Semiparametric and nonparametric BvM theorems ask when some low-dimensional functional has a Gaussian posterior limit even though the full parameter is infinite-dimensional.
Functionals and influence functions
The target is often a functional \(\psi(P)\), not the whole distribution.
Definition. A functional \(\psi\) is pathwise differentiable at \(P_0\) if there is an influence function \(\tilde\psi\in L_0^2(P_0)\) such that along regular paths, \[ \frac{\mathrm{d}}{\mathrm{d}t}\psi(P_t)\bigg|_{t=0} =P_0\tilde\psi s, \] where \(s\) is the score of the path.
Remark. The efficient variance bound is \(P_0\tilde\psi^2\).
Semiparametric BvM
A functional BvM says the posterior of the functional is asymptotically normal and efficient.
Theorem. Under posterior contraction of the nuisance parameter, sufficient regularity of \(\psi\), and a no-bias condition, the posterior of \[ \sqrt n(\psi(P)-\hat\psi) \] converges to \(N(0,P_0\tilde\psi^2)\).
Proof. Expand \(\psi(P)-\psi(P_0)\) into its linear influence-function term plus a remainder. Posterior contraction controls the remainder. The likelihood shifts the posterior in the least favorable direction, and the linear term obeys the ordinary Gaussian limit.
Nonparametric failure modes
The full posterior on an infinite-dimensional parameter usually cannot be Gaussian in strong norms.
Remark. Different functionals can have different regularity. Smooth linear functionals may satisfy BvM while the full density or regression function does not.
Credible intervals for functionals
When semiparametric BvM holds, Bayesian credible intervals can be efficient confidence intervals.
Corollary. Posterior quantile intervals for a regular functional have asymptotic frequentist coverage \(1-\alpha\) and length determined by the efficient variance.
Proof. The posterior quantiles converge to Gaussian quantiles with variance \(P_0\tilde\psi^2/n\), and the efficient estimator has the same asymptotic sampling law.
Pathwise differentiable functionals
Definition. Let \(\psi(P)\in\mathbb R\). It is pathwise differentiable at \(P_0\) relative to tangent space \(\mathcal T\subset L_0^2(P_0)\) if there exists \(\widetilde\psi\in\overline{\mathcal T}\) such that for every regular path \[ \frac{\mathrm{d}P_t}{\mathrm{d}P_0} =1+tg+o(t) \quad(g\in\mathcal T), \] \[ \left. \frac{\mathrm{d}}{\mathrm{d}t} \psi(P_t) \right|_{t=0} =P_0(\widetilde\psi g). \] The minimum-norm representer \(\widetilde\psi\) is the efficient influence function.
Least-favorable local perturbation
Theorem (semiparametric BvM template). Suppose a posterior concentrates around nuisance parameter \(\eta_0\), and the likelihood admits a uniform expansion along a least-favorable path: \[ \ell_n(\eta_{t/\sqrt n}) -\ell_n(\eta) =t\Delta_{n,\psi} -\frac12t^2V_\psi +o_{P_0}(1), \] where \[ \Delta_{n,\psi} =\frac1{\sqrt n}\sum_{i=1}^n \widetilde\psi(X_i) \Rightarrow N(0,V_\psi), \qquad V_\psi=P_0\widetilde\psi^2. \] If the prior is stable under this local change of variables and nuisance bias is \(o(n^{-1/2})\), then \[ \Pi\left( \sqrt n\{\psi(\eta)-\widehat\psi_n\} \in\cdot\mid X^{(n)} \right) \rightsquigarrow N(0,V_\psi), \] where \[ \widehat\psi_n =\psi(\eta_0)+ \frac1n\sum_i\widetilde\psi(X_i) +o_P(n^{-1/2}). \]
Proof outline. Evaluate the posterior Laplace transform of the centered functional. Shift the integration variable along the least-favorable path by \(t/\sqrt n\). The likelihood ratio contributes \[ t\Delta_{n,\psi}+\frac12t^2V_\psi \] after centering, while prior stability makes the transformed prior ratio tend to one. Uniform remainder and posterior localization permit integration. Convergence of Laplace transforms identifies the Gaussian posterior limit.
