6,008 research outputs found
Importance sampling schemes for evidence approximation in mixture models
The marginal likelihood is a central tool for drawing Bayesian inference
about the number of components in mixture models. It is often approximated
since the exact form is unavailable. A bias in the approximation may be due to
an incomplete exploration by a simulated Markov chain (e.g., a Gibbs sequence)
of the collection of posterior modes, a phenomenon also known as lack of label
switching, as all possible label permutations must be simulated by a chain in
order to converge and hence overcome the bias. In an importance sampling
approach, imposing label switching to the importance function results in an
exponential increase of the computational cost with the number of components.
In this paper, two importance sampling schemes are proposed through choices for
the importance function; a MLE proposal and a Rao-Blackwellised importance
function. The second scheme is called dual importance sampling. We demonstrate
that this dual importance sampling is a valid estimator of the evidence and
moreover show that the statistical efficiency of estimates increases. To reduce
the induced high demand in computation, the original importance function is
approximated but a suitable approximation can produce an estimate with the same
precision and with reduced computational workload.Comment: 24 pages, 5 figure
Gaussian process hyper-parameter estimation using parallel asymptotically independent Markov sampling
Gaussian process emulators of computationally expensive computer codes
provide fast statistical approximations to model physical processes. The
training of these surrogates depends on the set of design points chosen to run
the simulator. Due to computational cost, such training set is bound to be
limited and quantifying the resulting uncertainty in the hyper-parameters of
the emulator by uni-modal distributions is likely to induce bias. In order to
quantify this uncertainty, this paper proposes a computationally efficient
sampler based on an extension of Asymptotically Independent Markov Sampling, a
recently developed algorithm for Bayesian inference. Structural uncertainty of
the emulator is obtained as a by-product of the Bayesian treatment of the
hyper-parameters. Additionally, the user can choose to perform stochastic
optimisation to sample from a neighbourhood of the Maximum a Posteriori
estimate, even in the presence of multimodality. Model uncertainty is also
acknowledged through numerical stabilisation measures by including a nugget
term in the formulation of the probability model. The efficiency of the
proposed sampler is illustrated in examples where multi-modal distributions are
encountered. For the purpose of reproducibility, further development, and use
in other applications the code used to generate the examples is freely
available for download at https://github.com/agarbuno/paims_codesComment: Computational Statistics \& Data Analysis, Volume 103, November 201
Bayesian Modelling and Inference on Mixtures of Distributions.
bayesian models;
From here to infinity - sparse finite versus Dirichlet process mixtures in model-based clustering
In model-based-clustering mixture models are used to group data points into
clusters. A useful concept introduced for Gaussian mixtures by Malsiner Walli
et al (2016) are sparse finite mixtures, where the prior distribution on the
weight distribution of a mixture with components is chosen in such a way
that a priori the number of clusters in the data is random and is allowed to be
smaller than with high probability. The number of cluster is then inferred
a posteriori from the data.
The present paper makes the following contributions in the context of sparse
finite mixture modelling. First, it is illustrated that the concept of sparse
finite mixture is very generic and easily extended to cluster various types of
non-Gaussian data, in particular discrete data and continuous multivariate data
arising from non-Gaussian clusters. Second, sparse finite mixtures are compared
to Dirichlet process mixtures with respect to their ability to identify the
number of clusters. For both model classes, a random hyper prior is considered
for the parameters determining the weight distribution. By suitable matching of
these priors, it is shown that the choice of this hyper prior is far more
influential on the cluster solution than whether a sparse finite mixture or a
Dirichlet process mixture is taken into consideration.Comment: Accepted versio
Dealing with Label Switching in Mixture Models Under Genuine Multimodality
The fitting of finite mixture models is an ill-defined estimation problem as completely different parameterizations can induce similar mixture distributions. This leads to multiple modes in the likelihood which is a problem for frequentist maximum likelihood estimation, and complicates statistical inference of Markov chain Monte Carlo draws in Bayesian estimation. For the analysis of the posterior density of these draws a suitable separation into different modes is desirable. In addition, a unique labelling of the component specific estimates is necessary to solve the label
switching problem. This paper presents and compares two approaches to achieve these goals: relabelling under multimodality and constrained clustering. The algorithmic details are discussed and their application is demonstrated on artificial and real-world data
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