Geometric ergodicity of the Random Walk Metropolis with position-dependent proposal covariance

We consider a Metropolis-Hastings method with proposal kernel $\mathcal{N}(x,hG^{-1}(x))$, where $x$ is the current state. After discussing specific cases from the literature, we analyse the ergodicity properties of the resulting Markov chains. In one dimension we find that suitable choice of $G^{-1}(x)$ can change the ergodicity properties compared to the Random Walk Metropolis case $\mathcal{N}(x,h\Sigma)$, either for the better or worse. In higher dimensions we use a specific example to show that judicious choice of $G^{-1}(x)$ can produce a chain which will converge at a geometric rate to its limiting distribution when probability concentrates on an ever narrower ridge as $|x|$ grows, something which is not true for the Random Walk Metropolis.Comment: 15 pages + appendices, 4 figure

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

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

Gradient-free Hamiltonian Monte Carlo with Efficient Kernel Exponential Families

We propose Kernel Hamiltonian Monte Carlo (KMC), a gradient-free adaptive MCMC algorithm based on Hamiltonian Monte Carlo (HMC). On target densities where classical HMC is not an option due to intractable gradients, KMC adaptively learns the target's gradient structure by fitting an exponential family model in a Reproducing Kernel Hilbert Space. Computational costs are reduced by two novel efficient approximations to this gradient. While being asymptotically exact, KMC mimics HMC in terms of sampling efficiency, and offers substantial mixing improvements over state-of-the-art gradient free samplers. We support our claims with experimental studies on both toy and real-world applications, including Approximate Bayesian Computation and exact-approximate MCMC.Comment: 20 pages, 7 figure

Association between birthweight and Acute Lymphoblastic Leukemia in children, a systematic review

BackgroundBirthweight is normally determined by a range of genetic traits and exposures occurring within the intra-uterine environment. Some epidemiological studies have reported high birthweight as a risk factor of Acute lymphoblastic Leukemia (ALL). Other studies have however not demonstrated this relationship.ObjectivesThe objective of this review is to assess the association between birthweight and Acute Lymphoblastic Leukemia in children. Search methods We searched observational studies from Cochrane, MEDLINE, EMBASE, ISI Web of Science, BIOSIS, the allied and Complementary Medicine Database and National Research Register, ClinicalTrial.gov, WHO International Trials Registry Platform. Selection criteriaWe included case control and cohort studies assessing the association between birthweight and ALL in children. All particpants below the age of 18 years (children) with ALL were included in the analysis. The independent variable in this review was birthweight. Birthweight was categorised into two (2). Birthweight >4kg (experimental arm) and ≤ (control arm).Data collection and analysisTwo reviewers independently assessed identified studies through two stages of screening. First, titles and abstracts of all references identified through searches were screened and irrelevant studies were excluded. Also, full texts of potentially eligible studies were further assessed according to previously defined inclusion criteria. Studies that did not meet inclusion criteria were excluded and reasons for their exclusion were stated. All studies that met the inclusion criteria were included. Areas of disagreement were resolved by a third-party review. Two review authors double checked the studies independently.Main resultsOut of the 348 studies screened, 16 of them met the inclusion criteria. A total of 3650728 participant provided data for analysis in this review. These studies were published between 1987 and 2018. The age span of studies was similar across studies (roughly (0-18 years). The vast majority of ALL was diagnosed before 15 years. 14 of the included studies were case control studies and 2 of them were cohort studies. Figure 1 presents odd ratio estimates for effect of birthweight on ALL (≤ 4000g vs. > 4000g). There was a statistically significant positive relationship between high birthweight (birthweight > 4000g) and risk of ALL (16 studies, OR 0.81, 95% CI 0.77, 0.85). Authors' conclusionsOur study revealed a significant positive relationship between high birthweight and ALL. Several studies have demonstrated an association between factors such as: high pre-pregnancy weight and height; gestational age greater than 42 weeks; parity greater than and high birthweight. Therefore, public health programs and interventions aimed at reducing the incidence of these maternal factors can reduce the risk of high birthweight and lower the incidence of ALL

Adaptive MCMC for Bayesian variable selection in generalised linear models and survival models

Developing an efficient computational scheme for high-dimensional Bayesian variable selection in generalised linear models and survival models has always been a challenging problem due to the absence of closed-form solutions for the marginal likelihood. The RJMCMC approach can be employed to samples model and coefficients jointly, but effective design of the transdimensional jumps of RJMCMC can be challenge, making it hard to implement. Alternatively, the marginal likelihood can be derived using data-augmentation scheme e.g. Polya-gamma data argumentation for logistic regression) or through other estimation methods. However, suitable data-augmentation schemes are not available for every generalised linear and survival models, and using estimations such as Laplace approximation or correlated pseudo-marginal to derive marginal likelihood within a locally informed proposal can be computationally expensive in the "large n, large p" settings. In this paper, three main contributions are presented. Firstly, we present an extended Point-wise implementation of Adaptive Random Neighbourhood Informed proposal (PARNI) to efficiently sample models directly from the marginal posterior distribution in both generalised linear models and survival models. Secondly, in the light of the approximate Laplace approximation, we also describe an efficient and accurate estimation method for the marginal likelihood which involves adaptive parameters. Additionally, we describe a new method to adapt the algorithmic tuning parameters of the PARNI proposal by replacing the Rao-Blackwellised estimates with the combination of a warm-start estimate and an ergodic average. We present numerous numerical results from simulated data and 8 high-dimensional gene fine mapping data-sets to showcase the efficiency of the novel PARNI proposal compared to the baseline add-delete-swap proposal