5,304 research outputs found
Wasserstein convergence in Bayesian and frequentist deconvolution models
We study the multivariate deconvolution problem of recovering the distribution of a signal from independent and identically distributed observations additively contaminated with random errors (noise) from a known distribution. For errors with independent coordinates having ordinary smooth densities, we derive an inversion inequality relating the L1-Wasserstein distance between two distributions of the signal to the L1-distance between the corresponding mixture densities of the observations. This smoothing inequality outperforms existing inversion inequalities. As an application of the inversion inequality to the Bayesian framework, we consider 1-Wasserstein deconvolution with Laplace noise in dimension one using a Dirichlet process mixture of normal densities as a prior measure on the mixing distribution (or distribution of the signal). We construct an adaptive approximation of the sampling density by convolving the Laplace density with a well-chosen mixture of normal densities and show that the posterior measure concentrates around the sampling density at a nearly minimax rate, up to a log-factor, in the L1-distance. The same posterior law is also shown to automatically adapt to the unknown Sobolev regularity of the mixing density, thus leading to a new Bayesian adaptive estimation procedure for mixing distributions with regular densities under the L1-Wasserstein metric. We illustrate utility of the inversion inequality also in a frequentist setting by showing that an appropriate isotone approximation of the classical kernel deconvolution estimator attains the minimax rate of convergence for 1-Wasserstein deconvolution in any dimension d≥1, when only a tail condition is required on the latent mixing density and we derive sharp lower bounds for these problems
Minimax Estimation of Kernel Mean Embeddings
In this paper, we study the minimax estimation of the Bochner integral
also called as the kernel
mean embedding, based on random samples drawn i.i.d.~from , where
is a positive definite
kernel. Various estimators (including the empirical estimator),
of are studied in the literature wherein all of
them satisfy with
being the reproducing kernel Hilbert space induced by . The
main contribution of the paper is in showing that the above mentioned rate of
is minimax in and
-norms over the class of discrete measures and
the class of measures that has an infinitely differentiable density, with
being a continuous translation-invariant kernel on . The
interesting aspect of this result is that the minimax rate is independent of
the smoothness of the kernel and the density of (if it exists). This result
has practical consequences in statistical applications as the mean embedding
has been widely employed in non-parametric hypothesis testing, density
estimation, causal inference and feature selection, through its relation to
energy distance (and distance covariance)
Bayesian adaptation
In the need for low assumption inferential methods in infinite-dimensional
settings, Bayesian adaptive estimation via a prior distribution that does not
depend on the regularity of the function to be estimated nor on the sample size
is valuable. We elucidate relationships among the main approaches followed to
design priors for minimax-optimal rate-adaptive estimation meanwhile shedding
light on the underlying ideas.Comment: 20 pages, Propositions 3 and 5 adde
Optimal graphon estimation in cut distance
Consider the twin problems of estimating the connection probability matrix of
an inhomogeneous random graph and the graphon of a W-random graph. We establish
the minimax estimation rates with respect to the cut metric for classes of
block constant matrices and step function graphons. Surprisingly, our results
imply that, from the minimax point of view, the raw data, that is, the
adjacency matrix of the observed graph, is already optimal and more involved
procedures cannot improve the convergence rates for this metric. This
phenomenon contrasts with optimal rates of convergence with respect to other
classical distances for graphons such as the l 1 or l 2 metrics
A Note on Minimax Testing and Confidence Intervals in Moment Inequality Models
This note uses a simple example to show how moment inequality models used in
the empirical economics literature lead to general minimax relative efficiency
comparisons. The main point is that such models involve inference on a low
dimensional parameter, which leads naturally to a definition of "distance"
that, in full generality, would be arbitrary in minimax testing problems. This
definition of distance is justified by the fact that it leads to a duality
between minimaxity of confidence intervals and tests, which does not hold for
other definitions of distance. Thus, the use of moment inequalities for
inference in a low dimensional parametric model places additional structure on
the testing problem, which leads to stronger conclusions regarding minimax
relative efficiency than would otherwise be possible
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