2,804 research outputs found
Fully Adaptive Gaussian Mixture Metropolis-Hastings Algorithm
Markov Chain Monte Carlo methods are widely used in signal processing and
communications for statistical inference and stochastic optimization. In this
work, we introduce an efficient adaptive Metropolis-Hastings algorithm to draw
samples from generic multi-modal and multi-dimensional target distributions.
The proposal density is a mixture of Gaussian densities with all parameters
(weights, mean vectors and covariance matrices) updated using all the
previously generated samples applying simple recursive rules. Numerical results
for the one and two-dimensional cases are provided
Adaptive Incremental Mixture Markov chain Monte Carlo
We propose Adaptive Incremental Mixture Markov chain Monte Carlo (AIMM), a
novel approach to sample from challenging probability distributions defined on
a general state-space. While adaptive MCMC methods usually update a parametric
proposal kernel with a global rule, AIMM locally adapts a semiparametric
kernel. AIMM is based on an independent Metropolis-Hastings proposal
distribution which takes the form of a finite mixture of Gaussian
distributions. Central to this approach is the idea that the proposal
distribution adapts to the target by locally adding a mixture component when
the discrepancy between the proposal mixture and the target is deemed to be too
large. As a result, the number of components in the mixture proposal is not
fixed in advance. Theoretically, we prove that there exists a process that can
be made arbitrarily close to AIMM and that converges to the correct target
distribution. We also illustrate that it performs well in practice in a variety
of challenging situations, including high-dimensional and multimodal target
distributions
Ensemble Transport Adaptive Importance Sampling
Markov chain Monte Carlo methods are a powerful and commonly used family of
numerical methods for sampling from complex probability distributions. As
applications of these methods increase in size and complexity, the need for
efficient methods increases. In this paper, we present a particle ensemble
algorithm. At each iteration, an importance sampling proposal distribution is
formed using an ensemble of particles. A stratified sample is taken from this
distribution and weighted under the posterior, a state-of-the-art ensemble
transport resampling method is then used to create an evenly weighted sample
ready for the next iteration. We demonstrate that this ensemble transport
adaptive importance sampling (ETAIS) method outperforms MCMC methods with
equivalent proposal distributions for low dimensional problems, and in fact
shows better than linear improvements in convergence rates with respect to the
number of ensemble members. We also introduce a new resampling strategy,
multinomial transformation (MT), which while not as accurate as the ensemble
transport resampler, is substantially less costly for large ensemble sizes, and
can then be used in conjunction with ETAIS for complex problems. We also focus
on how algorithmic parameters regarding the mixture proposal can be quickly
tuned to optimise performance. In particular, we demonstrate this methodology's
superior sampling for multimodal problems, such as those arising from inference
for mixture models, and for problems with expensive likelihoods requiring the
solution of a differential equation, for which speed-ups of orders of magnitude
are demonstrated. Likelihood evaluations of the ensemble could be computed in a
distributed manner, suggesting that this methodology is a good candidate for
parallel Bayesian computations
An Adaptive Interacting Wang-Landau Algorithm for Automatic Density Exploration
While statisticians are well-accustomed to performing exploratory analysis in
the modeling stage of an analysis, the notion of conducting preliminary
general-purpose exploratory analysis in the Monte Carlo stage (or more
generally, the model-fitting stage) of an analysis is an area which we feel
deserves much further attention. Towards this aim, this paper proposes a
general-purpose algorithm for automatic density exploration. The proposed
exploration algorithm combines and expands upon components from various
adaptive Markov chain Monte Carlo methods, with the Wang-Landau algorithm at
its heart. Additionally, the algorithm is run on interacting parallel chains --
a feature which both decreases computational cost as well as stabilizes the
algorithm, improving its ability to explore the density. Performance is studied
in several applications. Through a Bayesian variable selection example, the
authors demonstrate the convergence gains obtained with interacting chains. The
ability of the algorithm's adaptive proposal to induce mode-jumping is
illustrated through a trimodal density and a Bayesian mixture modeling
application. Lastly, through a 2D Ising model, the authors demonstrate the
ability of the algorithm to overcome the high correlations encountered in
spatial models.Comment: 33 pages, 20 figures (the supplementary materials are included as
appendices
Parallel hierarchical sampling:a general-purpose class of multiple-chains MCMC algorithms
This paper introduces the Parallel Hierarchical Sampler (PHS), a class of Markov chain Monte Carlo algorithms using several interacting chains having the same target distribution but different mixing properties. Unlike any single-chain MCMC algorithm, upon reaching stationarity one of the PHS chains, which we call the “mother” chain, attains exact Monte Carlo sampling of the target distribution of interest. We empirically show that this translates in a dramatic improvement in the sampler’s performance with respect to single-chain MCMC algorithms. Convergence of the PHS joint transition kernel is proved and its relationships with single-chain samplers, Parallel Tempering (PT) and variable augmentation algorithms are discussed. We then provide two illustrative examples comparing the accuracy of PHS with
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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