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In many articles regarding computational models of some particular phenomenon, there seems to be a consensus: "the smaller the number of 'free parameters' in the model, the better". So, what is meant by "free parameter", and why is it less desirable to model something with such a parameter?

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A free parameter is one that can be adjusted to make the model fit the data. If I make a model that says $A$ is proportional to $B$, there is one free parameter, the proportionality constant. If my model has a specific value of the proportionality constant, there are no free parameters.If I say that $A$ is a quadratic function of $B$, there are three free parameters, $a,b,c$ in $A=aB^2+bB+c$. That makes it easier to fit the data, even if my model is not correct, so it is a less impressive.

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The following marvellous quote from Freeman Dyson's "A meeting with Enrico Fermi" contains both an example for your first question and an answer to your second.

[Enrico Fermi] delivered his verdict in a quiet, even voice. . . . "To reach your calculated results, you had to introduce arbitrary cut-off procedures that are not based either on solid physics or on solid mathematics."

In desperation I asked Fermi whether he was not impressed by the agreement between our calculated numbers and his measured numbers. He replied, "How many arbitrary parameters did you use for your calculations?" I thought for a moment about our cut-off procedures and said, "Four." He said, "I remember my friend Johnny von Neumann used to say, with four parameters I can fit an elephant, and with five I can make him wiggle his trunk." With that, the conversation was over. . . .

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