Why Use Models?

In my previous post, I defined a model as one entity that represents another and can predict some properties of the second.  The ability to predict is the most important aspect of a model.  It indicates not only how to use a model, but also to measure how good a model is.

The question of whether to use models is really somewhat silly, because we innately use hundreds of models every day.  There is a model relating to which hand will move and how far, according to neural impulses sent from you brain.  There is a model of what will happen when you turn the tap.  You have a mental model of the roads in your community and how congested they will be as a function of time.  Your world is filled with models. 

So the question in the title is really more about understanding what models are, so one can use them purposefully.  Whether to use a model comes back to the definition: What are you trying to predict?  How will it benefit you to know the answer before you run the experiment? 

There are three main reasons for using a model:

1. To make predictions where direct experimentation is impractical;
2. To generalize the results of experimentation; or
3. To abstract and generalize other models.

These approaches to modelling reflect roughly the disciplines of engineering, science and mathematics, in that order.  An engineer will use a model if it helps him get the job done.  A scientist is in the business of producing models without necessarily concerning himself how it is used.  The mathematician is the same in that respect, but constructs models in the absence of experimental data.

My description of the engineer provides a testable assertion on when one should use a model.  If it is more practical (i.e. cheaper) to make a direct measurement, then developing and using a model would be a waste of resources.  But there are myriad instances where building and using models is cheaper than running the experiment directly.  If someone had thought to put a scale model of the Tacoma Narrows Bridge (rev. 1) in a wind tunnel, that would have saved a pile of money.

Science is interesting, because it produces models without necessarily wanting to use them.  The ability to make predictions is certainly worth money.  By itself, the raw data obtained from experimentation is hopefully worth more than the effort required to obtain it.  Sufficiently dense sampling spanning the region of interest might support the interpolation of any point of interest.  But such a model is verbose and therefore not very efficient.  A terse summarization of experimental data makes it much more useful.  (More on this in "How good is a model?".)

Early mathematicians examined the commonalities between models to produce abstract models unconnected with experimental data.  The discipline evolved to the creation of models for which there may not be any practical application.

The observation that scientists are more-or-less in the business of producing models suggests another important use for models: To convey a description of an entity from one person to another.  Since this involves communication, a secondary criterion for the quality of a model is its conciseness.

We may want to distinguish using a model from merely transferring it: A person receiving a model may want to run the model to make predictions, and that constitutes its actual use.  Merely sending or receiving a model doesn't really amount to using it.

To summarize, the decision of whether to use a model (as opposed to direct measurement) should be driven by economic considerations.  The model is superior if it provides a prediction (i.e. an answer) more cheaply than direct measurement.  In cases where direct measurement is impossible, its cost can be taken as infinite -- in which case using the model is clearly advantageous.