Subjective Probability and Geometry: Three Metric Theorems Concerning Random Quantities

Affine properties are more general than metric ones because they are independent of the choice of a coordinate system. Nevertheless, a metric, that is to say, a scalar product which takes each pair of vectors and returns a real number, is meaningful when $n$ vectors, which are all unit vectors and orthogonal to each other, constitute a basis for the $n$-dimensional vector space $\mathcal{A}$. In such a space $n$ events $E_i$, $i = 1, \ldots, n$, whose Cartesian coordinates turn out to be $x^i$, are represented in a linear form. A metric is also meaningful when we transfer on a straight line the $n$-dimensional structure of $\mathcal{A}$ into which the constituents of the partition determined by $E_1, \ldots, E_n$ are visualized. The dot product of two vectors of the $n$-dimensional real space $\mathbb{R}^n$ is invariant: of these two vectors the former represents the possible values for a given random quantity, while the latter represents the corresponding probabilities which are assigned to them in a subjective fashion. We deduce these original results, which are the foundation of our next and extensive study concerning the formulation of a geometric, well-organized and original theory of random quantities, from pioneering works which deal with a specific geometric interpretation of probability concept, unlike the most part of the current ones which are pleased to keep the real and deep meaning of probability notion a secret because they consider a success to give a uniquely determined answer to a problem even when it is indeterminate. Therefore, we believe that it is inevitable that our references limit themselves to these pioneering works.