12,084 research outputs found
Adaptive Gaussian Process Approximation for Bayesian Inference with Expensive Likelihood Functions
We consider Bayesian inference problems with computationally intensive
likelihood functions. We propose a Gaussian process (GP) based method to
approximate the joint distribution of the unknown parameters and the data. In
particular, we write the joint density approximately as a product of an
approximate posterior density and an exponentiated GP surrogate. We then
provide an adaptive algorithm to construct such an approximation, where an
active learning method is used to choose the design points. With numerical
examples, we illustrate that the proposed method has competitive performance
against existing approaches for Bayesian computation
GPS-ABC: Gaussian Process Surrogate Approximate Bayesian Computation
Scientists often express their understanding of the world through a
computationally demanding simulation program. Analyzing the posterior
distribution of the parameters given observations (the inverse problem) can be
extremely challenging. The Approximate Bayesian Computation (ABC) framework is
the standard statistical tool to handle these likelihood free problems, but
they require a very large number of simulations. In this work we develop two
new ABC sampling algorithms that significantly reduce the number of simulations
necessary for posterior inference. Both algorithms use confidence estimates for
the accept probability in the Metropolis Hastings step to adaptively choose the
number of necessary simulations. Our GPS-ABC algorithm stores the information
obtained from every simulation in a Gaussian process which acts as a surrogate
function for the simulated statistics. Experiments on a challenging realistic
biological problem illustrate the potential of these algorithms
Accelerating Asymptotically Exact MCMC for Computationally Intensive Models via Local Approximations
We construct a new framework for accelerating Markov chain Monte Carlo in
posterior sampling problems where standard methods are limited by the
computational cost of the likelihood, or of numerical models embedded therein.
Our approach introduces local approximations of these models into the
Metropolis-Hastings kernel, borrowing ideas from deterministic approximation
theory, optimization, and experimental design. Previous efforts at integrating
approximate models into inference typically sacrifice either the sampler's
exactness or efficiency; our work seeks to address these limitations by
exploiting useful convergence characteristics of local approximations. We prove
the ergodicity of our approximate Markov chain, showing that it samples
asymptotically from the \emph{exact} posterior distribution of interest. We
describe variations of the algorithm that employ either local polynomial
approximations or local Gaussian process regressors. Our theoretical results
reinforce the key observation underlying this paper: when the likelihood has
some \emph{local} regularity, the number of model evaluations per MCMC step can
be greatly reduced without biasing the Monte Carlo average. Numerical
experiments demonstrate multiple order-of-magnitude reductions in the number of
forward model evaluations used in representative ODE and PDE inference
problems, with both synthetic and real data.Comment: A major update of the theory and example
Adaptive Multiple Importance Sampling for Gaussian Processes
In applications of Gaussian processes where quantification of uncertainty is
a strict requirement, it is necessary to accurately characterize the posterior
distribution over Gaussian process covariance parameters. Normally, this is
done by means of standard Markov chain Monte Carlo (MCMC) algorithms. Motivated
by the issues related to the complexity of calculating the marginal likelihood
that can make MCMC algorithms inefficient, this paper develops an alternative
inference framework based on Adaptive Multiple Importance Sampling (AMIS). This
paper studies the application of AMIS in the case of a Gaussian likelihood, and
proposes the Pseudo-Marginal AMIS for non-Gaussian likelihoods, where the
marginal likelihood is unbiasedly estimated. The results suggest that the
proposed framework outperforms MCMC-based inference of covariance parameters in
a wide range of scenarios and remains competitive for moderately large
dimensional parameter spaces.Comment: 27 page
Hamiltonian Monte Carlo Acceleration Using Surrogate Functions with Random Bases
For big data analysis, high computational cost for Bayesian methods often
limits their applications in practice. In recent years, there have been many
attempts to improve computational efficiency of Bayesian inference. Here we
propose an efficient and scalable computational technique for a
state-of-the-art Markov Chain Monte Carlo (MCMC) methods, namely, Hamiltonian
Monte Carlo (HMC). The key idea is to explore and exploit the structure and
regularity in parameter space for the underlying probabilistic model to
construct an effective approximation of its geometric properties. To this end,
we build a surrogate function to approximate the target distribution using
properly chosen random bases and an efficient optimization process. The
resulting method provides a flexible, scalable, and efficient sampling
algorithm, which converges to the correct target distribution. We show that by
choosing the basis functions and optimization process differently, our method
can be related to other approaches for the construction of surrogate functions
such as generalized additive models or Gaussian process models. Experiments
based on simulated and real data show that our approach leads to substantially
more efficient sampling algorithms compared to existing state-of-the art
methods
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