Some probability distributions can be described by relevant parameters, such as the mean and standard deviation for the case of Gaussians. This parameterization defines a manifold in which probability functions can be studied from a geometrical perspective. Information Geometry studies probability functions as points in this parameter-defined space, applying differential geometry. Probability functions can also be described by their mean square error, which can be approximated by a second-degree polynomial. In this contribution, we describe the characterization of probability functions in terms of the coefficients of second-degree polynomials that approximate their mean square error. The parameters of this polynomial define a manifold, approximated by a second-degree polynomial, in which probability distributions from different families can be compared by computing the arc length of the points linked to the distributions. One of the advantages of this approach is that the probability distributions can be compared in a more geometrical perspective. In this contribution, we describe the geometry of the induced manifold, and at the same time, we compare this manifold with the common structures from Information Geometry such as the Fisher–Rao distance. We offer empirical evidence that the characterization of probability distributions based on their mean square error can be of relevance not only for comparing them but also to gain a different look at the relation between probability distributions.
Garduño et al. (2026) studied this question.