a similar example, in which we are given ‘exact scientific estimates’ of probable outcomes, avoiding the above ambiguity. This might be a good candidate experimental question.

Kahneman’s question is not without its own subtleties, though. It concerns the efficacy of ‘programs to combat disease’. It seems to me that if I was told that a vaccine would save 1/3 of the lives, I would suppose that it had been widely tested, and that the ‘scientific’ estimate was well founded. On the other hand, if I was told that there was a 2/3 chance of the vaccine being ineffective I would suppose that it hadn’t been tested adequately, and the ‘scientific’ estimate was really just an informed guess. In this case, I would expect the estimate of efficacy to be revised in the light of new information. It could even be that while some scientist has made an honest estimate based on the information that they have, some other scientist (or technician) already knows that the vaccine is ineffective. A program based on such a vaccine would be more complicated and ‘risky’ than one based on a well-founded estimate, and so I would be reluctant to recommend it. (Ideally, I would want to know a lot more about how the estimates were arrived at, but if pressed for a quick decision, this is what I would do.)

Could the framing make a difference? In one case, we are told that ‘scientifically’, 200 people will be saved. But scientific conclusions always depend on assumptions, so really one should say ‘if …. then 200 will be saved’. My experience is that otherwise the outcome should not be expected, and that saving 200 is the best that should be expected. In the other case we are told that ‘400 will die’. This seems to me to be a very odd thing to say. From a logical perspective one would like to understand the circumstances in which someone would put it like this. I would be suspicious, and might well (‘irrationally’) avoid a program described in that way.

Addenda

The example also shows a common failing, in assuming that the utility is proportional to lives lost. Suppose that when we are told that lives will be ‘saved’ we assume that we will get credit, then we might take the utility from saving lives to be number of lives saved, but with a limit of ‘kudos’ at 250 lives saved. In this case, it is rational to save 200 ‘for sure’, as the expected credit from taking a risk is very much lower. On the other hand, if we are told that 400 lives will be ‘lost’ we might assume that we will be blamed, and take the utility to be minus the lives lost, limited at -10. In this case it is rational to take a risk, as we have some chance of avoiding the worst case utility, whereas if we went for the sure option we would be certain to suffer the worst case.

These kind of asymmetric utilities may be just the kind that experts experience. More study required?

 

Dave Marsay

Are fananciers really stupid?

The New Scientist (30 March 2013) has the following question, under the heading ‘Stupid is as stupid does’:

Jack is looking at Anne but Anne is looking at George. Jack is married but George is not. Is a married person looking at an unmarried person?

Possible answers are: “yes”, “no” or “cannot be determined”.

You might want to think about this before scrolling down.

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It is claimed that while ‘the vast majority’ (presumably including financiers, whose thinking is being criticised) think the answer is “cannot be determined”,

careful deduction shows that the answer is “yes”.

Similar views are expressed at  a learning blog and at a Physics blog, although the ‘careful deductions’ are not given. Would you like to think again?

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Now I have a confession to make. My first impression is that the closest of the admissible answers is ‘cannot be determined’, and having thought carefully for a while, I have not changed my mind. Am I stupid? (Based on this evidence!) You might like to think about this before scrolling down.

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Some people object that the term ‘is married’ may not be well-defined, but that is not my concern. Suppose that one has a definition of marriage that is as complete and precise as possible. What is the correct answer? Does that change your thinking?

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Okay, here are some candidate answers that I would prefer, if allowed:

  1. There are cases in which the answer cannot be determined.
  2. It is not possible to prove that there are not cases in which the answer cannot be determined. (So that the answer could actually be “yes”, but we cannot know that it is “yes”.)

Either way, it cannot be proved that there is a complete and precise way of determining the answer, but for different reasons. I lean towards the first answer, but am not sure. Which it is is not a logical or mathematical question, but a question about ‘reality’, so one should ask a Physicist. My reasoning follows … .

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Suppose that Anne marries Henry who dies while out in space, with a high relative velocity and acceleration. Then to answer yes we must at least be able to determine a unique time in Anne’s time-frame in which Henry dies, or else (it seems to me) there will be a period of time in which Anne’s status is indeterminate. It is not just that we do not know what Anne’s status is; she has no ‘objective’ status.

If there is some experiment which really proves that there is no possible ‘objective’ time (and I am not sure that there is) then am I not right? Even if there is no such experiment, one cannot determine the truth of physical theories, only fail to disprove them. So either way, am I not right?

