|Top Models. Let's sexualize kids. |
What could possibly go wrong?
Recall that in his Nobel laureate speech, "The Pretence of Knowledge," Friedrich August von Hayek (the father, we may suppose, of Freddy September) pointed to organized complexity as a major issue in economics and similar fields:
Organized complexity... means that the character of the structures showing it depends not only on the properties of the individual elements of which they are composed, and the relative frequency with which they occur, but also on the manner in which the individual elements are connected with each other. In the explanation of the working of such structures we can for this reason not replace the information about the individual elements by statistical information, but require full information about each element if from our theory we are to derive specific predictions about individual events. Without such specific information about the individual elements we shall be confined to what on another occasion I have called mere pattern predictions - predictions of some of the general attributes of the structures that will form themselves, but not containing specific statements about the individual elements of which the structures will be made up. [Emph. added]
The part that depends on "the manner in which the individual elements are connected with each other" is the formal cause. In modern parlance this is sometimes called "emergent properties" because the whole system has the property while the individual elements do not.
In other words, it's the form (pattern, organization) that is the key to intelligibility. Given a set of interconnected elements X1, X2,... Xn, we cannot legitimately replace the specific Xs with X-bar as we may in cases of disorganized complexity. At best we would obtain only statistical conclusions about the entire system -- as we do in fact regarding quantum mechanics.
When there are only a few elements in the system, the scientist introduces simplifications: infinite Euclidean space, ideal gasses, perfectly elastic collisions, and the like. Arrhenius' law relating CO2 to temperature assumes the atmosphere extends to infinity. TOF read a joke - he has forgotten where - about using models to predict the SuperBowl, which is a sort of football game sometimes (but not this past year) played by two teams. In the punchline, the physicist says, "consider each player to be a perfectly elastic sphere on an infinite Euclidean field..." Mathematics tends to become ornery when bumping up against boundary values* and it is precisely at the extremes where many models pop their suspenders.
(*) boundary values. TOF has his old college text, Fourier Series and Boundary Value Problems, by Ruel Churchill, which he will someday nerve himself to re-read.
Such simplifications may be illuminating precisely because they isolate and simplify certain aspects of the total system. (This is what we mean in Latin by abstractio.) No one ever dropped an actual cannon ball in an actual vacuum, but by thought-experimenting the motion of heavy objects in a theoretical vacuum, late medieval early modern physicists came "close enough" to modeling the motion of heavy objects in the air, a plenum which provides little resistance to cannonballs, water balloons, or ex-boyfriend's suitcases. But when the plenum becomes thicker -- dropping a ball in Jell-O™ -- resistance becomes a non-trivial factor in the model. The key to the success of Early Modern Science was to shift the discourse from the real world to an ideal world.
But no one supposes that dropping leaves, birds, or air balloons from the Tower of Pisa invalidates Benedetti's law of falling bodies* precisely because we realize that these are outside the boundaries of the model.
(*) What, did you think Galileo was first to do this?
|Three faces. Are you certain?|
The Three Faces of UncertaintyEt al. and Walker described three dimensions to uncertainty.
The Uncertainty of Where. In Part II, we considered where in the model uncertainties lie: in the
- model structure,
- model execution,
- parameters, and
The Uncertainty of Degree:
|Trucking up the knowledge line|
1. Determinacy: Everything is perfectly known. Yeah. Like that'll happen.The Uncertainty of Kind. The third dimension of uncertainty is two-fold:
2. Statistical Uncertainty: Any uncertainty in which a statistical expression of the uncertainty can be formulated. This is what scientists usually mean by "uncertainty" and includes such things as measurement uncertainty, sampling, confidence intervals, et al. It is the stable from which the wild p-value rides the night deceiving the unwary into unwarranted certainty. To rise to this level of uncertainty requires that:
- the functional relationships in the model are good descriptions of the phenomena being simulated, and
- the data used to calibrate the model are representative of circumstances to which the model will be applied.
3. Scenario Uncertainty. There is a range of possible outcomes, but the mechanisms leading to these outcomes are not well understood and therefore, it's not possible to formulate the probability of any particular outcome. Statistical uncertainty sinks into scenario uncertainty where a continuum of outcomes expressed stochastically changes to a range of discrete possibilities of unknown likelihoods.
These scenarios are "plausible descriptions of how the system and/or its driving forces may develop in the future." Typically, they involve the context or "environment" of the model and include such things as future technology changes, public attitudes, commodity prices, etc. Many forecasts of the economic effects legislation have come to grief because people have changed their economic behavior as a result of incentives built into the law. Scenario assumptions are usually unverifiable -- since they mostly involve the future. Climate models, for example, may use several scenarios for future CO2 emissions ranging from "no further increase" through "continues as today" to "increases exponentially." How much temperature change to forecast will depend on which of these scenarios eventually unfolds. Unlike physical processes, behavioral processes do not follow mathematical laws.
