A Bayesian palaeoenvironmental transfer function model for acidified lakes

A Bayesian approach to palaeoecological environmental reconstruction deriving from the unimodal responses generally exhibited by organisms to an environmental gradient is described. The approach uses Bayesian model selection to calculate a collection of probability-weighted, species-specific response curves (SRCs) for each taxon within a training set, with an explicit treatment for zero abundances. These SRCs are used to reconstruct the environmental variable from sub-fossilised assemblages. The approach enables a substantial increase in computational efficiency (several orders of magnitude) over existing Bayesian methodologies. The model is developed from the Surface Water Acidification Programme (SWAP) training set and is demonstrated to exhibit comparable predictive power to existing Weighted Averaging and Maximum Likelihood methodologies, though with improvements in bias; the additional explanatory power of the Bayesian approach lies in an explicit calculation of uncertainty for each individual reconstruction. The model is applied to reconstruct the Holocene acidification history of the Round Loch of Glenhead, including a reconstruction of recent recovery derived from sediment trap data.The Bayesian reconstructions display similar trends to conventional (Weighted Averaging Partial Least Squares) reconstructions but provide a better reconstruction of extreme pH and are more sensitive to small changes in diatom assemblages. The validity of the posteriors as an apparently meaningful representation of assemblage-specific uncertainty and the high computational efficiency of the approach open up the possibility of highly constrained multiproxy reconstructions

Bayesian optimization of the PC algorithm for learning Gaussian Bayesian networks

The PC algorithm is a popular method for learning the structure of Gaussian Bayesian networks. It carries out statistical tests to determine absent edges in the network. It is hence governed by two parameters: (i) The type of test, and (ii) its significance level. These parameters are usually set to values recommended by an expert. Nevertheless, such an approach can suffer from human bias, leading to suboptimal reconstruction results. In this paper we consider a more principled approach for choosing these parameters in an automatic way. For this we optimize a reconstruction score evaluated on a set of different Gaussian Bayesian networks. This objective is expensive to evaluate and lacks a closed-form expression, which means that Bayesian optimization (BO) is a natural choice. BO methods use a model to guide the search and are hence able to exploit smoothness properties of the objective surface. We show that the parameters found by a BO method outperform those found by a random search strategy and the expert recommendation. Importantly, we have found that an often overlooked statistical test provides the best over-all reconstruction results

Bayesian Estimation of Intensity Surfaces on the Sphere via Needlet Shrinkage and Selection

This paper describes an approach for Bayesian modeling in spherical datasets. Our method is based upon a recent construction called the needlet, which is a particular form of spherical wavelet with many favorable statistical and computational properties. We perform shrinkage and selection of needlet coefficients, focusing on two main alternatives: empirical-Bayes thresholding, and Bayesian local shrinkage rules. We study the performance of the proposed methodology both on simulated data and on two real data sets: one involving the cosmic microwave background radiation, and one involving the reconstruction of a global news intensity surface inferred from published Reuters articles in August, 1996. The fully Bayesian approach based on robust, sparse shrinkage priors seems to outperform other alternatives.Business Administratio

Sampling constrained probability distributions using Spherical Augmentation

Statistical models with constrained probability distributions are abundant in machine learning. Some examples include regression models with norm constraints (e.g., Lasso), probit, many copula models, and latent Dirichlet allocation (LDA). Bayesian inference involving probability distributions confined to constrained domains could be quite challenging for commonly used sampling algorithms. In this paper, we propose a novel augmentation technique that handles a wide range of constraints by mapping the constrained domain to a sphere in the augmented space. By moving freely on the surface of this sphere, sampling algorithms handle constraints implicitly and generate proposals that remain within boundaries when mapped back to the original space. Our proposed method, called {Spherical Augmentation}, provides a mathematically natural and computationally efficient framework for sampling from constrained probability distributions. We show the advantages of our method over state-of-the-art sampling algorithms, such as exact Hamiltonian Monte Carlo, using several examples including truncated Gaussian distributions, Bayesian Lasso, Bayesian bridge regression, reconstruction of quantized stationary Gaussian process, and LDA for topic modeling.Comment: 41 pages, 13 figure

A Bayesian Approach to Manifold Topology Reconstruction

In this paper, we investigate the problem of statistical reconstruction of piecewise linear manifold topology. Given a noisy, probably undersampled point cloud from a one- or two-manifold, the algorithm reconstructs an approximated most likely mesh in a Bayesian sense from which the sample might have been taken. We incorporate statistical priors on the object geometry to improve the reconstruction quality if additional knowledge about the class of original shapes is available. The priors can be formulated analytically or learned from example geometry with known manifold tessellation. The statistical objective function is approximated by a linear programming / integer programming problem, for which a globally optimal solution is found. We apply the algorithm to a set of 2D and 3D reconstruction examples, demon-strating that a statistics-based manifold reconstruction is feasible, and still yields plausible results in situations where sampling conditions are violated

Bayesian multi-modal model comparison: a case study on the generators of the spike and the wave in generalized spike–wave complexes

We present a novel approach to assess the networks involved in the generation of spontaneous pathological brain activity based on multi-modal imaging data. We propose to use probabilistic fMRI-constrained EEG source reconstruction as a complement to EEG-correlated fMRI analysis to disambiguate between networks that co-occur at the fMRI time resolution. The method is based on Bayesian model comparison, where the different models correspond to different combinations of fMRI-activated (or deactivated) cortical clusters. By computing the model evidence (or marginal likelihood) of each and every candidate source space partition, we can infer the most probable set of fMRI regions that has generated a given EEG scalp data window. We illustrate the method using EEG-correlated fMRI data acquired in a patient with ictal generalized spike–wave (GSW) discharges, to examine whether different networks are involved in the generation of the spike and the wave components, respectively. To this effect, we compared a family of 128 EEG source models, based on the combinations of seven regions haemodynamically involved (deactivated) during a prolonged ictal GSW discharge, namely: bilateral precuneus, bilateral medial frontal gyrus, bilateral middle temporal gyrus, and right cuneus. Bayesian model comparison has revealed the most likely model associated with the spike component to consist of a prefrontal region and bilateral temporal–parietal regions and the most likely model associated with the wave component to comprise the same temporal–parietal regions only. The result supports the hypothesis of different neurophysiological mechanisms underlying the generation of the spike versus wave components of GSW discharges