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  "body": "\n## Statement\n\nLet $\\set{P_\\theta : \\theta \\in \\Theta}$ be a [[parametric model|Q0DXKAQXE1]] with $\\Theta \\subseteq \\R^k$ open, and let $X_1, X_2, \\dots$ be [[i.i.d.|0AJN7WHBJZ]] with law $P_{\\theta_0}$ for some $\\theta_0 \\in \\Theta$.\nAssume the following.\n\n1. The model is differentiable in quadratic mean at $\\theta_0$, with nonsingular [[Fisher information|G1Z3WSTE8Q]] matrix $I(\\theta_0)$.\n2. For every $\\varepsilon \u003e 0$ there are tests $\\varphi_n = \\varphi_n(X_1, \\dots, X_n)$ with $E_{\\theta_0}[\\varphi_n] \\to 0$ and $\\sup_{\\norm{\\theta - \\theta_0} \\ge \\varepsilon} E_\\theta[1 - \\varphi_n] \\to 0$.\n3. The [[prior|00NBBSXRQS]] has a density with respect to Lebesgue measure that is continuous and positive in a neighbourhood of $\\theta_0$.\n\nLet $\\hat{\\theta}_n$ be an efficient estimator, that is, $\\sqrt{n}(\\hat{\\theta}_n - \\theta_0) = \\frac{1}{\\sqrt{n}} \\sum_{i=1}^{n} I(\\theta_0)^{-1} \\dot{\\ell}_{\\theta_0}(X_i) + o_P(1)$, where $\\dot{\\ell}_{\\theta_0}$ is the score function.\nThen the [[posterior distribution|0KJ7ZS8AJA]] $\\Pi_n$ of $\\theta$ given $X_1, \\dots, X_n$ satisfies\n$$\\norm{\\Pi_n - \\mathcal{N}\\left(\\hat{\\theta}_n, \\tfrac{1}{n} I(\\theta_0)^{-1}\\right)}_{TV} \\to 0$$\nin $P_{\\theta_0}$-[[probability|0EFDDXTVD6]], where $\\norm{\\cdot}_{TV}$ is the [[total variation distance|3SZESKX3N5]].\n\n## Remarks\n\nThe proof expands the log-likelihood near $\\theta_0$: by local asymptotic normality it is, up to $o_P(1)$, a quadratic function of $\\sqrt{n}(\\theta - \\theta_0)$.\nThe tests of hypothesis 2 show that the posterior mass outside shrinking neighbourhoods of $\\theta_0$ vanishes, and hypothesis 3 makes the prior nearly constant on these neighbourhoods.\nSo the posterior is close to the normalised exponential of a quadratic form, that is, to a [[multivariate normal|0G558GJQMW]] law.\nUnder the usual regularity conditions the maximum likelihood estimator is efficient and can serve as $\\hat{\\theta}_n$.\nExample: for i.i.d. Bernoulli($p_0$) data with $0 \u003c p_0 \u003c 1$ and any beta prior, the posterior of $p$ is approximately $\\mathcal{N}(\\hat{p}_n, \\hat{p}_n(1 - \\hat{p}_n)/n)$ for large $n$, with $\\hat{p}_n$ the proportion of successes.\nThe prior must charge every neighbourhood of $\\theta_0$: a prior that vanishes near $\\theta_0$ gives a posterior that stays away from $\\theta_0$, and the conclusion fails.\nThe theorem can fail in nonparametric models.\n\n## Explanations\n\n### L1 human\n\nWith enough data, the prior stops mattering much.\nThe posterior then looks like a bell curve centred at the best classical estimate, with the spread that classical theory predicts.\nSo, in large samples, Bayesian credible intervals and frequentist confidence intervals nearly agree.\n\n### L2 human\n\nThe theorem is the bridge between [[Bayesian inference|0JWRJX2RMP]] and frequentist asymptotics in regular parametric models.\nIt implies that an equal-tailed $95\\%$ credible interval for a coordinate of $\\theta$ has frequentist coverage tending to $95\\%$.\nUnder additional integrability conditions, the posterior mean is also asymptotically efficient, like the maximum likelihood estimator.\nIts hypotheses are the same local asymptotic normality conditions that drive the asymptotic theory of the maximum likelihood estimator.\n",
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