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'''Model selection''' is the task of selecting a [[mathematical model]] from a set of potential models, given evidence. In its most basic forms, this is one of the fundamental tasks of scientific inquery: determining the principle behind a series of observations is often linked directly to a mathematical model predicting those observations. For example, when [[Gallileo]] performed his [[Aristotelian_theory_of_gravity|inclined plane]] experiments, he demonstrated the motion of the balls fit the parabola predicted by his model.
Of the endless string of possible models that could have produced the data, how can one even begin to choose the correct model? The mathematical approach commonly taken decides between a series of given models, it is still necessary to choose this set of models before beginning. When possible, simple models such as [[polynomials]] or [[quadric|quadrics]] are used as a starting point.
Once the set of possible models are selected, the mathematical analysis allows us to determine the best of these models. What is meant by best is controversial. A good model selection technique will balance goodness of fit and complexity. More complex models will be better able to adapt their shape to fit the data (for example, a sixth-order polynomial can exactly fit six points), but the additional parameters may not represent anything useful. (Perhaps those six points are really just randomly distributed about a line.) Goodness of fit is generally determined in the [[chi-square]] sense. The complexity is generally measured by counting the number of [[free parameters]] in the model.
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