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Andrew Gelman
Choices is statistical graphics
Graphics are a central part of statistical graphics but are
typically neglected in statistical theory. We view graphics
as a form of model checking, with interesting and surprising
visual patterns corresponding to aspects of the data that are
unexpected, that is were not likely to occur given the assumed
model (which might be implicit). At the same time, graphics
are an increasingly prevalent part of journalism. "Infographics"
often have the goal of aesthetic appeal to draw a casual
viewer in deeper, while "statistical graphics" often have the goal
to reveal patterns for viewers who are already interested in
the problem. We discuss all of these points in the context
of graphics we have published for our research over the years.
Download the slides (a previous version of the slides presented at the seminar)
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Harrison Zhou
Asymptotic Normality and Optimalities in Estimation of Large Gaussian Graphical Model
The Gaussian graphical model, a popular paradigm for
studying relationship among variables in a wide range of
applications, has attracted great attention in recent years.
This talk considers a fundamental question: When is it
possible to estimate low-dimensional parameters at
parametric square-root rate in a large Gaussian graphical
model? A novel regression approach is proposed to obtain
asymptotically efficient estimation of each entry of a precision
matrix under a sparseness condition relative to the sample
size. When the precision matrix is not sufficiently sparse,
or equivalently the sample size is not sufficiently large,
a lower bound is established to show that it is no longer
possible to achieve the parametric rate in the estimation
of each entry. This lower bound result, which provides
an answer to the delicate sample size question, is established
with a novel construction of a subset of sparse precision
matrices in an application of Le Cam's Lemma. Moreover,
the proposed estimator is proven to have optimal convergence
rate when the parametric rate cannot be achieved, under
a minimal sample requirement.
The proposed estimator is applied to test the presence of
an edge in the Gaussian graphical model or to recover
the support of the entire model, to obtain adaptive rate-optimal
estimation of the entire precision matrix as measured by
the matrix lq operator norm, and to make inference in
latent variables in the graphical model. All these are achieved
under a sparsity condition on the precision matrix and a side
condition on the range of its spectrum. This significantly relaxes
the commonly imposed uniform signal strength condition
on the precision matrix, irrepresentable condition on
the Hessian tensor operator of the covariance matrix or
the l1 constraint on the precision matrix. Numerical results
confirm our theoretical findings. The ROC curve of the proposed
algorithm, Asymptotic Normal Thresholding (ANT), for support
recovery significantly outperforms that of the popular
GLasso algorithm.
Download the slides
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