One of the problems of polynomial regression is the interpretation of the parameters. If we want to know how  changes per unit change of ; we cannot just check the value of , because  and higher coefficients, if present, have an effect on such a quantity. Therefore, the  coefficients are no longer slopes, they are something else. In the previous example,  is positive and hence the curve begins with a positive slope, but  is negative and hence, after a while, the line begins to curve downwards. So, it is like we have two forces at play, one pushing the line up and the other down. The interplay depends on the value of . When  (on the original scale or 2 on the centered scale), the dominating contribution comes from , and when  dominates.

The problem of interpreting the parameters is not just a mathematical problem. If this were the case, we could solve it by careful inspection and understanding of the model. The problem is that, in many cases, the parameters are not translated to meaningful quantities in our domain knowledge. We cannot relate them with the metabolic rate of a cell or the energy emitted by a distant galaxy or the number of bedrooms in a house. They are just nobs we can tweak to improve the fit but without a clear physical meaning. In practice, most people will agree that polynomials of order higher than 2 or 3 are not generally very useful models and alternatives are preferred, such as Gaussian processes, which is the main subject of Chapter 7, Gaussian Processes.