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  "body": "\n## Definition\n\nLet $\\mathcal{P} = \\set{P_\\theta : \\theta \\in \\Theta}$ be a [[statistical model|JZ6R7J36K5]] indexed by a parameter space $\\Theta$.\nThe model is *identifiable* if the map $\\theta \\mapsto P_\\theta$ from $\\Theta$ to the set of probability measures is [[injective|0PVG1CRAJN]].\nThat is, for all $\\theta_1, \\theta_2 \\in \\Theta$, if $P_{\\theta_1} = P_{\\theta_2}$, then $\\theta_1 = \\theta_2$.\n\n## Remarks\n\nExample: the normal model $P_{(\\mu, \\sigma^2)} = \\mathcal{N}(\\mu, \\sigma^2)$ with $\\mu \\in \\R$ and $\\sigma^2 \u003e 0$ is identifiable, because the mean and the variance of the law recover $\\mu$ and $\\sigma^2$.\nNon-example: let $P_{(a, b)} = \\mathcal{N}(a + b, 1)$ with $(a, b) \\in \\R^2$.\nThen $P_{(1, 0)} = P_{(0, 1)}$ while $(1, 0) \\ne (0, 1)$, so the model is not identifiable.\nA function $\\tau(\\theta)$ is identifiable when $P_{\\theta_1} = P_{\\theta_2}$ implies $\\tau(\\theta_1) = \\tau(\\theta_2)$; in the non-example, $\\tau(a, b) = a + b$ is identifiable.\n",
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