60,677 research outputs found
Dimension-free Mixing for High-dimensional Bayesian Variable Selection
Yang et al. (2016) proved that the symmetric random walk Metropolis--Hastings
algorithm for Bayesian variable selection is rapidly mixing under mild
high-dimensional assumptions. We propose a novel MCMC sampler using an informed
proposal scheme, which we prove achieves a much faster mixing time that is
independent of the number of covariates, under the same assumptions. To the
best of our knowledge, this is the first high-dimensional result which
rigorously shows that the mixing rate of informed MCMC methods can be fast
enough to offset the computational cost of local posterior evaluation.
Motivated by the theoretical analysis of our sampler, we further propose a new
approach called "two-stage drift condition" to studying convergence rates of
Markov chains on general state spaces, which can be useful for obtaining tight
complexity bounds in high-dimensional settings. The practical advantages of our
algorithm are illustrated by both simulation studies and real data analysis
Spike-and-Slab Priors for Function Selection in Structured Additive Regression Models
Structured additive regression provides a general framework for complex
Gaussian and non-Gaussian regression models, with predictors comprising
arbitrary combinations of nonlinear functions and surfaces, spatial effects,
varying coefficients, random effects and further regression terms. The large
flexibility of structured additive regression makes function selection a
challenging and important task, aiming at (1) selecting the relevant
covariates, (2) choosing an appropriate and parsimonious representation of the
impact of covariates on the predictor and (3) determining the required
interactions. We propose a spike-and-slab prior structure for function
selection that allows to include or exclude single coefficients as well as
blocks of coefficients representing specific model terms. A novel
multiplicative parameter expansion is required to obtain good mixing and
convergence properties in a Markov chain Monte Carlo simulation approach and is
shown to induce desirable shrinkage properties. In simulation studies and with
(real) benchmark classification data, we investigate sensitivity to
hyperparameter settings and compare performance to competitors. The flexibility
and applicability of our approach are demonstrated in an additive piecewise
exponential model with time-varying effects for right-censored survival times
of intensive care patients with sepsis. Geoadditive and additive mixed logit
model applications are discussed in an extensive appendix
Foundational principles for large scale inference: Illustrations through correlation mining
When can reliable inference be drawn in the "Big Data" context? This paper
presents a framework for answering this fundamental question in the context of
correlation mining, with implications for general large scale inference. In
large scale data applications like genomics, connectomics, and eco-informatics
the dataset is often variable-rich but sample-starved: a regime where the
number of acquired samples (statistical replicates) is far fewer than the
number of observed variables (genes, neurons, voxels, or chemical
constituents). Much of recent work has focused on understanding the
computational complexity of proposed methods for "Big Data." Sample complexity
however has received relatively less attention, especially in the setting when
the sample size is fixed, and the dimension grows without bound. To
address this gap, we develop a unified statistical framework that explicitly
quantifies the sample complexity of various inferential tasks. Sampling regimes
can be divided into several categories: 1) the classical asymptotic regime
where the variable dimension is fixed and the sample size goes to infinity; 2)
the mixed asymptotic regime where both variable dimension and sample size go to
infinity at comparable rates; 3) the purely high dimensional asymptotic regime
where the variable dimension goes to infinity and the sample size is fixed.
Each regime has its niche but only the latter regime applies to exa-scale data
dimension. We illustrate this high dimensional framework for the problem of
correlation mining, where it is the matrix of pairwise and partial correlations
among the variables that are of interest. We demonstrate various regimes of
correlation mining based on the unifying perspective of high dimensional
learning rates and sample complexity for different structured covariance models
and different inference tasks
Decorrelation of Neutral Vector Variables: Theory and Applications
In this paper, we propose novel strategies for neutral vector variable
decorrelation. Two fundamental invertible transformations, namely serial
nonlinear transformation and parallel nonlinear transformation, are proposed to
carry out the decorrelation. For a neutral vector variable, which is not
multivariate Gaussian distributed, the conventional principal component
analysis (PCA) cannot yield mutually independent scalar variables. With the two
proposed transformations, a highly negatively correlated neutral vector can be
transformed to a set of mutually independent scalar variables with the same
degrees of freedom. We also evaluate the decorrelation performances for the
vectors generated from a single Dirichlet distribution and a mixture of
Dirichlet distributions. The mutual independence is verified with the distance
correlation measurement. The advantages of the proposed decorrelation
strategies are intensively studied and demonstrated with synthesized data and
practical application evaluations
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