Variety of Probabilistic Independence

Question

We may consider a probability space on $n$ outcomes as a point in $\mathbb{R}^{n+1}$ by assigning outcomes to standard basis vectors, and picking a point with its coordinate on each basis vector equal to the probability of the corresponding outcomes. The point then belongs to the probability simplex: $\Delta^n$, which is a regular $n$-simplex given as the convex hull of the standard basis vectors. This gives us a new geometric representation of independence. Let's consider all probability spaces where our outcomes are $(a,b,c...)$ for $1\leq a\leq A$, $1\leq b\leq B$ and so on, with the coordinates of our outcomes mutually independent. This set is the intersection of $\Delta^{ABC...}$ with an affine variety $V$ in $\mathbb{C}^{ABC...+1}$. $V$ is the zero set of equations $\ [p(a_0,b_1,c_1,...)+p(a_1,b_1,c_1,...)...]p(a_0,b_0,c_0,...)-[p(a_0,b_0,c_0,...)+p(a_1,b_0,c_0,...)...]p(a_0,b_1,c_1,...)$, where $p(\cdot)$ denotes probability. What can we say about $V$? Its dimension is obvious. Is it connected? Does it have singular points? Does it have a nice parametrization?

Answer 1

$V$ is the affine cone over a Segre variety. The Segre variety itself is isomorphic to a product of projective spaces (more or less by definition) so in particular is connected and smooth.

This sort of connection between probability and algebraic geometry goes under the name algebraic statistics. You can check out, for example, Lectures on Algebraic Statistics by Drton, Sturmfels, and Sullivant.