ترغب بنشر مسار تعليمي؟ اضغط هنا

High-dimensional Bayesian model selection by proximal nested sampling

84   0   0.0 ( 0 )
 نشر من قبل Xiaohao Cai
 تاريخ النشر 2021
والبحث باللغة English




اسأل ChatGPT حول البحث

Imaging methods often rely on Bayesian statistical inference strategies to solve difficult imaging problems. Applying Bayesian methodology to imaging requires the specification of a likelihood function and a prior distribution, which define the Bayesian statistical model from which the posterior distribution of the image is derived. Specifying a suitable model for a specific application can be very challenging, particularly when there is no reliable ground truth data available. Bayesian model selection provides a framework for selecting the most appropriate model directly from the observed data, without reference to ground truth data. However, Bayesian model selection requires the computation of the marginal likelihood (Bayesian evidence), which is computationally challenging, prohibiting its use in high-dimensional imaging problems. In this work we present the proximal nested sampling methodology to objectively compare alternative Bayesian imaging models, without reference to ground truth data. The methodology is based on nested sampling, a Monte Carlo approach specialised for model comparison, and exploits proximal Markov chain Monte Carlo techniques to scale efficiently to large problems and to tackle models that are log-concave and not necessarily smooth (e.g., involving L1 or total-variation priors). The proposed approach can be applied computationally to problems of dimension O(10^6) and beyond, making it suitable for high-dimensional inverse imaging problems. It is validated on large Gaussian models, for which the likelihood is available analytically, and subsequently illustrated on a range of imaging problems where it is used to analyse different choices for the sparsifying dictionary and measurement model.



قيم البحث

اقرأ أيضاً

In this paper we review the concepts of Bayesian evidence and Bayes factors, also known as log odds ratios, and their application to model selection. The theory is presented along with a discussion of analytic, approximate and numerical techniques. S pecific attention is paid to the Laplace approximation, variational Bayes, importance sampling, thermodynamic integration, and nested sampling and its recent variants. Analogies to statistical physics, from which many of these techniques originate, are discussed in order to provide readers with deeper insights that may lead to new techniques. The utility of Bayesian model testing in the domain sciences is demonstrated by presenting four specific practical examples considered within the context of signal processing in the areas of signal detection, sensor characterization, scientific model selection and molecular force characterization.
Sampling errors in nested sampling parameter estimation differ from those in Bayesian evidence calculation, but have been little studied in the literature. This paper provides the first explanation of the two main sources of sampling errors in nested sampling parameter estimation, and presents a new diagrammatic representation for the process. We find no current method can accurately measure the parameter estimation errors of a single nested sampling run, and propose a method for doing so using a new algorithm for dividing nested sampling runs. We empirically verify our conclusions and the accuracy of our new method.
We consider the problem of variable selection in high-dimensional settings with missing observations among the covariates. To address this relatively understudied problem, we propose a new synergistic procedure -- adaptive Bayesian SLOPE -- which eff ectively combines the SLOPE method (sorted $l_1$ regularization) together with the Spike-and-Slab LASSO method. We position our approach within a Bayesian framework which allows for simultaneous variable selection and parameter estimation, despite the missing values. As with the Spike-and-Slab LASSO, the coefficients are regarded as arising from a hierarchical model consisting of two groups: (1) the spike for the inactive and (2) the slab for the active. However, instead of assigning independent spike priors for each covariate, here we deploy a joint SLOPE spike prior which takes into account the ordering of coefficient magnitudes in order to control for false discoveries. Through extensive simulations, we demonstrate satisfactory performance in terms of power, FDR and estimation bias under a wide range of scenarios. Finally, we analyze a real dataset consisting of patients from Paris hospitals who underwent a severe trauma, where we show excellent performance in predicting platelet levels. Our methodology has been implemented in C++ and wrapped into an R package ABSLOPE for public use.
We develop a Bayesian variable selection method, called SVEN, based on a hierarchical Gaussian linear model with priors placed on the regression coefficients as well as on the model space. Sparsity is achieved by using degenerate spike priors on inac tive variables, whereas Gaussian slab priors are placed on the coefficients for the important predictors making the posterior probability of a model available in explicit form (up to a normalizing constant). The strong model selection consistency is shown to be attained when the number of predictors grows nearly exponentially with the sample size and even when the norm of mean effects solely due to the unimportant variables diverge, which is a novel attractive feature. An appealing byproduct of SVEN is the construction of novel model weight adjusted prediction intervals. Embedding a unique model based screening and using fast Cholesky updates, SVEN produces a highly scalable computational framework to explore gigantic model spaces, rapidly identify the regions of high posterior probabilities and make fast inference and prediction. A temperature schedule guided by our model selection consistency derivations is used to further mitigate multimodal posterior distributions. The performance of SVEN is demonstrated through a number of simulation experiments and a real data example from a genome wide association study with over half a million markers.
Yang et al. (2016) proved that the symmetric random walk Metropolis--Hastings algorithm for Bayesian variable selection is rapidly mixing under mild high-dimensional assumptions. We propose a novel MCMC sampler using an informed proposal scheme, whic h we prove achieves a much faster mixing time that is independent of the number of covariates, under the same assumptions. To the best of our knowledge, this is the first high-dimensional result which rigorously shows that the mixing rate of informed MCMC methods can be fast enough to offset the computational cost of local posterior evaluation. Motivated by the theoretical analysis of our sampler, we further propose a new approach called two-stage drift condition to studying convergence rates of Markov chains on general state spaces, which can be useful for obtaining tight complexity bounds in high-dimensional settings. The practical advantages of our algorithm are illustrated by both simulation studies and real data analysis.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا