10,099 research outputs found
On an adaptive preconditioned Crank-Nicolson MCMC algorithm for infinite dimensional Bayesian inferences
Many scientific and engineering problems require to perform Bayesian
inferences for unknowns of infinite dimension. In such problems, many standard
Markov Chain Monte Carlo (MCMC) algorithms become arbitrary slow under the mesh
refinement, which is referred to as being dimension dependent. To this end, a
family of dimensional independent MCMC algorithms, known as the preconditioned
Crank-Nicolson (pCN) methods, were proposed to sample the infinite dimensional
parameters. In this work we develop an adaptive version of the pCN algorithm,
where the covariance operator of the proposal distribution is adjusted based on
sampling history to improve the simulation efficiency. We show that the
proposed algorithm satisfies an important ergodicity condition under some mild
assumptions. Finally we provide numerical examples to demonstrate the
performance of the proposed method
Exploiting the Statistics of Learning and Inference
When dealing with datasets containing a billion instances or with simulations
that require a supercomputer to execute, computational resources become part of
the equation. We can improve the efficiency of learning and inference by
exploiting their inherent statistical nature. We propose algorithms that
exploit the redundancy of data relative to a model by subsampling data-cases
for every update and reasoning about the uncertainty created in this process.
In the context of learning we propose to test for the probability that a
stochastically estimated gradient points more than 180 degrees in the wrong
direction. In the context of MCMC sampling we use stochastic gradients to
improve the efficiency of MCMC updates, and hypothesis tests based on adaptive
mini-batches to decide whether to accept or reject a proposed parameter update.
Finally, we argue that in the context of likelihood free MCMC one needs to
store all the information revealed by all simulations, for instance in a
Gaussian process. We conclude that Bayesian methods will remain to play a
crucial role in the era of big data and big simulations, but only if we
overcome a number of computational challenges.Comment: Proceedings of the NIPS workshop on "Probabilistic Models for Big
Data
The Barker proposal: Combining robustness and efficiency in gradient-based MCMC
There is a tension between robustness and efficiency when designing Markov chain Monte Carlo (MCMC) sampling algorithms. Here we focus on robustness with respect to tuning parameters, showing that more sophisticated algorithms tend to be more sensitive to the choice of step-size parameter and less robust to heterogeneity of the distribution of interest. We characterise this phenomenon by studying the behaviour of spectral gaps as an increasingly poor step-size is chosen for the algorithm. Motivated by these considerations, we propose a novel and simple gradient-based MCMC algorithm, inspired by the classical Barker accept-reject rule, with improved robustness properties. Extensive theoretical results, dealing with robustness to tuning, geometric ergodicity and scaling with dimension, suggest that the novel scheme combines the robustness of simple schemes with the efficiency of gradient-based ones. We show numerically that this type of robustness is particularly beneficial in the context of adaptive MCMC, giving examples where our proposed scheme significantly outperforms state-of-the-art alternatives
Adaptive MC^3 and Gibbs algorithms for Bayesian Model Averaging in Linear Regression Models
The MC (Madigan and York, 1995) and Gibbs (George and McCulloch, 1997)
samplers are the most widely implemented algorithms for Bayesian Model
Averaging (BMA) in linear regression models. These samplers draw a variable at
random in each iteration using uniform selection probabilities and then propose
to update that variable. This may be computationally inefficient if the number
of variables is large and many variables are redundant. In this work, we
introduce adaptive versions of these samplers that retain their simplicity in
implementation and reduce the selection probabilities of the many redundant
variables. The improvements in efficiency for the adaptive samplers are
illustrated in real and simulated datasets
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