31,137 research outputs found
Free energy Sequential Monte Carlo, application to mixture modelling
We introduce a new class of Sequential Monte Carlo (SMC) methods, which we
call free energy SMC. This class is inspired by free energy methods, which
originate from Physics, and where one samples from a biased distribution such
that a given function of the state is forced to be
uniformly distributed over a given interval. From an initial sequence of
distributions of interest, and a particular choice of ,
a free energy SMC sampler computes sequentially a sequence of biased
distributions with the following properties: (a) the
marginal distribution of with respect to is
approximatively uniform over a specified interval, and (b)
and have the same conditional distribution with respect to . We
apply our methodology to mixture posterior distributions, which are highly
multimodal. In the mixture context, forcing certain hyper-parameters to higher
values greatly faciliates mode swapping, and makes it possible to recover a
symetric output. We illustrate our approach with univariate and bivariate
Gaussian mixtures and two real-world datasets.Comment: presented at "Bayesian Statistics 9" (Valencia meetings, 4-8 June
2010, Benidorm
Thermostat-assisted continuously-tempered Hamiltonian Monte Carlo for Bayesian learning
We propose a new sampling method, the thermostat-assisted
continuously-tempered Hamiltonian Monte Carlo, for Bayesian learning on large
datasets and multimodal distributions. It simulates the Nos\'e-Hoover dynamics
of a continuously-tempered Hamiltonian system built on the distribution of
interest. A significant advantage of this method is that it is not only able to
efficiently draw representative i.i.d. samples when the distribution contains
multiple isolated modes, but capable of adaptively neutralising the noise
arising from mini-batches and maintaining accurate sampling. While the
properties of this method have been studied using synthetic distributions,
experiments on three real datasets also demonstrated the gain of performance
over several strong baselines with various types of neural networks plunged in
Two adaptive rejection sampling schemes for probability density functions log-convex tails
Monte Carlo methods are often necessary for the implementation of optimal
Bayesian estimators. A fundamental technique that can be used to generate
samples from virtually any target probability distribution is the so-called
rejection sampling method, which generates candidate samples from a proposal
distribution and then accepts them or not by testing the ratio of the target
and proposal densities. The class of adaptive rejection sampling (ARS)
algorithms is particularly interesting because they can achieve high acceptance
rates. However, the standard ARS method can only be used with log-concave
target densities. For this reason, many generalizations have been proposed.
In this work, we investigate two different adaptive schemes that can be used
to draw exactly from a large family of univariate probability density functions
(pdf's), not necessarily log-concave, possibly multimodal and with tails of
arbitrary concavity. These techniques are adaptive in the sense that every time
a candidate sample is rejected, the acceptance rate is improved. The two
proposed algorithms can work properly when the target pdf is multimodal, with
first and second derivatives analytically intractable, and when the tails are
log-convex in a infinite domain. Therefore, they can be applied in a number of
scenarios in which the other generalizations of the standard ARS fail. Two
illustrative numerical examples are shown
Nonparametric Hierarchical Clustering of Functional Data
In this paper, we deal with the problem of curves clustering. We propose a
nonparametric method which partitions the curves into clusters and discretizes
the dimensions of the curve points into intervals. The cross-product of these
partitions forms a data-grid which is obtained using a Bayesian model selection
approach while making no assumptions regarding the curves. Finally, a
post-processing technique, aiming at reducing the number of clusters in order
to improve the interpretability of the clustering, is proposed. It consists in
optimally merging the clusters step by step, which corresponds to an
agglomerative hierarchical classification whose dissimilarity measure is the
variation of the criterion. Interestingly this measure is none other than the
sum of the Kullback-Leibler divergences between clusters distributions before
and after the merges. The practical interest of the approach for functional
data exploratory analysis is presented and compared with an alternative
approach on an artificial and a real world data set
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