890 research outputs found
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
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
Consistent estimation of the filtering and marginal smoothing distributions in nonparametric hidden Markov models
In this paper, we consider the filtering and smoothing recursions in
nonparametric finite state space hidden Markov models (HMMs) when the
parameters of the model are unknown and replaced by estimators. We provide an
explicit and time uniform control of the filtering and smoothing errors in
total variation norm as a function of the parameter estimation errors. We prove
that the risk for the filtering and smoothing errors may be uniformly upper
bounded by the risk of the estimators. It has been proved very recently that
statistical inference for finite state space nonparametric HMMs is possible. We
study how the recent spectral methods developed in the parametric setting may
be extended to the nonparametric framework and we give explicit upper bounds
for the L2-risk of the nonparametric spectral estimators. When the observation
space is compact, this provides explicit rates for the filtering and smoothing
errors in total variation norm. The performance of the spectral method is
assessed with simulated data for both the estimation of the (nonparametric)
conditional distribution of the observations and the estimation of the marginal
smoothing distributions.Comment: 27 pages, 2 figures. arXiv admin note: text overlap with
arXiv:1501.0478
Fundamental limits for learning hidden Markov model parameters
We study the frontier between learnable and unlearnable hidden Markov models
(HMMs). HMMs are flexible tools for clustering dependent data coming from
unknown populations. The model parameters are known to be identifiable as soon
as the clusters are distinct and the hidden chain is ergodic with a full rank
transition matrix. In the limit as any one of these conditions fails, it
becomes impossible to identify parameters. For a chain with two hidden states
we prove nonasymptotic minimax upper and lower bounds, matching up to
constants, which exhibit thresholds at which the parameters become learnable
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