Enlightenment, please. The link to finance is that the New Scientist article says that

Employees leaving logic at the office door helped cause the financial crisis.

I agree, but it seems to me (after Keynes) that it was their use of the kind of ‘classical’ logic that is implicitly assumed in the article that is at fault. Being married is a relation, not a proposition about Anne. Anne has no state or attributes from which her marital status can be determined, any more than terms such as crash, recession, money supply, inflation, inequality, value or ‘the will of the people’ have any correspondence in real economies.  Unless you know different?

Dave Marsay

Haldane’s The dog and the Frisbee

Andrew Haldane The dog and the Frisbee

Haldane argues in favour of simplified regulation. I find the conclusions reasonable, but have some quibbles about the details of the argument. My own view is that much of our financial problems have been due – at least in part – to a misrepresentation of the associated mathematics, and so I am keen to ensure that we avoid similar misunderstandings in the future. I see this as a primary responsibility of ‘regulators’, viewed in the round.

The paper starts with a variation of Ashby’s ball-catching observation, involving dog and a Frisbee instead of a man and a ball: you don’t need to estimate the position of the Frisbee or be an expert in aerodynamics: a simple, natural, heuristic will do. He applies this analogy to financial regulation, but it is somewhat flawed. When catching a Frisbee one relies on the Frisbee behaving normally, but in financial regulation one is concerned with what had seemed to be abnormal, such as the crisis period of 2007/8.

It is noted of Game theory that

John von Neumann and Oskar Morgenstern established that optimal decision-making involved probabilistically-weighting all possible future outcomes.

In apparent contrast

Many of the dominant figures in 20th century economics – from Keynes to Hayek, from Simon to Friedman – placed imperfections in information and knowledge centre-stage. Uncertainty was for them the normal state of decision-making affairs.

“It is not what we know, but what we do not know which we must always address, to avoid major failures, catastrophes and panics.”

The Game Theory thinking is characterised as ignoring the possibility of uncertainty, which – from a mathematical point of view – seems an absurd misreading. Theories can only ever have conditional conclusions: any unconditional misinterpretation goes beyond the proper bounds. The paper – rightly – rejects the conclusions of two-player zero-sum static game theory. But its critique of such a theory is much less thorough than von Neumann and Morgenstern’s own (e.g. their 4.3.3) and fails to identify which conditions are violated by economics. More worryingly, it seems to invite the reader to accept them, as here:

The choice of optimal decision-making strategy depends importantly on the degree of uncertainty about the environment – in statistical terms, model uncertainty. A key factor determining that uncertainty is the length of the sample over which the model is estimated. Other things equal, the smaller the sample, the greater the model uncertainty and the better the performance of simple, heuristic strategies.

This seems to suggest that – contra game theory – we could ‘in principle’ establish a sound model, if only we had enough data. Yet:

Einstein wrote that: “The problems that exist in the world today cannot be solved by the level of thinking that created them”.

There seems a non-sequitur here: if new thinking is repeatedly being applied then surely the nature of the system will continually be changing? Or is it proposed that the ‘new thinking’ will yield a final solution, eliminating uncertainty? If it is the case that ‘new thinking’ is repeatedly being applied then the regularity conditions of basic game theory (e.g. at 4.6.3 and 11.1.1) are not met (as discussed at 2.2.3). It is certainly not an unconditional conclusion that the methods of game theory apply to economies beyond the short-run, and experience would seem to show that such an assumption would be false.

The paper recommends the use of heuristics, by which it presumably means what Gigernezer means: methods that ignore some of the data. Thus, for example, all formal methods are heuristics since they ignore intuition.  But a dog catching a Frisbeee only has its own experience, which it is using, and so presumably – by this definition – is not actually using a heuristic either. In 2006 most financial and economics methods were heuristics in the sense that they ignored the lessons identified by von Neumann and Morgenstern. Gigerenzer’s definition seems hardly helpful. The dictionary definition relates to learning on one’s own, ignoring others. The economic problem, it seems to me, was of paying too much atention to the wrong people, and too little to those such as von Neumann and Morgenstern – and Keynes.   

