Metaphors for complexity and uncertainty

Metaphors

According to psychologists we often tell ourselves stories to rationalize away complexity and uncertainty. It would be good to have some stories that reveal the complexity and uncertainty, as an antidote.

Route Planning and the Normandy Invasion

I noticed in the 80s (before SATNAV) that not everyone thinks of a common task like route-planning in the same way. Some are pragmatic, picking the best route and only considering alternatives if that one is blocked. Others have heuristics that allow for uncertainty, such as preferring routes that have ready detours, just in case. There was some discussion among geek-dom as to whether a pure Bayesian approach would be adequate or even better. Logically, it should depend on the nature of the problem. If blocks are probabilistic with learn-able distributions, and the goal is to minimise average journey times, then the assumptions of Bayes hold and hence it shouldn’t be possible to beat Bayes consistently.

One day I discovered that a friend of the family who had been strongly anti-Bayes had been involved in planning the D-day invasion, and I realised that here was a good example of a complex problem rich in uncertainties, showing the benefits of a more principled approach to uncertainty in planning. I published a brief paper (also here ), which may be helpful. It was almost a decade before I was faced with a planning situation similarly rich with uncertainty.

Navigation

Routine journeys can be thought of as a single entity, with the usual habits of driving to keep in lane and away from the car in front. If the road is blocked one may need to divert, calling for some navigation. The routine epoch is interrupted by ‘higher-level’ considerations. If one has always optimised for journey time one will never have explored the by-ways. If one occasionally uses alternative, slightly slower, routes, one will be in a better position to find a good alternative when you have to divert. Thus optimising for journey time when all goes well is at the expense of robustness: coping well with exceptional circumstances. (This is a form of the Boyd uncertainty principle.)

A more interesting ‘change of epoch’ occurred a few years back. An unprecedented police shoot-out on the M5 near Bristol caused chaos and was widely publicised. The next weekend my partner and I were about to drive down the same stretch of motorway when there were reports of a copy-cat shoot-out. Traffic jams over a large area were inevitable, but we thought that if we were quick enough and took a wide-enough loop around, we should be able to make it.

SAT-NAVs had only just become fairly common, and the previous weekend had shown up their weakness in this situation: everyone gets diverted the same way, so the SAT-NAV sends people into big queues, while others could divert around. This week-end most drivers knew that, and so we expected many to be taking wider detours. But how wide? Too narrow, and one gets into a jam. Too wide and one is too slow, and gets into jams at the far end. Thus the probability of a road actually being jammed depended on the extent to which drivers expected it to be jammed: an example of Keynes’ reflexive probability. It also an example where the existence of meaningful ‘prior probabilities’ is doubtful: the recent popularity of SAT-NAVs and the previous incident made any decision-making based on experience of dubious validity.

This is just the kind of situation for which some of my colleagues criticise ‘the mathematical approach’, so just to add to the fun I drive while my partner, who teaches ‘decision mathematics’ advised. Contrary to what some might have expected, we took a 100-mile right-hook detour, just getting through some choke points in the nick of time, thus having a lot more fun with only about a 20 minute delay from using the motorway. I noticed, though, that rather than use one of the standard decision mathematical methods she used the theory. I wonder if some of the criticisms of mathematics are when people apply a ‘mathematical’ method without considering the theory: that is not mathematics!  

Drunken walks

A man walks along the cliff edge to the pub most evenings. His partner will not let him go if it is too windy, or the forecast is poor. The landlord calls a taxi if it is too windy at closing time.

One night two walkers comment on how dangerous the walk along the cliff is. They are ignored. The drinker walks home and off the cliff.

The cliff had been unstable but had been buttressed. Some had questioned the reliability of the contractors used, but the local authorities had given assurances that the cliff was now stable. And yet the work had been poor and the cliff had collapsed, so that the drinker had followed the path to his death.

Games field

A man notices that different things are going on as he passes a games field. He decides that he can spend 10 hours over the next 10 years observing what is going on, in an attempt to work out ‘the rules of the game’. If spends 600 1 minute intervals selected at random from games over the 10 years, he may come to have a very good idea of what games are played when, but a poor idea of the rules of any one game. On the other hand if he observes the first 10 hours of play he may form a good view of the rules of the current game, but have no idea of how games are selected. This is an example of the organizational and entropic uncertainty principles, generalizations of Heisenberg’s better-known uncertainty principle.

Particle Physics

Quantum theories arose from a recognition of the limits of the classical view, and were developed by thinkers who corresponded with Whitehead and Smuts, for example, on general logical issues. The similarities can be seen in the Bohm interpretation, for example. Temporarily stable complexes interacting with and within stable fields have dynamics that follow stable rules until previously separate complexes come to interact, in which case one has a ‘collapse of the wave function’ and a discontinuity in the dynamics. These periods of relative homogenaity correspond to Whitehead’s epochs, and the mechanism for change is essentially Cybernetic. In this formalism particles have precise positions and momentum; uncertainty is measurement uncertainty.

The Bohm interpretation only applies when one has quantum equilibrium, so that higher-level epochs are fixed, and consequential changes bounded. Otherwise one has greater uncertainty. 

Quantum Cognition

 Quantum cognition notices certain ‘irrationalities’ about human cognition, and proposes a model ‘inspired by’ quantum mechanics. It is mentioned here because the inspiration seems sound, even if the details are questionable.

Under categorization-decision it notes that asking about a relevant factor can affect categorization. This seems reasonable, as it affects the context.

Under memory-process disassociation it notes that people are more likely to recognize that something had been seen on a list if they were asked about a specific list. Taking this to extremes, people may be more likely to recognize that they have met someone at some conference if a specific conference is named. This seems reasonable. Unpacking effects is similar. The questions in conceptual combinations are similar, but the contexts and results quite different.

Under the Linda problem it notes mis-ordering of probabilities in a context where probability would be hard to judge and possibly discriminatory. Respondents may have reported likelihoods instead: the two are often confused. This would be a ‘representativeness type mechanism’.

There seem to be two things going on here: the experimental subjects might not be good at estimating, and the experimenter might not be very good at analysis. Quantum probability appears to be an attempt to avoid:

  • Problems that arise when the analyst ignores the conditionality of probabilities. 
  • Problems that arise when experimental settings or terminology (e.g. probability and likelihood) are confused.
  • Variation of performance with ‘focus’, such as when a specific list is mentioned.

Quantum probability seems to be a heuristic that gives a more moderate result, thus compensating for the above effects. It seems more natural to take account of them more directly and specifically.

Anyone?

Dave Marsay

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About Dave Marsay
Mathematician with an interest in 'good' reasoning.

3 Responses to Metaphors for complexity and uncertainty

  1. Pingback: Reasoning under uncertainty methods | djmarsay

  2. Pingback: Decoding Reality | djmarsay

  3. Pingback: Examples of Uncertainty in Real Decisions | djmarsay

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