Information-geometric Markov Chain Monte Carlo methods using Diffusions

Recent work incorporating geometric ideas in Markov chain Monte Carlo is reviewed in order to highlight these advances and their possible application in a range of domains beyond Statistics. A full exposition of Markov chains and their use in Monte Carlo simulation for Statistical inference and molecular dynamics is provided, with particular emphasis on methods based on Langevin diffusions. After this geometric concepts in Markov chain Monte Carlo are introduced. A full derivation of the Langevin diffusion on a Riemannian manifold is given, together with a discussion of appropriate Riemannian metric choice for different problems. A survey of applications is provided, and some open questions are discussed.Comment: 22 pages, 2 figure

The coefficient of determination R2 and intra-class correlation coefficient from generalized linear mixed-effects models revisited and expanded

The coefficient of determinationR2quantifies the proportion of varianceexplained by a statistical model and is an important summary statisticof biological interest. However, estimatingR2for generalized linear mixedmodels (GLMMs) remains challenging. We have previously introduced a ver-sion ofR2that we calledR2GLMMfor Poisson and binomial GLMMs, but notfor other distributional families. Similarly, we earlier discussed how to estimateintra-class correlation coefficients (ICCs) using Poisson and binomial GLMMs.Inthis paper, we generalize our methodsto allothernon-Gaussian distributions,in particular to negative binomial and gamma distributions that are commonlyused formodellingbiological data. Whileexpanding ourapproach,we highlighttwo useful concepts for biologists, Jensen’s inequality and the delta method,both of which help us in understanding the properties of GLMMs. Jensen’sinequality has important implications for biologically meaningful interpretationof GLMMs, whereas the delta method allows a general derivation of varianceassociated with non-Gaussian distributions. We also discuss some special con-siderations for binomial GLMMs with binary or proportion data. We illustratethe implementation of our extension by worked examples from the field of ecol-ogy and evolution in theRenvironment. However, our method can be usedacross disciplines and regardless of statistical environments

Kinetic energy choice in Hamiltonian/hybrid Monte Carlo

We consider how different choices of kinetic energy in Hamiltonian Monte Carlo affect algorithm performance. To this end, we introduce two quantities which can be easily evaluated, the composite gradient and the implicit noise. Results are established on integrator stability and geometric convergence, and we show that choices of kinetic energy that result in heavy-tailed momentum distributions can exhibit an undesirable negligible moves property, which we define. A general efficiency-robustness trade off is outlined, and implementations which rely on approximate gradients are also discussed. Two numerical studies illustrate our theoretical findings, showing that the standard choice which results in a Gaussian momentum distribution is not always optimal in terms of either robustness or efficiency.Comment: 15 pages (+7 page supplement, included here as an appendix), 2 figures (+1 in supplement

Rose Effect and the Euro: The Magic is Gone

This paper presents an updated meta-analysis of the effect of currency unions on trade, focusing on the Euro area. Using meta-regression methods such as funnel asymmetry test, evidence for strong publication bias is found. The estimated underlying effect for non-Euro studies reaches about 50%. However, the Euro's trade promoting effect corrected for publication bias is insignificant. The Rose effect literature shows signs of the economics research cycle: reported t-statistic is a quadratic function of publication year. Explanatory meta-regression (robust fixed effects and random effects) suggests that some authors produce predictable results. Interestingly, proxies for authors' IT skills were also found significant.Rose effect; Trade; Currency union; Euro; Meta-analysis; Publication bias

Bounding stationary averages of polynomial diffusions via semidefinite programming

We introduce an algorithm based on semidefinite programming that yields increasing (resp. decreasing) sequences of lower (resp. upper) bounds on polynomial stationary averages of diffusions with polynomial drift vector and diffusion coefficients. The bounds are obtained by optimising an objective, determined by the stationary average of interest, over the set of real vectors defined by certain linear equalities and semidefinite inequalities which are satisfied by the moments of any stationary measure of the diffusion. We exemplify the use of the approach through several applications: a Bayesian inference problem; the computation of Lyapunov exponents of linear ordinary differential equations perturbed by multiplicative white noise; and a reliability problem from structural mechanics. Additionally, we prove that the bounds converge to the infimum and supremum of the set of stationary averages for certain SDEs associated with the computation of the Lyapunov exponents, and we provide numerical evidence of convergence in more general settings

Age Differences in Intra-Individual Variability in Simple and Choice Reaction Time: Systematic Review and Meta-Analysis

Intra-individual variability in reaction time (RT IIV) is considered to be an index of central nervous system functioning. Such variability is elevated in neurodegenerative diseases or following traumatic brain injury. It has also been suggested to increase with age in healthy ageing.To investigate and quantify age differences in RT IIV in healthy ageing; to examine the effect of different tasks and procedures; to compare raw and mean-adjusted measures of RT IIV.Four electronic databases: PsycINFO, Medline, Web of Science and EMBASE, and hand searching of reference lists of relevant studies.English language journal articles, books or book chapters, containing quantitative empirical data on simple and/or choice RT IIV. Samples had to include younger (under 60 years) and older (60 years and above) human adults.Studies were evaluated in terms of sample representativeness and data treatment. Relevant data were extracted, using a specially-designed form, from the published report or obtained directly from the study authors. Age-group differences in raw and RT-mean-adjusted measures of simple and choice RT IIV were quantified using random effects meta-analyses.Older adults (60+ years) had greater RT IIV than younger (20-39) and middle-aged (40-59) adults. Age effects were larger in choice RT tasks than in simple RT tasks. For all measures of RT IIV, effect sizes were larger for the comparisons between older and younger adults than between older and middle-aged adults, indicating that the age-related increases in RT IIV are not limited to old age. Effect sizes were also larger for raw than for RT-mean-adjusted RT IIV measures.RT IIV is greater among older adults. Some (but not all) of the age-related increases in RT IIV are accounted for by the increased RT means

INLA or MCMC? A Tutorial and Comparative Evaluation for Spatial Prediction in log-Gaussian Cox Processes

We investigate two options for performing Bayesian inference on spatial log-Gaussian Cox processes assuming a spatially continuous latent field: Markov chain Monte Carlo (MCMC) and the integrated nested Laplace approximation (INLA). We first describe the device of approximating a spatially continuous Gaussian field by a Gaussian Markov random field on a discrete lattice, and present a simulation study showing that, with careful choice of parameter values, small neighbourhood sizes can give excellent approximations. We then introduce the spatial log-Gaussian Cox process and describe MCMC and INLA methods for spatial prediction within this model class. We report the results of a simulation study in which we compare MALA and the technique of approximating the continuous latent field by a discrete one, followed by approximate Bayesian inference via INLA over a selection of 18 simulated scenarios. The results question the notion that the latter technique is both significantly faster and more robust than MCMC in this setting; 100,000 iterations of the MALA algorithm running in 20 minutes on a desktop PC delivered greater predictive accuracy than the default \verb=INLA= strategy, which ran in 4 minutes and gave comparative performance to the full Laplace approximation which ran in 39 minutes.Comment: This replaces the previous version of the report. The new version includes results from an additional simulation study, and corrects an error in the implementation of the INLA-based method

Precise large deviations for dependent regularly varying sequences

We study a precise large deviation principle for a stationary regularly varying sequence of random variables. This principle extends the classical results of A.V. Nagaev (1969) and S.V. Nagaev (1979) for iid regularly varying sequences. The proof uses an idea of Jakubowski (1993,1997) in the context of centra limit theorems with infinite variance stable limits. We illustrate the principle for \sv\ models, functions of a Markov chain satisfying a polynomial drift condition and solutions of linear and non-linear stochastic recurrence equations