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Two books that I recently read, pursuing my hobby of reading every non-technical book on Probability, were Laws of the Game by Manfred Eigen and Ruthild Winkler (1975) and Randomnicity by Anastasios Tsonis (2008). Both make the point that the way mathematical probability (as opposed to statistics) is typically used in science, engineering and the quantitative social sciences is via “rules-plus-dice” models. Consider, for instance, the Galton-Watson branching process or the M/M/1 queue or the voter model. The definition of such a process involves first a framework describing what a state of the process is and how it can evolve, and second a specification of “rules” for how it actually does evolve, the rules having both deterministic and random (metaphorically, “dice”) components. No-one would deny that such models are informative and empirically accurate in some settings. But what is the big picture here – how extensive a range of phenomena can be usefully studied this way? Does our applied probability community suffer from the “if all you have is a hammer, then everything looks like a nail” syndrome, continuing to develop rules-plus-dice models simply because it's the only methodology we know?
“Complex networks” is one topic where controversy has arisen. One can devise various probability models that reproduce some features (in particular, power law degree distributions) of various real-world networks, but what does one learn from this exercise? Critics such as John Doyle at Cal Tech (see, for example, the 2008 paper with David Alderson Contrasting views of complexity and their implications for network-centric infrastructures, which can be found online) argue “complexity arises in highly evolved biological and technological systems primarily to provide mechanisms to create robustness... This view of complexity is fundamentally different from the dominant perspective, which downplays function, constraints and tradeoffs and tends to minimize the role of organization and design”. Continuing, they argue that the “organized complexity” one sees in much of the real world is essentially different from the “disorganized complexity” produced by rules-plus-dice probability models.
Instead of engaging “highly evolved systems”, let me turn to four more familiar everyday topics:
These are not primarily mathematical objects, but there is a lot of (non-numerical) data concerning such objects, so we can choose to look at them with a statistical or mathematical eye. I am inclined to argue, following Doyle, that a priori one expects such objects to have some “organized complexity” which would not be usefully represented by simple rules-plus-dice models. Is there an alternative? Well, ever since Shannon we have understood the concept of entropy rate of natural language (or any discrete data stream) as both interesting in itself and useful for a specific purpose (data compression). And coastlines are claimed as an iconic example of sets with a fractal dimension. Now my point is: given appropriate data, to talk about or to estimate entropy rate or fractal dimension does not depend on inventing some specific, and typically unrealistic, rules-plus-dice model. Instead the relevant mathematical theory associates a numerical statistic (entropy rate; fractal dimension) to each process in some large class of processes (stationary ergodic processes; random fractals, suitably defined). Such a statistic is interesting because it relates to theoretical aspects of the behavior of the process but can also be estimated from data.
These two topics prompt me to muse about other contexts where it would be more fruitful to take this approach. That is, to study a phenomenon by considering some broad class of processes, and to view real-world data as arising from some initially unspecified process in the broad class, rather than jumping immediately to some more structured rules-plus-dice model. Finance models based on “no arbitrage” might be viewed as one such context. I have not spotted any such approach within the huge current literature on social networks (where models tend to make some explicit and unrealistic “random links” assumption). Some of my own current work, on abstracted models of road networks in the plane, does take this approach, but within a rather specialized context. Of course, we have many well-studied mathematically defined classes of processes (Gaussian processes, Markov chains, etc.), but can you think of other real-world structures with plentiful data (such as the 4 bullet-pointed above) to which this approach might be tried? If so, let me know!
David Aldous, Berkeley
Editors’ note: This is the second installment of a regular opinion column.