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  "body": "\n## Definition\n\nLet $\\Delta_{m-1} = \\set{p \\in \\R^m : p_i \\ge 0, \\ \\sum_{i=1}^m p_i = 1}$ be the probability simplex of distributions on $m$ outcomes.\nAn *algebraic statistical model* is a [[statistical model|K3AMMSY8DD]] $\\mathcal M \\subseteq \\Delta_{m-1}$ of one of the following forms:\n\n1. the image $\\mathcal M = \\varphi(\\Theta)$ of an open parameter set $\\Theta \\subseteq \\R^d$ under a polynomial or rational map $\\varphi \\colon \\Theta \\to \\Delta_{m-1}$, or\n2. the set $\\mathcal M = \\set{p \\in \\Delta_{m-1} : f(p) = 0 \\text{ for all } f \\in F}$ for a set $F$ of polynomials in the [[polynomial ring|0SDC934R52]] $\\R[p_1, \\dots, p_m]$.\n\n*Algebraic statistics* is the study of statistical models and methods through the [[ideals|096Z9MP2ME]] of polynomials vanishing on such models and the [[algebraic varieties|HV6YBS0YDH]] they define.\n\n## Remarks\n\nExample: the independence model for two binary variables is the image of $(a, b) \\in (0, 1)^2$ under $\\varphi(a, b) = (ab, a(1-b), (1-a)b, (1-a)(1-b))$.\nIt equals the set of $p = (p_{11}, p_{12}, p_{21}, p_{22}) \\in \\Delta_3$ with all $p_{ij} \u003e 0$ and $p_{11} p_{22} - p_{12} p_{21} = 0$.\nOne inclusion: $ab \\cdot (1-a)(1-b) - a(1-b) \\cdot (1-a)b = 0$.\nConversely, 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.\n\nNon-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.\nIt 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 $\\lambda$.\n",
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