5,240 research outputs found
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
Bayesian factorizations of big sparse tensors
It has become routine to collect data that are structured as multiway arrays
(tensors). There is an enormous literature on low rank and sparse matrix
factorizations, but limited consideration of extensions to the tensor case in
statistics. The most common low rank tensor factorization relies on parallel
factor analysis (PARAFAC), which expresses a rank tensor as a sum of rank
one tensors. When observations are only available for a tiny subset of the
cells of a big tensor, the low rank assumption is not sufficient and PARAFAC
has poor performance. We induce an additional layer of dimension reduction by
allowing the effective rank to vary across dimensions of the table. For
concreteness, we focus on a contingency table application. Taking a Bayesian
approach, we place priors on terms in the factorization and develop an
efficient Gibbs sampler for posterior computation. Theory is provided showing
posterior concentration rates in high-dimensional settings, and the methods are
shown to have excellent performance in simulations and several real data
applications
Sparse Linear Identifiable Multivariate Modeling
In this paper we consider sparse and identifiable linear latent variable
(factor) and linear Bayesian network models for parsimonious analysis of
multivariate data. We propose a computationally efficient method for joint
parameter and model inference, and model comparison. It consists of a fully
Bayesian hierarchy for sparse models using slab and spike priors (two-component
delta-function and continuous mixtures), non-Gaussian latent factors and a
stochastic search over the ordering of the variables. The framework, which we
call SLIM (Sparse Linear Identifiable Multivariate modeling), is validated and
bench-marked on artificial and real biological data sets. SLIM is closest in
spirit to LiNGAM (Shimizu et al., 2006), but differs substantially in
inference, Bayesian network structure learning and model comparison.
Experimentally, SLIM performs equally well or better than LiNGAM with
comparable computational complexity. We attribute this mainly to the stochastic
search strategy used, and to parsimony (sparsity and identifiability), which is
an explicit part of the model. We propose two extensions to the basic i.i.d.
linear framework: non-linear dependence on observed variables, called SNIM
(Sparse Non-linear Identifiable Multivariate modeling) and allowing for
correlations between latent variables, called CSLIM (Correlated SLIM), for the
temporal and/or spatial data. The source code and scripts are available from
http://cogsys.imm.dtu.dk/slim/.Comment: 45 pages, 17 figure
Fast Exact Bayesian Inference for Sparse Signals in the Normal Sequence Model
We consider exact algorithms for Bayesian inference with model selection
priors (including spike-and-slab priors) in the sparse normal sequence model.
Because the best existing exact algorithm becomes numerically unstable for
sample sizes over n=500, there has been much attention for alternative
approaches like approximate algorithms (Gibbs sampling, variational Bayes,
etc.), shrinkage priors (e.g. the Horseshoe prior and the Spike-and-Slab LASSO)
or empirical Bayesian methods. However, by introducing algorithmic ideas from
online sequential prediction, we show that exact calculations are feasible for
much larger sample sizes: for general model selection priors we reach n=25000,
and for certain spike-and-slab priors we can easily reach n=100000. We further
prove a de Finetti-like result for finite sample sizes that characterizes
exactly which model selection priors can be expressed as spike-and-slab priors.
The computational speed and numerical accuracy of the proposed methods are
demonstrated in experiments on simulated data, on a differential gene
expression data set, and to compare the effect of multiple hyper-parameter
settings in the beta-binomial prior. In our experimental evaluation we compute
guaranteed bounds on the numerical accuracy of all new algorithms, which shows
that the proposed methods are numerically reliable whereas an alternative based
on long division is not
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