The implication of the paper and Gigerenzer is, I think, that a heuristic is a set method that is used, rather than solving a problem from first principles. This is clearly a good idea, provided that the method incorporates a check that whatever principles that it relies upon do in fact hold in the case at hand. (This is what economists have often neglecte to do.) If set methods are used as meta-heuristics to identify the appropriate heuristics for particular cases, then one has something like recognition-primed decision-making. It could be argued that the financial community had such meta-heuristics, which led to the crash: the adoption of heuristics as such seems not to be a solution. Instead one needs to appreciate what kind of heuristic are appropriate when. Game theory shows us that the probabilistic heuristics are ill-founded when there is significant innovation, as there was both prior, through and immediately after 2007/8. In so far as economics and finance are games, some events are game-changers. The problem is not the proper application of mathematical game theory, but the ‘pragmatic’ application of a simplistic version: playing the game as it appears to be unless and until it changes. An unstated possible deduction from the paper is surely that such ‘pragmatic’ approaches are inadequate. For mutable games, strategy needs to take place at a higher level than it does for fixed games: it is not just that different strategies are required, but that ‘strategy’ has a different meaning: it should at least recognize the possibility of a change to a seemingly established status quo.

If we take an analogy with a dog and a Frisbee, and consider Frisbee catching to be a statistically regular problem, then the conditions of simple game theory may be met, and it is also possible to establish statistically that a heuristic (method) is adequate. But if there is innovation in the situation then we cannot rely on any simplistic theory or on any learnt methods. Instead we need a more principled approach, such as that of Keynes or Ashby,  considering the conditionality and looking out for potential game-changers. The key is not just simpler regulation, but regulation that is less reliant on conditions that we expect to hold but for which, on maturer reflection, are not totally reliable. In practice this may necessitate a mature on-going debate to adjust the regime to potential game-changers as they emerge.

See Also

Ariel Rubinstein opines that:

classical game theory deals with situations where people are fully rational.

Yet von Neumann and Morgenstern (4.1.2) note that:

the rules of rational behaviour must provide definitely for the possibility of irrational conduct on the part of others.

Indeed, in a paradigmatic zero-sum two person game, if the other person players rationally (according to game theory) then your expected return is the same irrespective of how you play. Thus it is of the essence that you consider potential non-rational plays. I take it, then, that game theory as reflected in economics is a very simplified – indeed an over-simplified – version. It is presumably this distorted version that Haldane’s criticism’s properly apply to.

Dave Marsay

Haldane’s Tails of the Unexpected

A. Haldane, B. Nelson Tails of the unexpected,  The Credit Crisis Five Years On: Unpacking the Crisis conference, University of Edinburgh Business School, 8-9 June 2012

The credit crisis is blamed on a simplistic belief in ‘the Normal Distribution’ and its ‘thin tails’, understating risk. Complexity and chaos theories point to greater risks, as does the work of Taleb.

Modern weather forecasting is pointed to as good relevant practice, where one can spot trouble brewing. Robust and resilient regulatory mechanisms need to be employed. It is no good relying on statistics like VaR (Value at Risk) that assume a normal distribution. The Bank of England is developing an approach based on these ideas.

Comment

Risk arises when the statistical distribution of the future can be calculated or is known. Uncertainty arises when this distribution is incalculable, perhaps unknown.

While the paper acknowledges Keynes’ economics and Knightian uncertainty, it overlooks Keynes’ Treatise on Probability, which underpins his economics.

Much of modern econometric theory is … underpinned by the assumption of randomness in variables and estimated error terms.

Keynes was critical of this assumption, and of this model:

Economics … shift[ed] from models of Classical determinism to statistical laws. … Evgeny Slutsky (1927) and Ragnar Frisch (1933) … divided the dynamics of the economy into two elements: an irregular random element or impulse and a regular systematic element or propagation mechanism. This impulse/propagation paradigm remains the centrepiece of macro-economics to this day.

Keynes pointed out that such assumptions could only be validated empirically and (as the current paper also does) in the Treatise he cited Lexis’s falsification.

The paper cites a game of paper/scissors/stone which Sotheby’s thought was a simple game of chance but which Christie’s saw  as an opportunity for strategizing – and won millions of dollars. Apparently Christie’s consulted some 11 year old girls, but they might equally well have been familiar with Shannon‘s machine for defeating strategy-impaired humans. With this in mind, it is not clear why the paper characterises uncertainty a merly being about unknown probability distributions, as distinct from Keynes’ more radical position, that there is no such distribution. 

The paper is critical of nerds, who apparently ‘like to show off’.  But to me the problem is not the show-offs, but those who don’t know as much as they think they know. They pay too little attention to the theory, not too much. The girls and Shannon seem okay to me: it is those nerds who see everything as the product of randomness or a game of chance who are the problem.

