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6 Model Selection, Comparison, and AveragingNicolaus Copernicus (1473–1543): Polish astrologer, ecclesiastical lawyer, and blasphemer.Famous for his heliocentric model of the solar system, Copernicus argued for replacingthe geocentric model, because the heliocentric model was more “harmonious.” is positioneventually lead (decades later) to Galileo’s famous disharmony with, and trial by, the Church.is story has become a fable of science’s triumph over ideology and superstition. ButCopernicus’ justification looks poor to us now, ideology aside. ere are two problems: themodel was neither particularly harmonious nor more accurate than the geocentric model.e Copernican model was very complicated. In fact, it had similar epicycle clutter as thePtolemaic model (FIGURE 6.1). Copernicus had moved the Sun to the center, but since hestill used perfect circles for orbits, he still needed epicycles. And so “harmony” doesn’t quitedescribe the model’s appearance. Just like the Ptolemaic model, the Copernican model waseffectively a Fourier series, a means of approximating periodic functions. is leads to thesecond problem: the heliocentric model made exactly the same predictions as the geocentricmodel. Equivalent approximations can be constructed whether the Earth is stationary orrather moving. So there was no reason to prefer it on the basis of accuracy alone.Copernicus didn’t appeal just to some vague harmony, though. He also argued for thesuperiority of his model on the basis of needing fewer causes: “We thus follow Nature, whoproducing nothing in vain or superfluous oen prefers to endow one cause with many effects.”70 And it was true that a heliocentric model required fewer circles and epicycles tomake the same predictions as a geocentric model. In this sense, it was simpler than the geocentricmodel.Scholars oen prefer simpler theories. is preference is oen le vague—a kind of aestheticpreference. Other times we retreat to pragmatism, preferring simpler models becausethey are easier to work with. Frequently, scientists cite a loose principle known as OCK-HAM’S RAZOR: models with fewer assumptions are to be preferred. In the case of Copernicusand Ptolemy, the razor makes a clear recommendation. It cannot guarantee that Copernicuswas right (he wasn’t, aer all), but since the heliocentric and geocentric models make thesame predictions, at least the razor offers a clear resolution to the dilemma. But the razorcan be hard to use more generally, because usually we must choose among models that differin both their accuracy and their simplicity. How are we to trade these different criteriaagainst one another? e razor offers no guidance.is chapter describes some of the most commonly used tools for coping with this tradeoff.Some notion of simplicity usually features in all of these tools, and so each is commonlycompared to Ockham’s razor. But each tool is equally about improving predictive accuracy.So they are not like the razor, because they explicitly tradeoff accuracy and simplicity.173