Generalized Maximum Entropy |
Reparameterization of the errors in a regression equation is the process of specifying a support for the errors, observation by observation. If a two-point support is used, the error for the tth observation is reparameterized by setting , where and are the upper and lower bounds for the tth error , and and represent the weight associated with the point and . The error distribution is usually chosen to be symmetric, centered around zero, and the same across observations so that , where R is the support value chosen for the problem (Golan, Judge, and Miller; 1996).
The generalized maximum entropy (GME) formulation was proposed for the ill-posed or underdetermined case where there is insufficient data to estimate the model with traditional methods. is reparameterized by defining a support for (and a set of weights in the cross entropy case), which defines a prior distribution for .
In the simplest case, each is reparameterized as , where and represent the probabilities ranging from [0,1] for each , and and represent the lower and upper bounds placed on . The support points, and , are usually distributed symmetrically around the most likely value for based on some prior knowledge.
With these reparameterizations, the GME estimation problem is
where y denotes the column vector of length T of the dependent variable; denotes the matrix of observations of the independent variables; p denotes the LK column vector of weights associated with the points in Z; w denotes the LT column vector of weights associated with the points in V; , , and are K-, L-, and T-dimensional column vectors, respectively, of ones; and and are and dimensional identity matrices.
These equations can be rewritten using set notation as follows:
The subscript l denotes the support point (l=1, 2, ..., L), k denotes the parameter (k=1, 2, ..., K), and t denotes the observation (t=1, 2, ..., T).
The GME objective is strictly concave; therefore, a unique solution exists. The optimal estimated probabilities, p and w, and the prior supports, Z and V, can be used to form the point estimates of the unknown parameters, , and the unknown errors, e.
Note: This procedure is experimental.