Do Buzz de Terence Tao
Autor: Renato Assuncao | Publicado: 2011-06-27T15:12:00-03:00

In order to use mathematical modelling in order to solve a real-world problem, one ideally would like to have three ingredients besides the actual mathematical analysis:

* A good mathematical model. This is a mathematical construct which connects the observable data, the predicted outcome, and various unspecified parameters of the model to each other. In some cases, the model may be probabilistic instead of deterministic (thus the predicted outcome will be given as a random variable rather than as a fixed quantity).

* A good set of observable data.

* Good values for the parameters of the model.

For instance, if one wanted to work out the distance D to a distant galaxy, the model might be&nbsp;Hubble's law&nbsp;v = H D relating the distance to the recessional velocity v, the data might be the recessional velocity v (or, more realistically, a proxy for that velocity, such as the red shift), and the only parameter in this case would be the Hubble constant H. This is a particularly simple situation; of course, in general one would expect a much more complex model, a much larger set of data, and a large number of parameters. (And parameters need not be numerical; a model, for instance, could posit an unknown functional relationship between two observable quantities, in which case the function itself is the unknown parameter.)

As mentioned above, in ideal situations one has all three ingredients: a good model, good data, and good parameters. In this case the only remaining difficulty is a&nbsp;direct&nbsp;one, namely to solve the equations of the model with the given data and parameters to obtain the result. This type of situation pervades undergraduate homework exercises in applied mathematics and physics, and also accurately describes many mature areas of engineering (e.g. civil engineering or mechanical engineering) in which the model, data, and parameters are all well understood. One could also classify pure mathematics as being the quintessential example of this type of situation, since the models for mathematical foundations (e.g. the ZFC model for set theory) are incredibly well understood (to the point where we rarely even think of them as models any more), and one primarily works with well-formulated problems with precise hypotheses and data.

However, there are many situations in which one or more ingredients are missing. For instance, one may have a good model and good data, but the parameters of the model are initially unknown. In that case, one needs to first solve some sort of&nbsp;inverse problem&nbsp;to recover the parameters from existing sets of data (and their outcomes), before one can then solve the direct problem. In some cases, there are clever ways to gather and use the data so that various unknown parameters largely cancel themselves out, simplifying the task. (For instance, to test the efficiency of a drug, one can use a double-blind study in order to cancel out the numerous unknown parameters that affect both the control group and the experimental group equally.) Typically, one cannot solve for the parameters exactly, and so one must accept an increased range of error in one's predictions. This type of problem pervades undergraduate homework exercises in statistics, and accurately describes many mature sciences, such as physics, chemistry, materials science, and some of the life sciences.

Another common situation is when one has a good model and good parameters, but an incomplete or corrupted set of data. Here, one often has to clean up the data first using error-correcting techniques before proceeding (this often requires adding a mechanism for noise or corruption into the model itself, e.g. adding gaussian white noise to the measurements). This type of problem pervades undergraduate exercises in signal processing, and often arises in computer science and communications science.

In all of the above cases, mathematics can be utilised to great effect, though different types of mathematics are used for different situations (e.g. computational mathematics when one has a good model, data set, and parameters; statistics when one has good model and data set but unknown parameters; computer science, filtering, and compressed sensing when one has good model and parameters, but unknown data; and so forth). However, there is one important situation where the current state of mathematical sophistication is only of limited utility, and that is when it is the&nbsp;model&nbsp;which is unreliable. In this case, even having excellent data and knowledge of parameters may lead to error or a false sense of security; this for instance arose during the recent financial crisis, in which models based on independent gaussian fluctuations in various asset prices turned out to be totally incapable of describing tail events.

Nevertheless, there are still some ways in which mathematics can assist in this type of situation. For instance, one can mathematically test the&nbsp;robustness&nbsp;of a model by replacing it with other models and seeing the extent to which the results change. If it turns out that the results are largely unaffected, then this builds confidence that even a somewhat incorrect model may still yield usable and reasonably accurate results. At the other extreme, if the results turn out to be highly sensitive to the model assumptions, then even a model with a lot of theoretical justification would need to be heavily scrutinised by other means (e.g. cross-validation) before one would be confident enough to use it. Another use of mathematics in this context is to test the consistency of a model. For instance, if a model for a physical process leads to a non-physical consequence (e.g. if a partial differential equation used in the model leads to solutions that become infinite in finite time), this is evidence that the model needs to be modified or discarded before it can be used in applications.

It seems to me that one of the reasons why mathematicians working in different disciplines (e.g. mathematical physicists, mathematical biologists, mathematical signal processors, financial mathematicians, cryptologists, etc.) have difficulty communicating to each other mathematically is that their basic environment of model, data, and parameters are so different: a set of mathematical tools, principles, and intuition that works well in, say, a good model, good parameters, bad data environment may be totally inadequate or even misleading when working in, say, a bad model, bad parameters, good data environment. (And there are also other factors beyond these three that also significantly influence the mathematical environment and thus inhibit communication; for instance, problems with an active adversary, such as in cryptography or security, tend to be of a completely different nature than problems in the only adverse effects come from natural randomness, which is for instance the case in safety engineering.)