Wojciech Niemiro

Nonasymptotic bounds on the estimation error of MCMC algorithms

We address the problem of upper bounding the mean square error of MCMC estimators. Our analysis is non-asymptotic. We first establish a general result valid for essentially all ergodic Markov chains encountered in Bayesian computation and a possibly unbounded target function f: The bound is sharp in the sense that the leading term is exactly �2 as(P; f)=n, where �2 as(P; f) is the CLT asymptotic variance. Next, we proceed to specific assumptions and give explicit computable bounds for geometrically and polynomially ergodic Markov chains. As a corollary we provide results on confidence estimation.

Modeling exopeptidase activity from LC-MS data.

Recent studies demonstrate that the peptides in the serum of cancer patients that are generated (ex vivo) as a result of tumor protease activity can be used for the detection and classification of cancer. In this paper, we propose the first formal approach to modeling exopeptidase activity from liquid chromatography-mass spectrometry (LC-MS) samples. We design a statistical model of peptidome degradation and a Metropolis-Hastings algorithm for Bayesian inference of model parameters. The model is successfully validated on a real LC-MS dataset. Our findings support the hypotheses about disease-specific exopeptidase activity, which can lead to new diagnostic approach in clinical proteomics.

The Prevalence of Asthma and Declared Asthma in Poland on the Basis of ECAP Survey Using Correspondence Analysis

Results of epidemiological and public health surveys are often presented in the form of cross-classification tables. It is sometimes difficult to analyze data described in this way and to understand relations between variables. Graphical methods such as correspondence analysis are more convenient and useful. Our paper describes an application of correspondence analysis to epidemiological research. We apply the basic concepts of correspondence analysis like profiles, chi-square distance to medical data concerning prevalence of asthma. We aim at describing the relationship between asthma, region, and age. The data presented in this paper come from Epidemiology of Allergy in Poland (ECAP) survey in years 2006–2008. Correspondence analysis shows that there is a fundamental difference in the structure of age groups for people with symptoms compared to those who have declared asthma (regardless of the level of symptoms of asthma and the level of declaration). The variable which best differentiates declared asthma in all regions is “wheezing and whistling.” Correspondence analysis also shows significant differences between locations. Our analyses are performed in the R package “ca”.

A version of the Swendsen–Wang algorithm for restoration of images degraded by Poisson noise☆

Abstract An algorithm for restoration of images degraded by Poisson noise is proposed. The algorithm belongs to the family of Markov chain Monte Carlo methods with auxiliary variables. We explicitly use the fact that medical images consist of finitely many, often relatively few, grey-levels. The continuous scale of grey-levels is discretized in an adaptive way, so that a straightforward application of the Swendsen–Wang (Phys. Rev. Lett. 58 (1987) 86) algorithm becomes possible. Partial decoupling method due to Higdon (J. Am. Statist. Assoc. 93 (1998) 442, 585) is also incorporated into the algorithm. Simulation results suggest that the algorithm is reliable and efficient.

MCMC imputation in autologistic model

Abstract We consider statistical inference from incomplete sets of binary data. Our approach is based on the autologistic model, which is very flexible and well suited for medical applications. We propose a Bayesian approach, essentially using Monte Carlo techniques. The method developed in this paper is a special version of Gibbs sampler. We repeat intermittently the following two steps. First, missing values are generated from the predictive distribution. Second, unknown parametes are estimated from the completed data. The Monte Carlo method of computing maximum likelihood estimates due to Geyer and Thompson (J. R. Statist. Soc. B 54: 657–699, 1992) is modified to the Bayesian setting and missing data problems. We include results of some small scale simulation experiments. We artificially introduce missing values in a real data set and then use our algorithm to refill missings. The rate of correct imputations is quite satisfactory.

Tail events of simulated annealing Markov chains

We consider non-homogeneous Markov chains generated by the simulated annealing algorithm. We classify states according to asymptotic properties of trajectories. We identify recurrent and transient states. The set of recurrent states is partitioned into disjoint classes of asymptotically communicating states. These classes correspond to atoms of the tail sigma-field. The results are valid under the weak reversibility assumption of Hajek.

Geometric ergodicity of Rao and Teh’s algorithm for Markov jump processes and CTBNs

Rao and Teh (2013) introduced an efficient MCMC algorithm for sampling from the posterior distribution of a hidden Markov jump process. The algorithm is based on the idea of sampling virtual jumps. In the present paper we show that the Markov chain generated by Rao and Tehs algorithm is geometrically ergodic. To this end we establish a geometric drift condition towards a small set.

Sex estimation of upper long bones by selected measurements in a Radom (Poland) population from the 18th and 19th centuries AD

Abstract Several studies have shown that sex estimation methods based on measurements of the skeleton are specific to populations. Metric traits of the upper long bones have been reported as reliable indicators of sex. This study was designed to determine whether the four long bones can be used for the sex estimation of an historical skeletal population from Radom (Poland). The material used consists of the bones of 169 adult individuals (including 103 males and 66 females) from the 18th and 19th centuries. Twelve measurements were recovered from clavicle, humerus, radius and ulna. The initial comparison of males and females indicated significant differences in all measurements (p < 0.0001). The accuracy of sex estimation ranged from 68% to 84%. The best predictor for sex estimation of all the measurements in Radom’s population was the maximum length of the radius (84%), and the ulna (83%), and the vertical diameter of the humeral head (83%). The Generalized Linear Model (GLM) detected the strongest significant relationship between referential sex and the vertical diameter of the humeral head (p < 0.0001), followed by the maximal length of the ulna (p = 0.0117). In other measurements of the upper long bones, GLM did not detect statistically significant differences.

Adaptive Monte Carlo Maximum Likelihood

We consider Monte Carlo approximations to the maximum likelihood estimator in models with intractable norming constants. This paper deals with adaptive Monte Carlo algorithms, which adjust control parameters in the course of simulation. We examine asymptotics of adaptive importance sampling and a new algorithm, which uses resampling and MCMC. This algorithm is designed to reduce problems with degeneracy of importance weights. Our analysis is based on martingale limit theorems. We also describe how adaptive maximization algorithms of Newton-Raphson type can be combined with the resampling techniques. The paper includes results of a small scale simulation study in which we compare the performance of adaptive and non-adaptive Monte Carlo maximum likelihood algorithms.

Remarks on uniform convergence of random variables and statistics

Abstract Convergence in distribution, convergece in probability, and convergence almost surely, uniform with respect to a family of probability distributions, is considered. These concepts appeared to be appropriate tools for asymptotic theory of mathematical statistics and many partial results are scattered in the literature of the subject. The aim of this note is to present a unified review of the results in a general and abstract setup. We examine a few rather paradoxical examples which hopefully shed some light on the subtleties of the underlying definitions and the role of asymptotic approximations in statistics. A motivation for considering these problems is provided by their applications.