Given independent and identically distributed data from a compactly supported, $\alpha$-Hölder density $f$, we study estimation of the Fisher information of the Gaussian-smoothed density $f*\varphi_t$, where $\varphi_t$ is the density of $N(0, t)$. We derive the minimax rate including the sharp dependence on $t$ and show some simple, plug-in type estimators are optimal for $t > 0$, even though extra debiasing steps are widely employed in the literature to achieve the sharp rate in the unsmoothed ($t = 0$) case. Due to our result’s sharp characterization of the scaling in $t$, plug-in estimators of the mutual information and entropy are shown to achieve the parametric rate by way of the I-MMSE and de Bruijn’s identities.

The advent of large-scale inference has spurred reexamination of conventional statistical thinking. In a series of highly original articles, Efron persuasively illustrated the danger for downstream inference in assuming the veracity of a posited null distribution. In a Gaussian model for n many z-scores with at most$k \lt \frac {n}{2}$nonnulls, Efron suggests estimating the parameters of an empirical null$N(\theta , \sigma ^{2})$instead of assuming the theoretical null$N(0, 1)$. Looking to the robust statistics literature by viewing the nonnulls as outliers is unsatisfactory as the question of optimal rates is still open; even consistency is not known in the regime$k \asymp n$which is especially relevant to many large-scale inference applications. However, provably rate-optimal robust estimators have been developed in other models (e.g. Huber contamination) which appear quite close to Efron’s proposal. Notably, the impossibility of consistency when$k \asymp n$in these other models may suggest the same major weakness afflicts Efron’s popularly adopted recommendation. A sound evaluation thus requires a complete understanding of information-theoretic limits. We characterize the regime of k for which consistent estimation is possible, notably without imposing any assumptions at all on the nonnull effects. Unlike in other robust models, it is shown consistent estimation of the location parameter is possible if and only if$\frac {n}{2} {-} k = \omega (\sqrt {n})$, and of the scale parameter in the entire regime$k \lt \frac {n}{2}$. Furthermore, we establish sharp minimax rates and show estimators based on the empirical characteristic function are optimal by exploiting the Gaussian character of the data.

The normal means model is often studied under the assumption of a known variance. However, ignorance of the variance is a frequent issue in applications and basic theoretical questions still remain open in this setting. This article establishes that the sharp minimax rate of variance estimation in square error is $(\frac{\log\log n}{\log n})^2$ under arguably the most mild assumption imposed for identifiability: bounded means. The rate-optimal estimator proposed in this article achieves the optimal rate by estimating $O\left(\frac{\log n}{\log\log n}\right)$ cumulants and leveraging a variational representation of the noise variance in terms of the cumulants of the data distribution. The minimax lower bound involves a moment matching construction.

Sparse additive models are an attractive choice in circumstances calling for modelling flexibility in the face of high dimensionality. We study the signal detection problem and establish the minimax separation rate for the detection of a sparse additive signal. Our result is nonasymptotic and applicable to the general case where the univariate component functions belong to a generic reproducing kernel Hilbert space. Unlike the estimation theory, the minimax separation rate reveals a nontrivial interaction between sparsity and the choice of function space. We also investigate adaptation to sparsity and establish an adaptive testing rate for a generic function space; adaptation is possible in some spaces while others impose an unavoidable cost. Finally, adaptation to both sparsity and smoothness is studied in the setting of Sobolev space, and we correct some existing claims in the literature.

To develop a complete description of sensory encoding, it is necessary to account for trial-to-trial variability in cortical neurons. Using a linear model with terms corresponding to the visual stimulus, mouse running speed, and experimentally measured neuronal correlations, we modeled short term dynamics of L2/3 murine visual cortical neurons to evaluate the relative importance of each factor to neuronal variability within single trials. We find single trial predictions improve most when conditioning on the experimentally measured local correlations in comparison to predictions based on the stimulus or running speed. Specifically, accurate predictions are driven by positively co-varying and synchronously active functional groups of neurons. Including functional groups in the model enhances decoding accuracy of sensory information compared to a model that assumes neuronal independence. Functional groups, in encoding and decoding frameworks, provide an operational definition of Hebbian assemblies in which local correlations largely explain neuronal responses on individual trials.