No-bias condition
Proposition. Suppose a plug-in functional has expansion \[ \psi(\eta)-\psi(\eta_0) =P_0\{\widetilde\psi_\eta(\eta-\eta_0)\} +R(\eta,\eta_0), \] with \[ |R(\eta,\eta_0)| \lesssim d(\eta,\eta_0)^2. \] If the nuisance posterior contracts at rate \(\varepsilon_n\), then a sufficient remainder condition for root-\(n\) BvM is \[ \sqrt n\,\varepsilon_n^2\to0. \]
Proof. On the contraction set, \[ \sqrt n|R(\eta,\eta_0)| \lesssim\sqrt n\,\varepsilon_n^2=o(1). \] The first-order term is handled by the least-favorable score projection.
This sufficient condition is not always necessary; exact conjugacy, orthogonality, or sharper remainder structure can weaken it. But it explains why nuisance estimation at slower than \(n^{-1/4}\) often obstructs root-\(n\) functional inference.
Linear functionals in white noise
Example. In \[ Y_j=\theta_j+n^{-1/2}\varepsilon_j, \] consider \[ \psi(\theta)=\sum_{j\ge1}a_j\theta_j, \qquad (a_j)\in\ell^2. \] The efficient estimator is \[ \widehat\psi=\sum_ja_jY_j, \] with variance \[ \frac1n\sum_ja_j^2. \] Under independent Gaussian prior \(\theta_j\sim N(0,\tau_j^2)\), the posterior mean is \[ \sum_ja_j \frac{n\tau_j^2}{1+n\tau_j^2}Y_j. \] Its bias at truth is \[ -\sum_ja_j \frac{1}{1+n\tau_j^2}\theta_{0j}. \] A BvM theorem requires this bias to be \(o(n^{-1/2})\) and posterior variance to approach \(n^{-1}\sum_ja_j^2\).
Regular and irregular functionals
Proposition. In white noise, a linear functional with coefficients \((a_j)\notin\ell^2\) has no finite-variance influence function in the observation tangent space and is not root-\(n\) regular.
Proof. A candidate influence function corresponds to coefficient sequence \(a\), whose variance is \(\sum_ja_j^2\). If this diverges, the derivative is not a bounded linear functional on the \(L^2\) tangent space, so the Riesz representer does not exist there.
Point evaluation of an unrestricted nonparametric function is a standard irregular functional. Its posterior limit depends on smoothing and has a slower rate; imposing a Gaussian BvM statement at root-\(n\) scale is incorrect.
Prior invariance under local shifts
For a product prior on coefficients, a least-favorable shift changes many coordinates. Prior stability requires the log prior ratio \[ \log\frac{\pi(\theta-t a/\sqrt n)} {\pi(\theta)} \] to be \(o_P(1)\) uniformly on posterior-relevant sets after the appropriate nuisance correction. Thin tails, hard truncation, or highly informative shrinkage can violate this even when posterior contraction is optimal.
Weak nonparametric BvM
A whole-posterior BvM may hold in a negative Sobolev space \[ H^{-s} \] where white noise is tight, even though it fails in \(L^2\). The limiting Gaussian is then an infinite-dimensional Gaussian measure in the weak topology. Credible balls in that topology can have coverage, but mapping them to pointwise bands or \(L^2\) balls is not valid without additional arguments.
Bias correction and one-step posteriors
When the raw plug-in posterior is biased for \(\psi\), one may center draws or posterior summaries using an estimated influence-function correction: \[ \widehat\psi_{\mathrm{1step}} =\psi(\widehat\eta) +\mathbb P_n\widetilde\psi_{\widehat\eta}. \] Cross-fitting can make flexible nuisance estimation compatible with asymptotic linearity. Such a corrected distribution is a calibrated inferential construction; it need not be the pushforward of the original exact posterior.
Checklist for a functional BvM claim
- Is the functional pathwise differentiable?
- What is its efficient influence function and variance?
- In which metric does the nuisance posterior contract?
- Is the second-order remainder \(o(n^{-1/2})\)?
- Is the prior stable under the least-favorable shift?
- Is the centering estimator efficient?
- Is convergence pointwise in \(P_0\) or uniform over a class?
Only after these questions are answered do posterior quantiles inherit frequentist validity.