Discussion on the choice of $H_0$

When the conditions for MaxEnt PDF projection exist, then (3.5) holds. All reference hypotheses in class (3.5), determined by function $h({\bf z})$, produce exactly the same projected PDF, which will be the maximum entropy projected PDF. This follows from the fact that if (3.5) holds, then the J-function (2.3) is independent of ${\bf x}$ when ${\bf z}$ is fixed. This means that in the generation process (for generating samples of $G({\bf x})$), explained in Section 3.3, the manifold distribution (2.5) is in fact the uniform distribution. This is true regardless of which reference hypothesis is chosen within class (3.5).

It may seem difficult to choose a reference hypothesis by selecting the function $h({\bf z})$ in (3.5). Although any reference hypothesis in class (3.5) will result in the same projected PDF, there are some that are easier to analyze. After all, it is necessary to derive the distribution $p({\bf z}\vert H_0)$, preferably in closed form. Since we must also assume that the function $f(\;)$ in (3.4) exists, then we can restrict ourselves to the exponetial class (3.7). This class is determined by the type of norm in effect. The feature may also contain more than one norm. For example, the feature

$${\bf z}=\left[ \sum_{i=1}^N \vert x_i\vert, \; \sum_{i=1}^N x_i^2\right]$$

has two different norms. In this case, both the Laplacian and Gaussian reference hypotheses (See Table 3.1) could be used and would in fact produce the same result. For simple norms, there are canonical reference hypotheses (Table 3.1). In all practical problems we have encountered, either the canonical exponential or Gaussian have sufficed.

So, in summary, we recommend that you insure that the feature contains a simple energy statistic, such as found in Table 3.1, and use the corresponding canonical reference hypothesis. For additional discussion of the reference hypothesis, see Section 2.2.3.