Multiplicative random walk Metropolis-Hastings on the real line

In this article we propose multiplication based random walk Metropolis Hastings (MH) algorithm on the real line. We call it the random dive MH (RDMH) algorithm. This algorithm, even if simple to apply, was not studied earlier in Markov chain Monte Carlo literature. The associated kernel is shown to have standard properties like irreducibility, aperiodicity and Harris recurrence under some mild assumptions. These ensure basic convergence (ergodicity) of the kernel. Further the kernel is shown to be geometric ergodic for a large class of target densities on $\mathbb{R}$. This class even contains realistic target densities for which random walk or Langevin MH are not geometrically ergodic. Three simulation studies are given to demonstrate the mixing property and superiority of RDMH to standard MH algorithms on real line. A share-price return data is also analyzed and the results are compared with those available in the literature

Modelling and simulation framework for reactive transport of organic contaminants in bed-sediments using a pure java object - oriented paradigm

Numerical modelling and simulation of organic contaminant reactive transport in the environment is being increasingly relied upon for a wide range of tasks associated with risk-based decision-making, such as prediction of contaminant profiles, optimisation of remediation methods, and monitoring of changes resulting from an implemented remediation scheme. The lack of integration of multiple mechanistic models to a single modelling framework, however, has prevented the field of reactive transport modelling in bed-sediments from developing a cohesive understanding of contaminant fate and behaviour in the aquatic sediment environment. This paper will investigate the problems involved in the model integration process, discuss modelling and software development approaches, and present preliminary results from use of CORETRANS, a predictive modelling framework that simulates 1-dimensional organic contaminant reaction and transport in bed-sediments

Noisy Monte Carlo: Convergence of Markov chains with approximate transition kernels

Monte Carlo algorithms often aim to draw from a distribution $\pi$ by simulating a Markov chain with transition kernel $P$ such that $\pi$ is invariant under $P$. However, there are many situations for which it is impractical or impossible to draw from the transition kernel $P$. For instance, this is the case with massive datasets, where is it prohibitively expensive to calculate the likelihood and is also the case for intractable likelihood models arising from, for example, Gibbs random fields, such as those found in spatial statistics and network analysis. A natural approach in these cases is to replace $P$ by an approximation $\hat{P}$. Using theory from the stability of Markov chains we explore a variety of situations where it is possible to quantify how 'close' the chain given by the transition kernel $\hat{P}$ is to the chain given by $P$. We apply these results to several examples from spatial statistics and network analysis.Comment: This version: results extended to non-uniformly ergodic Markov chain

An appraisal of retained placentae in Ibadan: a five year review

Objectives: To determine the frequency of retained placenta at the University College Hospital Ibadan (UCH). and to describe the socio-demographic characteristics of the patients and examine the risk factors predisposing to retained placenta.Methods: This is a descriptive study covering a period of 5 years from January 1st 2002 to December 31st 2006. During the study period, 4980 deliveries took place at the University College Hospital, Ibadan and 106 cases of retained placenta were managed making the incidence 2.13 per cent of all births.Results: During the five year period, there were 106 patients with retained placenta; of these, 90 (84.9%) case notes were available for analysis. The mean age was 29.37 ± 4.99 years. First and second Para accounted for 52 per cent of the patients. Majority of the patient were unbooked for antenatal care in UCH with booked patients accounting for 27.8 per cent of the cases. The mean gestational age at delivery was 34.29 ± 6.02. Three patients presented to the hospital in shock of which 2 died on account ofsevere haemorrhagic shock. Fifty-eight patients (64.8%) presented with anaemia (packed cell volume less than 30 per cent) and 35 patients (38.8%) had blood transfusion ranging between 1-4 pints. 1 patient required hysterectomy on account of morbidly adherent placenta. Eleven patients (12.2%) had placenta retention in the past, 28 patients (31%) had a previous dilatation and curettage, 14 patients (15.5%) had previous caesarean sections and 47 patients (41.3%) had no known predisposing factorsConclusion: Retained placenta still remains a potentially life threatening condition in the tropics due to the associated haemorrhage, and other complications related to its removal. The incidence and severity may be decreased by health education, women empowerment and the provision of facilities for essential obstetric services by high skilled health care providers in ensuring a properly conducted delivery with active management of the third stage of labour

Exact simulation of diffusions

We describe a new, surprisingly simple algorithm, that simulates exact sample paths of a class of stochastic differential equations. It involves rejection sampling and, when applicable, returns the location of the path at a random collection of time instances. The path can then be completed without further reference to the dynamics of the target process

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

The size-Ramsey number of powers of paths

Given graphs $G$ and $H$ and a positive integer $q$, say that $G$ \emph{is $q$-Ramsey for} $H$, denoted $G\rightarrow (H)_q$, if every $q$-colouring of the edges of $G$ contains a monochromatic copy of $H$. The \emph{size-Ramsey number} \sr(H) of a graph $H$ is defined to be \sr(H)=\min\{|E(G)|\colon G\rightarrow (H)_2\}. Answering a question of Conlon, we prove that, for every fixed~$k$, we have \sr(P_n^k)=O(n), where~$P_n^k$ is the $k$th power of the $n$-vertex path $P_n$ (i.e., the graph with vertex set $V(P_n)$ and all edges $\{u,v\}$ such that the distance between $u$ and $v$ in $P_n$ is at most $k$). Our proof is probabilistic, but can also be made constructive.Most of the work for this paper was done during my PhD, which was half funded by EPSRC grant reference 1360036, and half by Merton College Oxford. The third author was partially supported by FAPESP (Proc.~2013/03447-6) and by CNPq (Proc.~459335/2014-6, 310974/2013-5). The fifth author was supported by FAPESP (Proc.~2013/11431-2, Proc.~2013/03447-6 and Proc.~2018/04876-1) and partially by CNPq (Proc.~459335/2014-6). This research was supported in part by CAPES (Finance Code 001). The collaboration of part of the authors was supported by a CAPES/DAAD PROBRAL grant (Proc.~430/15)

Annular pigment band on the posterior capsule following blunt ocular trauma: a case report

BACKGROUND: To report an unusual case of annular pigment band on the posterior capsule following blunt ocular trauma. CASE PRESENTATION: We describe an annular pigment band on the posterior capsule following blunt ocular trauma in a 28-year old male patient. Repeat examinations revealed no evidence of other signs of blunt ocular trauma or pigment dispersion syndrome in either eye. CONCLUSION: The annular pigment band in this case corresponds to the adherence of the hyaloideocapsulare ligament to the posterior capsule and reconfirms its rare visualization in the living eye. This finding may be an isolated sign of blunt ocular trauma and a compromised integrity of the vitreolenticular interface should be strongly suspected. We recommend careful documentation in context of future cataract surgery in these eyes

On the flexibility of the design of Multiple Try Metropolis schemes

The Multiple Try Metropolis (MTM) method is a generalization of the classical Metropolis-Hastings algorithm in which the next state of the chain is chosen among a set of samples, according to normalized weights. In the literature, several extensions have been proposed. In this work, we show and remark upon the flexibility of the design of MTM-type methods, fulfilling the detailed balance condition. We discuss several possibilities and show different numerical results