A prime example of model selection is presented in curve fitting.  Given a sample \(s\) consisting of a set of points in the plane \((x, y)\), we are asked to choose the curve that fits these data best. We assume that the models under consideration are of the form \(y = f(x) + \epsilon\), where \(\epsilon\) is a normal distribution with mean 0 and a fixed standard deviation, and where \(f\) is a polynomial function. Different models are characterized by polynomials of different degrees that have different numbers of parameters. Estimations fix the parameters of these polynomials. For example, for the 0-degree polynomial \(f(x) = c_{0}\) we estimate the constant \(\hat{c_{0}}\) for which the probability of the data is maximal, and for the 1-degree polynomial \(f(x) = c_{0} + c_{1}\, x\) we estimate the slope \(\hat{c_{1}}\) and the offset \(\hat{c_{0}}\). Now notice that for a total of \(n\) points, we can always find a polynomial of degree \(n\) that intersects with all points exactly, resulting in a comparatively high maximum likelihood \(P(s \mid \{\hat{c_{0}}, \ldots \hat{c_{n}} \})\). Applying the AIC, however, we will typically find that some model with a polynomial of degree \(k &lt; n\) is preferable. Although \(P(s \mid \{\hat{c_{0}}, \ldots \hat{c_{k}} \})\) will be somewhat lower, this is compensated for in the AIC by the smaller number of parameters.