One immediate question that came up in my mind with respect to Mr. Willard's idea was whether it could accommodate situations in which the target distribution was a censored distribution or a truncated distribution. The question becomes whether those derived distributions can be expressed as a transformed distribution such that FindFormula can find the proper transform. Unfortunately, at present it looks as if the answer is "no" for both. I think Mathematica's FindFormula could probably be tweaked so that at least some censored distributions would work; getting truncated distributions to work would be a lot harder.
Here's why.
A distribution censored to the interval [a,b] is really the equivalent of the transformed of that distribution clipped to a,b. Thus:
CensoredDistribution[{a,b},dist]===TransformedDistribution[Max[a,Min[b,x]],Distributed[x,dist]]
So, if the basis functions (TargetFunctions) in FindFormula included Min and Max, it should be able to find the right transformation. Unfortunately, if you evaluate this, you get the following error message.
FindFormula[
Map[{#, Clip[3 #, {2, 5}]} &, Range[0, 2, 0.1]], x, 3, All,
TargetFunctions -> {Plus, Times, Sqrt, Abs, Exp, Min, Max}]
FindFormula::nofun: {Plus,Times,Sqrt,Abs,Exp,Min,Max} is not a non empty list of functions supported by FindFormula. >>
So, what has to happen is that Mathematica's FindFormula needs to support Min and Max. I would not think this would be difficult to implement.
Truncation is another matter. In theory, one can express a truncated distribution as a transformed distribution if one allows piecewise transformations. We need a function that maps a value greater than any right truncation point to the rightmost point of the domain of the underlying distribution and that maps a value less than any left truncation point to the leftmost point of the domain of the underlying distribution. If the value is inside the truncation interval then we need to map that point such that its CDF in the truncated distribution as the same as the CDF of the target point in the underlying distribution. So, the following Mathematica code should get you the transformation:
Refine[Quantile[dist, CDF[TruncatedDistribution[{a, b}, dist], x]], a < x < b]
The problem then becomes that Mathematica's FindFormula function may not have the basis functions sufficient to find the transformation. Here's a relatively simple example showing the problem. Let's see if FindFormula could find the transform necessary to mimic a NormalDistribution truncated to the interval [1,3].
With[{dist = NormalDistribution[0, 1]},
Refine[Quantile[dist, CDF[TruncatedDistribution[{a, b}, dist], x]],
a < x < b]]
Here's the output you get. It's a conditional expression, but I am going to cheat and assume (correctly) that the condition is always true inside the interesting region. And I am going to simplify.
-Sqrt[2] InverseErfc[(2 (Erf[a/Sqrt[2]] - Erf[x/Sqrt[2]]))/(Erf[a/Sqrt[2]] - Erf[b/Sqrt[2]])]
FindFormula does not have the primitives that will ever let you find this exact formula. It does not have Erf or InverseErfc and these functions can not be decomposed into operations with the FindFormula primitives. So, we are going to have two problems. First, FindFormula does not understand piecewise functions. Second, FindFormula does not have primitives such as InverseErfc and Erf. While, conceivably Piecewise functions might be supported by a future FindFormula -- but how many pieces would you permit -- I have doubts that Mathematica if included gobs of special functions such as InverseErfc and Erf the whole system would work well. There might have to be some hierarchical system in which, if the regular functions didn't achieve some degree of fit, one went to special functions.
The short answer is that, as it stands, if the ProbabilityDistribution obtained in your formula is best represented as a censored or transformed distribution, the Willard method outlined above probably won't find it. This does NOT mean that the Willard method is bad -- as I've said, it is quite brilliant -- but it does seem to have limitations. More later.
