935 research outputs found
Consistency of a recursive estimate of mixing distributions
Mixture models have received considerable attention recently and Newton
[Sankhy\={a} Ser. A 64 (2002) 306--322] proposed a fast recursive algorithm for
estimating a mixing distribution. We prove almost sure consistency of this
recursive estimate in the weak topology under mild conditions on the family of
densities being mixed. This recursive estimate depends on the data ordering and
a permutation-invariant modification is proposed, which is an average of the
original over permutations of the data sequence. A Rao--Blackwell argument is
used to prove consistency in probability of this alternative estimate. Several
simulations are presented, comparing the finite-sample performance of the
recursive estimate and a Monte Carlo approximation to the permutation-invariant
alternative along with that of the nonparametric maximum likelihood estimate
and a nonparametric Bayes estimate.Comment: Published in at http://dx.doi.org/10.1214/08-AOS639 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
On nonparametric estimation of a mixing density via the predictive recursion algorithm
Nonparametric estimation of a mixing density based on observations from the
corresponding mixture is a challenging statistical problem. This paper surveys
the literature on a fast, recursive estimator based on the predictive recursion
algorithm. After introducing the algorithm and giving a few examples, I
summarize the available asymptotic convergence theory, describe an important
semiparametric extension, and highlight two interesting applications. I
conclude with a discussion of several recent developments in this area and some
open problems.Comment: 22 pages, 5 figures. Comments welcome at
https://www.researchers.one/article/2018-12-
Model-based clustering using non-parametric Hidden Markov Models
Thanks to their dependency structure, non-parametric Hidden Markov Models
(HMMs) are able to handle model-based clustering without specifying group
distributions. The aim of this work is to study the Bayes risk of clustering
when using HMMs and to propose associated clustering procedures. We first give
a result linking the Bayes risk of classification and the Bayes risk of
clustering, which we use to identify the key quantity determining the
difficulty of the clustering task. We also give a proof of this result in the
i.i.d. framework, which might be of independent interest. Then we study the
excess risk of the plugin classifier. All these results are shown to remain
valid in the online setting where observations are clustered sequentially.
Simulations illustrate our findings
Mixed LICORS: A Nonparametric Algorithm for Predictive State Reconstruction
We introduce 'mixed LICORS', an algorithm for learning nonlinear,
high-dimensional dynamics from spatio-temporal data, suitable for both
prediction and simulation. Mixed LICORS extends the recent LICORS algorithm
(Goerg and Shalizi, 2012) from hard clustering of predictive distributions to a
non-parametric, EM-like soft clustering. This retains the asymptotic predictive
optimality of LICORS, but, as we show in simulations, greatly improves
out-of-sample forecasts with limited data. The new method is implemented in the
publicly-available R package "LICORS"
(http://cran.r-project.org/web/packages/LICORS/).Comment: 11 pages; AISTATS 201
Modeling heterogeneity in random graphs through latent space models: a selective review
We present a selective review on probabilistic modeling of heterogeneity in
random graphs. We focus on latent space models and more particularly on
stochastic block models and their extensions that have undergone major
developments in the last five years
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