LAN is the local quadratic approximation that connects likelihood theory, Laplace approximation, and Gaussian posterior limits.
Local asymptotic normality
LAN studies likelihood ratios under \(n^{-1/2}\)-local perturbations.
Definition. A model is locally asymptotically normal at \(\theta_0\) with information \(I_{\theta_0}\) if for each fixed \(h\), \[ \log\frac{p_{\theta_0+h/\sqrt n}^{(n)}}{p_{\theta_0}^{(n)}}(X^{(n)}) =h^\top\Delta_n-\frac12h^\top I_{\theta_0}h+o_{P_{\theta_0}}(1), \] where \(\Delta_n\Rightarrow N(0,I_{\theta_0})\).
Remark. The random vector \(\Delta_n\) is the normalized score. The quadratic term is deterministic curvature.
Posterior in local coordinates
The posterior becomes approximately Gaussian after changing variables to \(h=\sqrt n(\theta-\theta_0)\).
Theorem. If the posterior localizes in \(n^{-1/2}\)-neighborhoods and the prior density is continuous and positive at \(\theta_0\), LAN implies that the local posterior density is approximately proportional to \[ \exp\{h^\top\Delta_n-\tfrac12h^\top I_{\theta_0}h\}. \]
Proof. Bayes formula in the \(h\)-coordinate multiplies the likelihood ratio by \(\pi(\theta_0+h/\sqrt n)\). Continuity of the prior makes this factor locally constant, while LAN gives the displayed quadratic likelihood ratio.
Completing the square
The LAN kernel is Gaussian after completing the square.
Proposition. The kernel \[ \exp\{h^\top\Delta-\tfrac12h^\top I h\} \] is proportional to the density of \(N(I^{-1}\Delta,I^{-1})\).
Proof. Complete the square: \[ h^\top\Delta-\frac12h^\top Ih =-\frac12(h-I^{-1}\Delta)^\top I(h-I^{-1}\Delta) +\frac12\Delta^\top I^{-1}\Delta. \] The last term does not depend on \(h\).
Laplace approximation as local LAN
In regular parametric models, Laplace approximation can be viewed as deterministic LAN around the posterior mode.
Remark. LAN is probabilistic and centered around the true parameter; Laplace approximation is analytic and centered around the mode. Under regularity, the two pictures agree to first order.
Differentiability in quadratic mean
Definition. A dominated model is differentiable in quadratic mean at \(\theta_0\) if there is \(\dot\ell_{\theta_0}\in L_0^2(P_{\theta_0})^d\) such that \[ \int \left[ \sqrt{p_{\theta_0+h}} -\sqrt{p_{\theta_0}} -\frac12h^\top\dot\ell_{\theta_0} \sqrt{p_{\theta_0}} \right]^2 \mathrm{d}\mu =o(\|h\|^2). \] The Fisher information is \[ I_{\theta_0} =P_{\theta_0} \dot\ell_{\theta_0}\dot\ell_{\theta_0}^\top. \]
Theorem (DQM implies LAN). For iid observations from a DQM model and fixed \(h\in\mathbb R^d\), \[ \log\frac{ p_{\theta_0+h/\sqrt n}^{(n)}(X^{(n)}) }{ p_{\theta_0}^{(n)}(X^{(n)}) } =h^\top\Delta_n -\frac12h^\top I_{\theta_0}h +o_{P_{\theta_0}}(1), \] where \[ \Delta_n =\frac1{\sqrt n}\sum_{i=1}^n \dot\ell_{\theta_0}(X_i) \Rightarrow N(0,I_{\theta_0}). \]
Proof outline. DQM expands the square-root likelihood ratio as \[ \sqrt{\frac{p_{\theta_0+h/\sqrt n}}{p_{\theta_0}}} =1+\frac{h^\top\dot\ell_{\theta_0}}{2\sqrt n} +r_{n,h}, \qquad P_{\theta_0}r_{n,h}^2=o(n^{-1}). \] Expand \(2\log(1+u)\) and sum over observations. The linear terms produce \(h^\top\Delta_n\); centering and the quadratic terms produce \(-h^\top I_{\theta_0}h/2\); remainder control follows from quadratic mean bounds and truncation. The multivariate CLT gives the score limit.
Local posterior coordinates
Proposition. Let \[ h=\sqrt n(\theta-\theta_0) \] and suppose the prior density is continuous and positive at \(\theta_0\). On every fixed compact set of \(h\), \[ \pi_n(h\mid X^{(n)}) \propto \exp\left\{ h^\top\Delta_n-\frac12h^\top I_{\theta_0}h +o_P(1) \right\}. \] Completing the square gives the Gaussian kernel \[ N\left(I_{\theta_0}^{-1}\Delta_n, I_{\theta_0}^{-1}\right). \]
Proof. The change of variables contributes a constant Jacobian \(n^{-d/2}\), irrelevant after normalization. LAN gives the likelihood term, while \[ \frac{\pi(\theta_0+h/\sqrt n)}{\pi(\theta_0)} \to1 \] uniformly on compact \(h\)-sets. Finally, \[ h^\top\Delta_n-\frac12h^\top Ih =-\frac12(h-I^{-1}\Delta_n)^\top I(h-I^{-1}\Delta_n) +\frac12\Delta_n^\top I^{-1}\Delta_n. \]
Efficient centering
Proposition. If the MLE is regular, \[ \sqrt n(\widehat\theta_n-\theta_0) =I_{\theta_0}^{-1}\Delta_n+o_P(1). \] Therefore the local Gaussian posterior may be centered at \(\widehat\theta_n\) instead of \(\theta_0+n^{-1/2}I^{-1}\Delta_n\).
Proof. Taylor-expand the score: \[ 0=\dot\ell_n(\widehat\theta_n) =\dot\ell_n(\theta_0) +\ddot\ell_n(\widetilde\theta_n) (\widehat\theta_n-\theta_0). \] Divide by \(n\), use \(n^{-1}\ddot\ell_n(\widetilde\theta_n)\to-I_{\theta_0}\), and solve.
Contiguous local alternatives
Theorem. Under \(\theta_n=\theta_0+h/\sqrt n\), \[ \Delta_n\Rightarrow N(I_{\theta_0}h,I_{\theta_0}). \]
Proof. LAN gives a joint Gaussian limit under \(P_{\theta_0}\) for the score and log likelihood ratio. Le Cam's third lemma tilts the score limit by covariance \(\operatorname{Cov}(\Delta,h^\top\Delta)=I_{\theta_0}h\).
Uniform LAN and dimension growth
Pointwise LAN for each fixed \(h\) is not enough to approximate a posterior whose mass ranges over growing sets. A BvM proof needs uniform control such as \[ \sup_{\|h\|\le M_n} \left| \ell_n(\theta_0+h/\sqrt n)-\ell_n(\theta_0) -h^\top\Delta_n+\frac12h^\top Ih \right| =o_P(1), \] for \(M_n\to\infty\) slowly, together with posterior localization inside that set. If dimension \(d=d_n\) grows, third-order likelihood remainders often impose conditions resembling \[ \frac{d_n^3}{n}\to0, \] though the exact power depends on model geometry and norm. Consistency alone gives no such uniform quadratic approximation.
Boundary contrast
Example. For \[ X_i\sim N(\theta,1),\qquad\theta\ge0, \] with true \(\theta_0=0\), local parameter \(h=\sqrt n\,\theta\) is constrained to \(h\ge0\). The local posterior is a normal kernel \[ \exp\{h\Delta_n-h^2/2\}1\{h\ge0\}, \] normalized as a truncated normal. Ordinary unconstrained BvM and symmetric credible intervals fail at the boundary.