4,760 research outputs found
Practical bounds on the error of Bayesian posterior approximations: A nonasymptotic approach
Bayesian inference typically requires the computation of an approximation to
the posterior distribution. An important requirement for an approximate
Bayesian inference algorithm is to output high-accuracy posterior mean and
uncertainty estimates. Classical Monte Carlo methods, particularly Markov Chain
Monte Carlo, remain the gold standard for approximate Bayesian inference
because they have a robust finite-sample theory and reliable convergence
diagnostics. However, alternative methods, which are more scalable or apply to
problems where Markov Chain Monte Carlo cannot be used, lack the same
finite-data approximation theory and tools for evaluating their accuracy. In
this work, we develop a flexible new approach to bounding the error of mean and
uncertainty estimates of scalable inference algorithms. Our strategy is to
control the estimation errors in terms of Wasserstein distance, then bound the
Wasserstein distance via a generalized notion of Fisher distance. Unlike
computing the Wasserstein distance, which requires access to the normalized
posterior distribution, the Fisher distance is tractable to compute because it
requires access only to the gradient of the log posterior density. We
demonstrate the usefulness of our Fisher distance approach by deriving bounds
on the Wasserstein error of the Laplace approximation and Hilbert coresets. We
anticipate that our approach will be applicable to many other approximate
inference methods such as the integrated Laplace approximation, variational
inference, and approximate Bayesian computationComment: 22 pages, 2 figure
String and Membrane Gaussian Processes
In this paper we introduce a novel framework for making exact nonparametric
Bayesian inference on latent functions, that is particularly suitable for Big
Data tasks. Firstly, we introduce a class of stochastic processes we refer to
as string Gaussian processes (string GPs), which are not to be mistaken for
Gaussian processes operating on text. We construct string GPs so that their
finite-dimensional marginals exhibit suitable local conditional independence
structures, which allow for scalable, distributed, and flexible nonparametric
Bayesian inference, without resorting to approximations, and while ensuring
some mild global regularity constraints. Furthermore, string GP priors
naturally cope with heterogeneous input data, and the gradient of the learned
latent function is readily available for explanatory analysis. Secondly, we
provide some theoretical results relating our approach to the standard GP
paradigm. In particular, we prove that some string GPs are Gaussian processes,
which provides a complementary global perspective on our framework. Finally, we
derive a scalable and distributed MCMC scheme for supervised learning tasks
under string GP priors. The proposed MCMC scheme has computational time
complexity and memory requirement , where
is the data size and the dimension of the input space. We illustrate the
efficacy of the proposed approach on several synthetic and real-world datasets,
including a dataset with millions input points and attributes.Comment: To appear in the Journal of Machine Learning Research (JMLR), Volume
1
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
Patterns of Scalable Bayesian Inference
Datasets are growing not just in size but in complexity, creating a demand
for rich models and quantification of uncertainty. Bayesian methods are an
excellent fit for this demand, but scaling Bayesian inference is a challenge.
In response to this challenge, there has been considerable recent work based on
varying assumptions about model structure, underlying computational resources,
and the importance of asymptotic correctness. As a result, there is a zoo of
ideas with few clear overarching principles.
In this paper, we seek to identify unifying principles, patterns, and
intuitions for scaling Bayesian inference. We review existing work on utilizing
modern computing resources with both MCMC and variational approximation
techniques. From this taxonomy of ideas, we characterize the general principles
that have proven successful for designing scalable inference procedures and
comment on the path forward
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