For a continuous variable, we may apply various distributions as approximations, but this is not so easily done with discrete scenarios which do not relate to one another by simple magnitude. Hence, the uncertainty "Which Scenario Will Unfold" cannot be folded into the statistically uncertainty and the reported statistical uncertainty will be smaller than the actual uncertainty. Instead of a projection of what will happen, the model produces a range of forecasts of what might happen under each of several discrete cases. Scenario uncertainty can be expressed:
4. Recognized Ignorance: Fundamental uncertainty about the mechanisms and functional relationships being studied. "Known unknowns." When neither functional relationships nor statistical properties are known, the scientific basis even for for scenario development is very weak. For example: do clouds increase temperatures or decrease it? This sort of ignorance may be either:
- as a range in the outcomes of an analysis due to different underlying assumptions (i.e., "If S, then X")
- as uncertainty about which changes and developments are relevant for the outcomes of interest, (i.e., have we considered the right scenarios?)
- as uncertainty about the levels of these relevant changes and developments
5. Total Ignorance: We don't even know what we don't know. "Unknown unknowns." Newton, for example, did not include c (speed of light) in his equations of motion.
- Reducible ignorance: which can be resolved by further research
- Irreducible ignorance: neither research nor development can provide sufficient knowledge about the essential relationships
- Epistemic uncertainty: Uncertainty due to imperfect knowledge. This might be reducible by further research.
- Ontological uncertainty: Uncertainty due to inherent variability in the system being modeled. This is especially true of systems concerning social, economic, and technological developments.
"The government are very keen on amassing statistics. They collect them, add them, raise them to the nth power, take the cube root and prepare wonderful diagrams. But you must never forget that every one of these figures comes in the first instance from the village watchman, who just puts down what he damn pleases."
-- Sir Josiah Stamp
|Error bars for #1 and #5 do not overlap any of the others.|
|Millikan's Oil Drop experiment|
Ontological uncertainty is related to what quality practitioners call "process capability." Process output will vary from time to time and from source to source. Sometimes this variation can be assigned to particular causes, but there is always a residuum of variation due to complex combinations of many causes that are impossible (or perhaps only impractical) to resolve further. This is sometimes called for pragmatic reasons "random variation."* This residual variation, remaining after all assignable causes have been accounted for, is called the process capability,
(*) random. But not because randomness is a cause of anything. It is because there is no one particular cause that accounts for the variation. The dice may show a 12 for many different reasons: orientation in the hand, angle of the throw, force of the throw, friction of the felt, etc., so that no matter how well-controlled any one causal factor is, the same results can occur from chance combinations of other causes.
Sources of ontological uncertainty include
- Inherent randomness of nature: the chaotic and unpredictable nature of natural processes. The ultimate example is (in the Copenhagen interpretation) quantum events.
- Human behaviour (behavioural variability): non-rational behaviour, discrepancies between what people say and what they actually do (cognitive dissonance), deviations from standard behavioural patterns (micro-level behaviour)
- Social, economic, and cultural dynamics (societal variability): the chaotic and unpredictable nature of societal processes (macro-level behaviour)
- Technological surprise: New developments or breakthroughs in technology or unexpected side-effects of technologies.
If the uncertainty drops to statistical uncertainty, the model may use a frequency distribution to represent it. However, it is important to distinguish uncertainty within a distribution and uncertainty about which distribution is in play ("between distributions"). That is, it one thing to sample (as most do in college "stats" classes) from a "normal distribution with μ=0 and σ=1" and quite another to know whether the mean really is 0 and the standard deviation really is 1 -- or that the data are adequately modeled by a normal distribution at all!*
(*) adequately modeled. The normal distribution runs to infinity in both directions. No real-world process is known to do so. Therefore, the model will fail somewhere in the extremes, if nowhere else. In fact, darn few stochastic processes show "a state of statistical control" enough to merit any statistical distribution at all.
For our (not) final installment -- Yes, TOF can hear the cries of relief -- we will take a tiptoe through the tulips, as it were, and look at some particular examples of uncertainties.
- Box, George E.P., William G. Hunter, J. Stuart Hunter. Statistics for Experimenters, Pt.IV “Building Models and Using Them.” (John Wiley & Sons, 1978)
- Curry, Judith and Peter Webster. “Climate Science and the Uncertainty Monster” Bull. Am. Met. Soc., V. 92, Issue 12 (December 2011)
- El-Haik, Basem and Kai Yang. "The components of complexity in engineering design," IIE Transactions (1999) 31, 925-934
- von Hayek, Friedrich August. "The Pretence of Knowledge," Lecture to the memory of Alfred Nobel, December 11, 1974
- Petersen, Arthur Caesar. "Simulating Nature" (dissertation, Vrije Universiteit, 2006)
- Ravetz, Jerome R. NUSAP - The Management of Uncertainty and Quality in Quantitative Information
- Swanson, Kyle L. "Emerging selection bias in large-scale climate change simulations," Geophysical Research Letters
- Turney, Jon. "A model world." aeon magazine (16 December 2013)
- Walker, W.E., et al. "Defining Uncertainty: A Conceptual Basis for Uncertainty Management in Model-Based Decision Support." Integrated Assessment (2003), Vol. 4, No. 1, pp. 5–17
- Weaver, Warren. "Science and Complexity," American Scientist, 36:536 (1948)