11,561 research outputs found
Introduction to finite mixtures
Mixture models have been around for over 150 years, as an intuitively simple
and practical tool for enriching the collection of probability distributions
available for modelling data. In this chapter we describe the basic ideas of
the subject, present several alternative representations and perspectives on
these models, and discuss some of the elements of inference about the unknowns
in the models. Our focus is on the simplest set-up, of finite mixture models,
but we discuss also how various simplifying assumptions can be relaxed to
generate the rich landscape of modelling and inference ideas traversed in the
rest of this book.Comment: 14 pages, 7 figures, A chapter prepared for the forthcoming Handbook
of Mixture Analysis. V2 corrects a small but important typographical error,
and makes other minor edits; V3 makes further minor corrections and updates
following review; V4 corrects algorithmic details in sec 4.1 and 4.2, and
removes typo
Sequential Bayesian inference for implicit hidden Markov models and current limitations
Hidden Markov models can describe time series arising in various fields of
science, by treating the data as noisy measurements of an arbitrarily complex
Markov process. Sequential Monte Carlo (SMC) methods have become standard tools
to estimate the hidden Markov process given the observations and a fixed
parameter value. We review some of the recent developments allowing the
inclusion of parameter uncertainty as well as model uncertainty. The
shortcomings of the currently available methodology are emphasised from an
algorithmic complexity perspective. The statistical objects of interest for
time series analysis are illustrated on a toy "Lotka-Volterra" model used in
population ecology. Some open challenges are discussed regarding the
scalability of the reviewed methodology to longer time series,
higher-dimensional state spaces and more flexible models.Comment: Review article written for ESAIM: proceedings and surveys. 25 pages,
10 figure
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
Identifiability of parameters in latent structure models with many observed variables
While hidden class models of various types arise in many statistical
applications, it is often difficult to establish the identifiability of their
parameters. Focusing on models in which there is some structure of independence
of some of the observed variables conditioned on hidden ones, we demonstrate a
general approach for establishing identifiability utilizing algebraic
arguments. A theorem of J. Kruskal for a simple latent-class model with finite
state space lies at the core of our results, though we apply it to a diverse
set of models. These include mixtures of both finite and nonparametric product
distributions, hidden Markov models and random graph mixture models, and lead
to a number of new results and improvements to old ones. In the parametric
setting, this approach indicates that for such models, the classical definition
of identifiability is typically too strong. Instead generic identifiability
holds, which implies that the set of nonidentifiable parameters has measure
zero, so that parameter inference is still meaningful. In particular, this
sheds light on the properties of finite mixtures of Bernoulli products, which
have been used for decades despite being known to have nonidentifiable
parameters. In the nonparametric setting, we again obtain identifiability only
when certain restrictions are placed on the distributions that are mixed, but
we explicitly describe the conditions.Comment: Published in at http://dx.doi.org/10.1214/09-AOS689 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A Nonparametric Bayesian Approach to Uncovering Rat Hippocampal Population Codes During Spatial Navigation
Rodent hippocampal population codes represent important spatial information
about the environment during navigation. Several computational methods have
been developed to uncover the neural representation of spatial topology
embedded in rodent hippocampal ensemble spike activity. Here we extend our
previous work and propose a nonparametric Bayesian approach to infer rat
hippocampal population codes during spatial navigation. To tackle the model
selection problem, we leverage a nonparametric Bayesian model. Specifically, to
analyze rat hippocampal ensemble spiking activity, we apply a hierarchical
Dirichlet process-hidden Markov model (HDP-HMM) using two Bayesian inference
methods, one based on Markov chain Monte Carlo (MCMC) and the other based on
variational Bayes (VB). We demonstrate the effectiveness of our Bayesian
approaches on recordings from a freely-behaving rat navigating in an open field
environment. We find that MCMC-based inference with Hamiltonian Monte Carlo
(HMC) hyperparameter sampling is flexible and efficient, and outperforms VB and
MCMC approaches with hyperparameters set by empirical Bayes
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