314 research outputs found
A Bernstein-Von Mises Theorem for discrete probability distributions
We investigate the asymptotic normality of the posterior distribution in the
discrete setting, when model dimension increases with sample size. We consider
a probability mass function on \mathbbm{N}\setminus \{0\} and a
sequence of truncation levels satisfying Let denote the maximum likelihood estimate of
and let denote the
-dimensional vector which -th coordinate is defined by \sqrt{n}
(\hat{\theta}_n(i)-\theta_0(i)) for We check that under mild
conditions on and on the sequence of prior probabilities on the
-dimensional simplices, after centering and rescaling, the variation
distance between the posterior distribution recentered around
and rescaled by and the -dimensional Gaussian distribution
converges in probability to
This theorem can be used to prove the asymptotic normality of Bayesian
estimators of Shannon and R\'{e}nyi entropies. The proofs are based on
concentration inequalities for centered and non-centered Chi-square (Pearson)
statistics. The latter allow to establish posterior concentration rates with
respect to Fisher distance rather than with respect to the Hellinger distance
as it is commonplace in non-parametric Bayesian statistics.Comment: Published in at http://dx.doi.org/10.1214/08-EJS262 the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org
About the posterior distribution in hidden Markov models with unknown number of states
We consider finite state space stationary hidden Markov models (HMMs) in the
situation where the number of hidden states is unknown. We provide a
frequentist asymptotic evaluation of Bayesian analysis methods. Our main result
gives posterior concentration rates for the marginal densities, that is for the
density of a fixed number of consecutive observations. Using conditions on the
prior, we are then able to define a consistent Bayesian estimator of the number
of hidden states. It is known that the likelihood ratio test statistic for
overfitted HMMs has a nonstandard behaviour and is unbounded. Our conditions on
the prior may be seen as a way to penalize parameters to avoid this phenomenon.
Inference of parameters is a much more difficult task than inference of
marginal densities, we still provide a precise description of the situation
when the observations are i.i.d. and we allow for possible hidden states.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ550 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Adaptive estimation of High-Dimensional Signal-to-Noise Ratios
We consider the equivalent problems of estimating the residual variance, the
proportion of explained variance and the signal strength in a
high-dimensional linear regression model with Gaussian random design. Our aim
is to understand the impact of not knowing the sparsity of the regression
parameter and not knowing the distribution of the design on minimax estimation
rates of . Depending on the sparsity of the regression parameter,
optimal estimators of either rely on estimating the regression parameter
or are based on U-type statistics, and have minimax rates depending on . In
the important situation where is unknown, we build an adaptive procedure
whose convergence rate simultaneously achieves the minimax risk over all up
to a logarithmic loss which we prove to be non avoidable. Finally, the
knowledge of the design distribution is shown to play a critical role. When the
distribution of the design is unknown, consistent estimation of explained
variance is indeed possible in much narrower regimes than for known design
distribution
Non parametric finite translation mixtures with dependent regime
In this paper we consider non parametric finite translation mixtures. We
prove that all the parameters of the model are identifiable as soon as the
matrix that defines the joint distribution of two consecutive latent variables
is non singular and the translation parameters are distinct. Under this
assumption, we provide a consistent estimator of the number of populations, of
the translation parameters and of the distribution of two consecutive latent
variables, which we prove to be asymptotically normally distributed under mild
dependency assumptions. We propose a non parametric estimator of the unknown
translated density. In case the latent variables form a Markov chain (Hidden
Markov models), we prove an oracle inequality leading to the fact that this
estimator is minimax adaptive over regularity classes of densities
Identifiability and consistent estimation of nonparametric translation hidden Markov models with general state space
This paper considers hidden Markov models where the observations are given as
the sum of a latent state which lies in a general state space and some
independent noise with unknown distribution. It is shown that these fully
nonparametric translation models are identifiable with respect to both the
distribution of the latent variables and the distribution of the noise, under
mostly a light tail assumption on the latent variables. Two nonparametric
estimation methods are proposed and we prove that the corresponding estimators
are consistent for the weak convergence topology. These results are illustrated
with numerical experiments
Malliavin Calculus for regularity structures: the case of gPAM
Malliavin calculus is implemented in the context of [M. Hairer, A theory of
regularity structures, Invent. Math. 2014]. This involves some constructions of
independent interest, notably an extension of the structure which accomodates a
robust, and purely deterministic, translation operator, in -directions,
between "models". In the concrete context of the generalized parabolic Anderson
model in 2D - one of the singular SPDEs discussed in the afore-mentioned
article - we establish existence of a density at positive times.Comment: Minor revision of [v1]. This version published in Journal of
Functional Analysis, Volume 272, Issue 1, 1 January 2017, Pages 363-41
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