A free parameter is one which is not pre-defined by the model, but which can be chosen or estimated based on theoretical ideas or experimental data. Other types of parameters include fixed and constrained.
- Fixed parameters are completely defined by the model; for example, if your model defines a parameter j = 5, then j is fixed.
- Constrained parameters are restricted to an interval but not completely defined. For instance, if your model defines the parameter k as between 0 and 1, it is a constrained parameter.
Examples of Free Parameters
If your model suggests that A is proportional to B (i.e. A = Bk), the proportionality constant k is a a free parameter. If, instead, our model stated that A = 2 * B, we would only have fixed parameters to work with.
If our model states that y and x are related by the polynomial equation y = ax2+ bx + c, then a, b, and c are all free parameters.
Why To Use Free Parameters—and Why To Avoid Them
We can always add free parameters to a model; and the more we add, the closer our model may seem to fit the data. But with enough arbitrary parameters, a model may be made to fit any data set; there is a big danger of free parameters moving us away from reality and into the realm of wishful thinking. In scientific modeling we try to keep the number of free parameters to a minimum.
If the number of free parameters in a model is the same as the number of distinct, separate statistics which it predicts, we call that model identified. If the number of free parameters in a model is greater than the number of predicted statistics, we call it underidentified. The number of unknowns is greater than the number of knowns, and the model is of little use.
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