If we compare the Slutsky Frisch model with Kuhn’s description of the development of science, then economics is assumed to develop in much the same way as normal science, but without ever undergoing anything like a (systemic) paradigm shift. Thus, while the model may be correct most of the time,  violations, such as in 2007/8, matter.

Attempts to fine-tune risk control may add to the probability of fat-tailed catastrophes. Constraining small bumps in the road may make a system, in particular a social system, more prone to systemic collapse. Why? Because if instead of being released in small bursts pressures are constrained and accumulate beneath the surface, they risk an eventual volcanic eruption.

 One can understand this reasoning by analogy with science: the more dominant a school which protects its core myths, the greater the reaction and impact when the myths are exposed. But in finance it may not be just ‘risk control’ that causes a problem. Any optimisation that is blind to the possibility of systemic change may tend to increase the chance of change (for good or ill) [E.g. Bohr Atomic Physics and Human Knowledge. Ox Bow Press 1958].

See Also

Previous posts on articles by or about Haldane, along similar lines:

My notes on:

Dave Marsay

NRC’s Assessing … Complex Models

Committee on Mathematical Foundations of Verification, Validation, and Uncertainty Quantification Board on Mathematical Sciences and Their Applications Assessing the Reliability of Complex Models: Mathematical and Statistical Foundations of Verification, Validation, and Uncertainty Quantification (US) NRC, 2012

The team were tasked to “examine practices for VVUQ of large-scale computational simulations”. Such simulations are complicated. The title seems misleading in using the term ‘complex’. The summary seems like a reasonable consensus summary of the state of the art in its focus area, and of research directions, with no surprises. But the main body does provide some ammunition for those who seek to emphasise deeper uncertainty issues, considering mathematics beyond computation.

Summary

Principles

Highlighted principles include:

    1. A validation assessment is well defined only in terms of specified quantities of interest (QOIs) and the accuracy needed for the intended use of the model.
    2. A validation assessment provides direct information about model accuracy only in the domain of applicability that is “covered” by the physical observations employed in the assessment.

Comments

The notion of a model here would be something like ‘all swans are white’. The first principle suggests that we need tolerance for what is regarded as ‘white’. The second principle suggests that if we have only considered British swans, we should restrict the domain of applicability of the model.

In effect, the model is being set within a justification, much as the conclusion of a mathematical theorem is linked to axioms by the proof. This is contrary to much school science practice, which simply teaches models: we need to understand the (empirical) theory. Typically, when we read ‘all swans are white’ we should understand that it really only means ‘all British swans are white-ish’.

Swans are relatively simple. The only problem is our limited observations of them. Economics, for example, is more complex. The quantities of interest are controversial, as are the relevant observations. Such complex situations seem beyond the intended scope of this report.

Research Topics

  1. Development of methods that help to define the “domain of applicability” of a model, including methods that help quantify the notions of near neighbors, interpolative predictions, and extrapolative predictions.
  2. Development of methods to assess model discrepancy and other sources of uncertainty in the case of rare events, especially when validation data do not include such events.

Comments

These topics are easier if one has an overarching theory of which the model is a specialisation, whose parameters are to be determined. In such cases the ‘domain of applicability’ could be based on an established classifying schema, and uncertainty could be probabilistic, drawing on established probabilistic models. The situation is more challenging, with broader uncertainties, where there is no such ruling theory, as in climate science.

Recommendations

  1. An effective VVUQ [verification, validation and uncertainty quantification] education should encourage students to confront and reflect on the ways that knowledge is acquired, used, and updated.
  2. The elements of probabilistic thinking, physical-systems modeling, and numerical methods and computing should become standard parts of the respective core curricula for scientists, engineers, and statisticians.

Comments

Most engineers and statisticians will be working pragmatically, assuming some ruling theory that guides their work. This report seems most suitable for them. Ideally, scientists acting as science advisors would also be working in such a way. However, surprises do happen, and scientists working on science should be actively doubting any supposed ruling theory. Thus it is sometimes vital to know the difference between a situation where an agreed theory should be regarded as, for example, ‘fit for government work’, and where it is not, particularly where extremes of complexity or uncertainty call for a more principled approach. In such cases it is not obvious that uncertainty can be quantified. For example, how does one put a number on ‘all swans are white’ when one has not been outside Britain?

As well as using mathematics to work out the implications of a ruling theory in a particular case, one needs to be able to use different mathematics to work out the implications of a particular case for theory.

Introduction

This cites Savage,  but in his terms it is implicitly addressing complicated but ‘small’ worlds rather than more complex ‘large’ ones, such as that of interest to climate science.

