Algebraic statistics

Remarks

Example: the independence model for two binary variables is the image of $(a, b) \in (0, 1)^2$ under $\varphi(a, b) = (ab, a(1-b), (1-a)b, (1-a)(1-b))$. It equals the set of $p = (p_{11}, p_{12}, p_{21}, p_{22}) \in \Delta_3$ with all $p_{ij} > 0$ and $p_{11} p_{22} - p_{12} p_{21} = 0$. One inclusion: $ab \cdot (1-a)(1-b) - a(1-b) \cdot (1-a)b = 0$. Conversely, such a $p$ is $\varphi(a, b)$ with $a = p_{11} + p_{12}$ and $b = p_{11} + p_{21}$, since $ab = p_{11}(p_{11} + p_{12} + p_{21}) + p_{12} p_{21} = p_{11}(p_{11} + p_{12} + p_{21} + p_{22}) = p_{11}$, and similarly for the other coordinates.

Non-example: the Poisson family $p_k = e^{-\lambda} \lambda^k / k!$, $k = 0, 1, 2, \dots$, is not an algebraic statistical model in this sense. It lives on infinitely many outcomes, so it is not a subset of a simplex $\Delta_{m-1}$, and its coordinates are not polynomial or rational in $\lambda$.