The augmented van Trees inequality

Abstract

We introduce an augmented form of the van Trees inequality, that yields uniformly tighter lower bounds on the minimax squared Bayes risk of estimators compared with the classical van Trees inequality. Our augmented inequality also accommodates prior distributions whose densities need not vanish at the boundaries of their supports. We demonstrate how this refinement can be utilised for elementary proofs of a number of minimax lower bounds for nonparametric estimands, that also often attain sharper constants than those obtained by the alternative Le Cam convergence of experiments theory and the classical van Trees inequality, and in some cases obtain exact constants. As an example, our augmented van Trees inequality can be used to obtain the asymptotic minimax pointwise mean squared error when estimating the regression function in the model with normal errors: when the regression function is univariate and differentiable with Lipschitz derivative we obtain this quantity up to a constant factor of $1.37$; and in the high dimensional regime with a Hölder smooth regression function of smoothness $β\in(0,2]$ we obtain exact constants. Both these results do not follow from an application of the classical van Trees inequality. The flexibility of our augmented van Trees inequality accommodates lower bounds for models beyond Gaussianity, loss functions beyond the squared error loss, and we are also able to incorporate this augmentation into generalised versions of the van Trees inequality for irregular models.

Figures (2)

• Figure 1: The lower bound for $\sup_{t\in T}\mathbb{E}_{P_t}[(\hat{t}(\boldsymbol{X})-t)^2]$ as given by the van Trees inequality (with optimal prior) \ref{['eq:vTcurve']} and the augmented van Trees inequality (with approximately optimal prior) \ref{['eq:AVT']}.
• Figure 2: The 'optimal' prior densities via the classical van Trees and the augmented van Trees inequalities in the setting of minimax Hölder function estimation in Section \ref{['sec:minimax']} with $(\beta,d)=(2,1)$. For the classical van Trees inequality the optimal prior is $\mu(t)=\cos^2({\pi t}/{2})$, with no augmentation. For the augmented van Trees inequality (AVT2) we obtain a lower bound by taking augmentation function $\alpha(t)=(1-|t|)^m$ for $m>0$ and the corresponding optimal prior \ref{['eq:mu-opt-g']}.

Theorems & Definitions (21)

• Theorem 1: Augmented van Trees inequality
• Theorem 2
• Example 3: Augmented van Trees 1
• Example 4: Augmented van Trees 2
• Theorem 5
• Theorem 6: Pointwise Hölder function estimation
• Theorem 7: Exact risk in high-dimensional regime
• Theorem 8: The augmented generalized van Trees inequality
• proof : Proof of Theorem \ref{['thm:gvt']}
• proof : Proof of Theorem \ref{['prop:gvt-simple']}