Sources of Uncertainty and Error

The general issue is whether formal validation of models of complex systems is actually feasible. This issue is both philosophical and practical and is discussed in greater depth in, for example, McWilliams (2007), Oreskes et al. (1994), and Stainforth et al. (2007).

There is a need to make decisions … before a complete UQ analysis will be available. … This does not mean that UQ can be ignored but rather that decisions need to be made in the face of only partial knowledge of the uncertainties involved. The “science” of these kinds of decisions is still evolving, and the various versions of decision analysis are certainly relevant.

Comment

 It seems that not all uncertainty is quantifiable, and that one needs to be able to make decisions in the face of such uncertainties.

In the case of ‘all swans are white’ the uncertainty arises because we have only looked in Britain. It is clear what can be done about this, even if we have no basis for assigning a number.

In the case of economics, even if we have a dominant theory we may be uncertainty because, for example, it has only been validated against the British economy for the last 10 years. We might not be able to put a number on the uncertainty, but it might be wise to look for more general theories, covering a broader range of countries and times, and then see how our dominant theory is situated within the broader theory. This might give us more confidence in some conclusions from the theory, even if we cannot assign a number. (One also needs to consider alternative theories.)

Model Validation and Prediction

Comparison with reality

In simple settings validation could be accomplished by directly comparing model results to physical measurements for the QOI  …

Findings

  1. Mathematical considerations alone cannot address the appropriateness of a model prediction in a new, untested setting. Quantifying uncertainties and assessing their reliability for a prediction require both statistical  and subject-matter reasoning.
  2. The idea of a domain of applicability is helpful for communicating the conditions for which predictions (with uncertainty) can be trusted. However, the mathematical foundations have not been established for defining such a domain or its boundaries.

Comment

I take the view that a situation that can be treated classically is not complex, only at most complicated. Complex situations may always contain elements that are surprising to us. Hence bullet 1 applies to complex situations too. The responsibility for dealing with complexities seems to be shifted from the mathematicians to the subject matter experts (SMEs). But if one is dealing with a new ‘setting’ one is dealing with dynamic complexity, of the kind that would be a crisis if the potential impact were serious. In such situations it may not be obvious which subject is the relevant one, or there may be more than one vital subject. SMEs may be unused to coping with complexity or with collaboration under crisis or near-crisis conditions. For example, climate science might need not only climatologists but also experts in dealing with uncertainty.

My view is that sometimes one can only assess the relevance and reliability of a model in a particular situation, that one needs particular experts in this, and that mathematics can help – but it is a different mathematics.

Next Steps in Practice, Research, and Education for Verification, Validation, and Uncertainty Quantification

 For validation, “domain of applicability” is recognized as an important concept, but how one defines this domain remains an open question. For predictions, characterizing how a model differs from reality, particularly in extrapolative regimes, is a pressing need. … advances in linking a model to reality will likely broaden the domain of applicability and improve confidence in extrapolative prediction.

Comment

As Keynes pointed out, in some complex situations one can only meaningfully predict in the short-term. Thus in early 2008 economic predictions were not in error, as short-term predictions. It is just that the uncertain long-term arrived. What is needed, therefore, is some long-term forecasting ability. This cannot be a prediction, in the sense of having a probability distribution, but it might be an effective anticipation, just as one might have anticipated that there were non-white swans in foreign parts. Different mathematics is needed.

My Summary

The report focusses on the complicatedness of the models. But I find it hard to think of a situation where one needs a complicated model and the actual situation is not complex. Usually, for example, the situation is ‘reflexive’ because the model is going to be used to inform interaction with the world, which will change it. Thus, the problem as I see it is how to model a situation that is uncertain and possibly complex. While the report does give some pointers it does not develop them.

The common sense view of modelling is that a model is based on observations. In fact – as the report notes – it tends to be based on observations plus assumptions, which are refined into a model, often iteratively. But the report seems to suppose that one’s initial assumptions will be ‘true’. But one can only say that the model fits one’s observations, not that it will continue to fit all possible observations, unless one can be sure that the situation is very constrained. That is, one cannot say that a scientific theory is unconditionally and absolutely true, but only ‘true to’ ones observations and assumptions.

The report is thus mainly for those who have a mature set of assumptions which they wish to refine, not those who expect the unexpected. It does briefly mention ‘rare events’, but it sees these as outliers on a probability distribution whereas I would see these more as challenging assumptions.

See Also

The better nature blog provides a view of science that is complimentary to this report.

My notes on science and uncertainty.

Dave Marsay