Gaussian shift experiments are the canonical local limits of smooth models. They separate information geometry from model-specific details: the parameter shifts a Gaussian mean, and the covariance or precision operator records the information available in each direction.
Finite-dimensional shifts
Definition. Let \(I\) be positive definite. The Gaussian shift experiment \[ \mathcal G_I=\{N_p(h,I^{-1}):h\in\mathbb R^p\} \] observes \[ Y=h+Z,\qquad Z\sim N_p(0,I^{-1}). \] Equivalently, the canonical statistic \[ \Delta=IY \] has law \(N_p(Ih,I)\).
Proposition. Relative to \(h=0\), \[ \log\frac{\mathrm{d}P_h}{\mathrm{d}P_0}(Y) = h^\top IY-\frac12h^\top Ih = h^\top\Delta-\frac12h^\top Ih. \]
Proof. The log density ratio is \[ -\frac12(Y-h)^\top I(Y-h) +\frac12Y^\top IY. \] Expanding the first quadratic form cancels \(Y^\top IY\) and leaves the stated linear and quadratic terms. \(\square\)
Whitening and information ellipsoids
Set \(W=I^{1/2}Y\) and \(u=I^{1/2}h\). Then \[ W=u+\varepsilon,\qquad\varepsilon\sim N_p(0,I_p). \] The information norm \[ \|h\|_I=(h^\top Ih)^{1/2} \] is the signal-to-noise distance from \(P_0\) to \(P_h\). Directions with larger Fisher information are easier to distinguish at the same Euclidean displacement.
Proposition. For \(P_h=N(h,I^{-1})\), \[ D_{\mathrm{KL}}(P_h\|P_0)=\frac12h^\top Ih, \] \[ \rho(P_h,P_0)=\exp\{-h^\top Ih/8\}, \] and \[ H^2(P_h,P_0)=2\{1-e^{-h^\top Ih/8}\}. \]
Quadratic-loss estimation
Theorem. Under loss \[ L(h,a)=(a-h)^\top W(a-h), \qquad W\succeq0, \] the equivariant estimator \(\delta(Y)=Y\) has constant risk \[ R(h,Y)=\operatorname{tr}(WI^{-1}). \] It is minimax.
Proof. Since \(Y-h=Z\), \[ R(h,Y)=E(Z^\top WZ)=\operatorname{tr}(WI^{-1}). \] For priors \(h\sim N(0,\tau^2I_p)\), the Bayes risk increases to this constant as \(\tau\to\infty\). Weak duality gives the matching minimax lower bound. \(\square\)
Minimaxity does not imply admissibility. Under ordinary squared loss and \(p\geq3\), \(Y\) is dominated by James–Stein shrinkage. This finite-dimensional fact later constrains claims about asymptotic admissibility.
Gaussian testing
Example (one direction). For testing \(h=0\) against \(h=h_1\), Neyman–Pearson rejects for large \(h_1^\top IY\). Under the null, \[ \frac{h_1^\top IY}{\sqrt{h_1^\top Ih_1}} \sim N(0,1), \] while under \(h_1\) its mean is \(\sqrt{h_1^\top Ih_1}\). The level-\(\alpha\) power is \[ 1-\Phi\{z_{1-\alpha}-\|h_1\|_I\}. \]
Example (two-sided composite test). For \(H_0:h=0\) against unrestricted \(h\), the likelihood-ratio statistic is \[ Y^\top IY=\Delta^\top I^{-1}\Delta. \] It has \(\chi_p^2\) law under the null and noncentral \(\chi_p^2(h^\top Ih)\) law under \(h\).
Normal priors and shrinkage
Proposition. Under prior \(h\sim N_p(m,V)\) and squared loss, the posterior is normal with covariance and mean \[ V_{\mathrm{post}}=(V^{-1}+I)^{-1}, \] \[ m_{\mathrm{post}} =(V^{-1}+I)^{-1}(V^{-1}m+IY). \] The posterior mean shrinks \(Y\) toward \(m\) according to prior precision relative to experimental information.
Hilbert-space Gaussian shifts
Definition. Let \(W\) be an isonormal Gaussian process on a Hilbert space \(\mathcal H\): \[ E\{W(g)W(k)\}=\langle g,k\rangle_{\mathcal H}. \] The Gaussian shift indexed by \(h\in\mathcal H\) has likelihood ratio \[ \frac{\mathrm{d}P_h}{\mathrm{d}P_0} = \exp\{W(h)-\tfrac12\|h\|_{\mathcal H}^2\}. \]
The Cameron–Martin space \(\mathcal H\) consists exactly of shifts absolutely continuous with respect to the base Gaussian law. Shifts outside it are mutually singular. In semiparametric theory, the tangent space and efficient score embed the local model into such a Gaussian Hilbert-space experiment.
Why the shift is canonical
In a LAN model, the central sequence has null limit \(N(0,I)\) and local mean \(Ih\). Its likelihood ratios converge to those displayed above. Solving a local decision problem in the original model is therefore asymptotically bounded by solving the corresponding problem in \(\mathcal G_I\).