64 research outputs found
Direct Ensemble Estimation of Density Functionals
Estimating density functionals of analog sources is an important problem in
statistical signal processing and information theory. Traditionally, estimating
these quantities requires either making parametric assumptions about the
underlying distributions or using non-parametric density estimation followed by
integration. In this paper we introduce a direct nonparametric approach which
bypasses the need for density estimation by using the error rates of k-NN
classifiers asdata-driven basis functions that can be combined to estimate a
range of density functionals. However, this method is subject to a non-trivial
bias that dramatically slows the rate of convergence in higher dimensions. To
overcome this limitation, we develop an ensemble method for estimating the
value of the basis function which, under some minor constraints on the
smoothness of the underlying distributions, achieves the parametric rate of
convergence regardless of data dimension.Comment: 5 page
Justification of Logarithmic Loss via the Benefit of Side Information
We consider a natural measure of relevance: the reduction in optimal
prediction risk in the presence of side information. For any given loss
function, this relevance measure captures the benefit of side information for
performing inference on a random variable under this loss function. When such a
measure satisfies a natural data processing property, and the random variable
of interest has alphabet size greater than two, we show that it is uniquely
characterized by the mutual information, and the corresponding loss function
coincides with logarithmic loss. In doing so, our work provides a new
characterization of mutual information, and justifies its use as a measure of
relevance. When the alphabet is binary, we characterize the only admissible
forms the measure of relevance can assume while obeying the specified data
processing property. Our results naturally extend to measuring causal influence
between stochastic processes, where we unify different causal-inference
measures in the literature as instantiations of directed information
Sharp Bounds for Generalized Uniformity Testing
We study the problem of generalized uniformity testing \cite{BC17} of a
discrete probability distribution: Given samples from a probability
distribution over an {\em unknown} discrete domain , we
want to distinguish, with probability at least , between the case that
is uniform on some {\em subset} of versus -far, in
total variation distance, from any such uniform distribution.
We establish tight bounds on the sample complexity of generalized uniformity
testing. In more detail, we present a computationally efficient tester whose
sample complexity is optimal, up to constant factors, and a matching
information-theoretic lower bound. Specifically, we show that the sample
complexity of generalized uniformity testing is
- …