109,216 research outputs found
The Kernel Interaction Trick: Fast Bayesian Discovery of Pairwise Interactions in High Dimensions
Discovering interaction effects on a response of interest is a fundamental
problem faced in biology, medicine, economics, and many other scientific
disciplines. In theory, Bayesian methods for discovering pairwise interactions
enjoy many benefits such as coherent uncertainty quantification, the ability to
incorporate background knowledge, and desirable shrinkage properties. In
practice, however, Bayesian methods are often computationally intractable for
even moderate-dimensional problems. Our key insight is that many hierarchical
models of practical interest admit a particular Gaussian process (GP)
representation; the GP allows us to capture the posterior with a vector of O(p)
kernel hyper-parameters rather than O(p^2) interactions and main effects. With
the implicit representation, we can run Markov chain Monte Carlo (MCMC) over
model hyper-parameters in time and memory linear in p per iteration. We focus
on sparsity-inducing models and show on datasets with a variety of covariate
behaviors that our method: (1) reduces runtime by orders of magnitude over
naive applications of MCMC, (2) provides lower Type I and Type II error
relative to state-of-the-art LASSO-based approaches, and (3) offers improved
computational scaling in high dimensions relative to existing Bayesian and
LASSO-based approaches.Comment: Accepted at ICML 2019. 20 pages, 4 figures, 3 table
Batch Bayesian Optimization via Local Penalization
The popularity of Bayesian optimization methods for efficient exploration of
parameter spaces has lead to a series of papers applying Gaussian processes as
surrogates in the optimization of functions. However, most proposed approaches
only allow the exploration of the parameter space to occur sequentially. Often,
it is desirable to simultaneously propose batches of parameter values to
explore. This is particularly the case when large parallel processing
facilities are available. These facilities could be computational or physical
facets of the process being optimized. E.g. in biological experiments many
experimental set ups allow several samples to be simultaneously processed.
Batch methods, however, require modeling of the interaction between the
evaluations in the batch, which can be expensive in complex scenarios. We
investigate a simple heuristic based on an estimate of the Lipschitz constant
that captures the most important aspect of this interaction (i.e. local
repulsion) at negligible computational overhead. The resulting algorithm
compares well, in running time, with much more elaborate alternatives. The
approach assumes that the function of interest, , is a Lipschitz continuous
function. A wrap-loop around the acquisition function is used to collect
batches of points of certain size minimizing the non-parallelizable
computational effort. The speed-up of our method with respect to previous
approaches is significant in a set of computationally expensive experiments.Comment: 11 pages, 10 figure
Syntactic Topic Models
The syntactic topic model (STM) is a Bayesian nonparametric model of language
that discovers latent distributions of words (topics) that are both
semantically and syntactically coherent. The STM models dependency parsed
corpora where sentences are grouped into documents. It assumes that each word
is drawn from a latent topic chosen by combining document-level features and
the local syntactic context. Each document has a distribution over latent
topics, as in topic models, which provides the semantic consistency. Each
element in the dependency parse tree also has a distribution over the topics of
its children, as in latent-state syntax models, which provides the syntactic
consistency. These distributions are convolved so that the topic of each word
is likely under both its document and syntactic context. We derive a fast
posterior inference algorithm based on variational methods. We report
qualitative and quantitative studies on both synthetic data and hand-parsed
documents. We show that the STM is a more predictive model of language than
current models based only on syntax or only on topics
Symbol Emergence in Robotics: A Survey
Humans can learn the use of language through physical interaction with their
environment and semiotic communication with other people. It is very important
to obtain a computational understanding of how humans can form a symbol system
and obtain semiotic skills through their autonomous mental development.
Recently, many studies have been conducted on the construction of robotic
systems and machine-learning methods that can learn the use of language through
embodied multimodal interaction with their environment and other systems.
Understanding human social interactions and developing a robot that can
smoothly communicate with human users in the long term, requires an
understanding of the dynamics of symbol systems and is crucially important. The
embodied cognition and social interaction of participants gradually change a
symbol system in a constructive manner. In this paper, we introduce a field of
research called symbol emergence in robotics (SER). SER is a constructive
approach towards an emergent symbol system. The emergent symbol system is
socially self-organized through both semiotic communications and physical
interactions with autonomous cognitive developmental agents, i.e., humans and
developmental robots. Specifically, we describe some state-of-art research
topics concerning SER, e.g., multimodal categorization, word discovery, and a
double articulation analysis, that enable a robot to obtain words and their
embodied meanings from raw sensory--motor information, including visual
information, haptic information, auditory information, and acoustic speech
signals, in a totally unsupervised manner. Finally, we suggest future
directions of research in SER.Comment: submitted to Advanced Robotic
Hidden Gibbs random fields model selection using Block Likelihood Information Criterion
Performing model selection between Gibbs random fields is a very challenging
task. Indeed, due to the Markovian dependence structure, the normalizing
constant of the fields cannot be computed using standard analytical or
numerical methods. Furthermore, such unobserved fields cannot be integrated out
and the likelihood evaluztion is a doubly intractable problem. This forms a
central issue to pick the model that best fits an observed data. We introduce a
new approximate version of the Bayesian Information Criterion. We partition the
lattice into continuous rectangular blocks and we approximate the probability
measure of the hidden Gibbs field by the product of some Gibbs distributions
over the blocks. On that basis, we estimate the likelihood and derive the Block
Likelihood Information Criterion (BLIC) that answers model choice questions
such as the selection of the dependency structure or the number of latent
states. We study the performances of BLIC for those questions. In addition, we
present a comparison with ABC algorithms to point out that the novel criterion
offers a better trade-off between time efficiency and reliable results
Bayesian Approximate Kernel Regression with Variable Selection
Nonlinear kernel regression models are often used in statistics and machine
learning because they are more accurate than linear models. Variable selection
for kernel regression models is a challenge partly because, unlike the linear
regression setting, there is no clear concept of an effect size for regression
coefficients. In this paper, we propose a novel framework that provides an
effect size analog of each explanatory variable for Bayesian kernel regression
models when the kernel is shift-invariant --- for example, the Gaussian kernel.
We use function analytic properties of shift-invariant reproducing kernel
Hilbert spaces (RKHS) to define a linear vector space that: (i) captures
nonlinear structure, and (ii) can be projected onto the original explanatory
variables. The projection onto the original explanatory variables serves as an
analog of effect sizes. The specific function analytic property we use is that
shift-invariant kernel functions can be approximated via random Fourier bases.
Based on the random Fourier expansion we propose a computationally efficient
class of Bayesian approximate kernel regression (BAKR) models for both
nonlinear regression and binary classification for which one can compute an
analog of effect sizes. We illustrate the utility of BAKR by examining two
important problems in statistical genetics: genomic selection (i.e. phenotypic
prediction) and association mapping (i.e. inference of significant variants or
loci). State-of-the-art methods for genomic selection and association mapping
are based on kernel regression and linear models, respectively. BAKR is the
first method that is competitive in both settings.Comment: 22 pages, 3 figures, 3 tables; theory added; new simulations
presented; references adde
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