Sunday, October 24, 2010

Trade Off #14: Precision vs Simplicity

When you describe something, the more precisely your model explains the given data, the more complicated it must be. Don't believe me? Lo, behold these examples, then:
  • In describing the path up to my apartment, I could say "there are stairs", or I could say "there are fourteen stairs"; vagueness is less precise but it is also simpler. The bottom line is that having to walk up any number of stairs is too many.
  • In fitting a model to data, one can explain more variance by including more free variables, at the cost of complication. There are plenty of ways to punish a model for having additional parameters and thus make the model earn each of its parameters through explanatory ability. (see here and here)
  • The failure of humans to adequately trade off precision and simplicity in certain contexts, like when we say that the prob of X and Y is greater than the prob of just X, is one of our well-documented cognitive biases. (see here)
There are some well-known incidents in the history of science that on first glance appear to be exceptions to this trade-off. For example, Kepler's idea of elliptical planetary orbits eliminated the need for astronomers to model extra epicycles, both simplifying and adding precision to our understanding of planetary motion.

But in the view of this committee, these precision-enabling paradigm shifts are especially complicated, involve the shifting of assumptions at a fundamental level, and only seem simple in distant hindsight. That's one reason why they are so hard to come upon.

(photo of spiral galaxy, which Johannes Kepler probably would have marveled at, goes to NASA's Marshall Center)