2,566 research outputs found
Inference and Mixture Modeling with the Elliptical Gamma Distribution
We study modeling and inference with the Elliptical Gamma Distribution (EGD).
We consider maximum likelihood (ML) estimation for EGD scatter matrices, a task
for which we develop new fixed-point algorithms. Our algorithms are efficient
and converge to global optima despite nonconvexity. Moreover, they turn out to
be much faster than both a well-known iterative algorithm of Kent & Tyler
(1991) and sophisticated manifold optimization algorithms. Subsequently, we
invoke our ML algorithms as subroutines for estimating parameters of a mixture
of EGDs. We illustrate our methods by applying them to model natural image
statistics---the proposed EGD mixture model yields the most parsimonious model
among several competing approaches.Comment: 23 pages, 11 figure
Automatic Differentiation Variational Inference
Probabilistic modeling is iterative. A scientist posits a simple model, fits
it to her data, refines it according to her analysis, and repeats. However,
fitting complex models to large data is a bottleneck in this process. Deriving
algorithms for new models can be both mathematically and computationally
challenging, which makes it difficult to efficiently cycle through the steps.
To this end, we develop automatic differentiation variational inference (ADVI).
Using our method, the scientist only provides a probabilistic model and a
dataset, nothing else. ADVI automatically derives an efficient variational
inference algorithm, freeing the scientist to refine and explore many models.
ADVI supports a broad class of models-no conjugacy assumptions are required. We
study ADVI across ten different models and apply it to a dataset with millions
of observations. ADVI is integrated into Stan, a probabilistic programming
system; it is available for immediate use
Simplified Pair Copula Constructions --- Limits and Extensions
So called pair copula constructions (PCCs), specifying multivariate
distributions only in terms of bivariate building blocks (pair copulas),
constitute a flexible class of dependence models. To keep them tractable for
inference and model selection, the simplifying assumption that copulas of
conditional distributions do not depend on the values of the variables which
they are conditioned on is popular. In this paper, we show for which classes of
distributions such a simplification is applicable, significantly extending the
discussion of Hob{\ae}k Haff et al. (2010). In particular, we show that the
only Archimedean copula in dimension d \geq 4 which is of the simplified type
is that based on the gamma Laplace transform or its extension, while the
Student-t copula is the only one arising from a scale mixture of Normals.
Further, we illustrate how PCCs can be adapted for situations where conditional
copulas depend on values which are conditioned on
Physiological Gaussian Process Priors for the Hemodynamics in fMRI Analysis
Background: Inference from fMRI data faces the challenge that the hemodynamic
system that relates neural activity to the observed BOLD fMRI signal is
unknown.
New Method: We propose a new Bayesian model for task fMRI data with the
following features: (i) joint estimation of brain activity and the underlying
hemodynamics, (ii) the hemodynamics is modeled nonparametrically with a
Gaussian process (GP) prior guided by physiological information and (iii) the
predicted BOLD is not necessarily generated by a linear time-invariant (LTI)
system. We place a GP prior directly on the predicted BOLD response, rather
than on the hemodynamic response function as in previous literature. This
allows us to incorporate physiological information via the GP prior mean in a
flexible way, and simultaneously gives us the nonparametric flexibility of the
GP.
Results: Results on simulated data show that the proposed model is able to
discriminate between active and non-active voxels also when the GP prior
deviates from the true hemodynamics. Our model finds time varying dynamics when
applied to real fMRI data.
Comparison with Existing Method(s): The proposed model is better at detecting
activity in simulated data than standard models, without inflating the false
positive rate. When applied to real fMRI data, our GP model in several cases
finds brain activity where previously proposed LTI models does not.
Conclusions: We have proposed a new non-linear model for the hemodynamics in
task fMRI, that is able to detect active voxels, and gives the opportunity to
ask new kinds of questions related to hemodynamics.Comment: 18 pages, 14 figure
Extremal t processes: Elliptical domain of attraction and a spectral representation
The extremal t process was proposed in the literature for modeling spatial
extremes within a copula framework based on the extreme value limit of
elliptical t distributions (Davison, Padoan and Ribatet (2012)). A major
drawback of this max-stable model was the lack of a spectral representation
such that for instance direct simulation was infeasible. The main contribution
of this note is to propose such a spectral construction for the extremal t
process. Interestingly, the extremal Gaussian process introduced by Schlather
(2002) appears as a special case. We further highlight the role of the extremal
t process as the maximum attractor for processes with finite-dimensional
elliptical distributions. All results naturally also hold within the
multivariate domain
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