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  "body": "\n## Definition\n\nLet $S$ be a set, the sample space, with a [[sigma-algebra|04WSW8T3SY]] $\\mathcal{S}$ on $S$.\nA *statistical model* on $(S, \\mathcal{S})$ is a pair $(S, \\mathcal{P})$, where $\\mathcal{P}$ is a set of [[probability measures|0VGMEBVW1M]] on $(S, \\mathcal{S})$.\nThe model is often written with an index set $\\Theta$ as $\\mathcal{P} = \\set{P_\\theta : \\theta \\in \\Theta}$, and $\\Theta$ is the parameter space.\n\n## Remarks\n\nExample: $S = \\set{0, 1}^n$ with all subsets measurable, and $P_p(x) = p^{k}(1 - p)^{n - k}$ where $k = x_1 + \\dots + x_n$, for $p \\in [0, 1]$.\nEach $P_p$ is a probability measure, because the binomial theorem gives $\\sum_{x} P_p(x) = (p + 1 - p)^n = 1$.\nThis is the model of $n$ independent tosses of a coin with unknown chance $p$ of heads.\nNon-example: the set of measures $\\set{c \\cdot P_{1/2} : c \u003e 0}$ is not a statistical model, because for $c \\ne 1$ the measure $c \\cdot P_{1/2}$ has total mass $c$, so it is not a probability measure.\nThe data are modelled as a value $x$ of a random element $X$ with law in $\\mathcal{P}$.\n",
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