Sufficient burn-in for Gibbs samplers for a hierarchical random effects model

We consider Gibbs and block Gibbs samplers for a Bayesian hierarchical version of the one-way random effects model. Drift and minorization conditions are established for the underlying Markov chains. The drift and minorization are used in conjunction with results from J. S. Rosenthal [J. Amer. Statist. Assoc. 90 (1995) 558-566] and G. O. Roberts and R. L. Tweedie [Stochastic Process. Appl. 80 (1999) 211-229] to construct analytical upper bounds on the distance to stationarity. These lead to upper bounds on the amount of burn-in that is required to get the chain within a prespecified (total variation) distance of the stationary distribution. The results are illustrated with a numerical example

Evaluation of Formal posterior distributions via Markov chain arguments

We consider evaluation of proper posterior distributions obtained from improper prior distributions. Our context is estimating a bounded function $\phi$ of a parameter when the loss is quadratic. If the posterior mean of $\phi$ is admissible for all bounded $\phi$, the posterior is strongly admissible. We give sufficient conditions for strong admissibility. These conditions involve the recurrence of a Markov chain associated with the estimation problem. We develop general sufficient conditions for recurrence of general state space Markov chains that are also of independent interest. Our main example concerns the $p$-dimensional multivariate normal distribution with mean vector $\theta$ when the prior distribution has the form $g(\|\theta\|^2) d\theta$ on the parameter space $\mathbb{R}^p$. Conditions on $g$ for strong admissibility of the posterior are provided.Comment: Published in at http://dx.doi.org/10.1214/07-AOS542 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Evaluating default priors with a generalization of Eaton's Markov chain

We consider evaluating improper priors in a formal Bayes setting according to the consequences of their use. Let Φ be a class of functions on the parameter space and consider estimating elements of Φ under quadratic loss. If the formal Bayes estimator of every function in Φ is admissible, then the prior is strongly admissible with respect to Φ. Eaton’s method for establishing strong admissibility is based on studying the stability properties of a particular Markov chain associated with the inferential setting. In previous work, this was handled differently depending upon whether ϕ ∈ Φ was bounded or unbounded. We consider a new Markov chain which allows us to unify and generalize existing approaches while simultaneously broadening the scope of their potential applicability. We use our general theory to investigate strong admissibility conditions for location models when the prior is Lebesgue measure and for the p-dimensional multivariate Normal distribution with unknown mean vector θ and a prior of the form ν(‖θ‖²)dθ

«Grey zone» of heart failure

The review is devoted to modern understanding of heart failure with mid-range ejection fraction. The formation of the paradigm of «two phenotypes» of heart failure began around the end of the last century. As a result of a number of large epidemiological studies on heart failure with preserved ejection fraction, so-called «grey zone» of ejection fraction values was formed in the range of about 40-50%. This situation arose because of the lack of clearly established level of normal ejection fraction and underlines imperfection of this parameter as the only classification criterion. But no more convenient «tool» for research work was offered. In the past decade, «grey zone» of heart failure has been actively explored by clinical epidemiologists and clinicians. Should we classify these patients as one of the existing phenotypes of heart failure or present them as a new, separate phenotype? Both the first and second decisions require information about the population «portrait» of subgroup, about their response to treatment, and presumptive pathophysiological mechanisms of heart failure. In 2016 European society of cardiology guidelines for the diagnosis and treatment of acute and chronic heart failure, heart failure with mid-range ejection fraction was determined as a separate subgroup to stimulate the search for such data. At the moment mid-range ejection fraction is known to be recorded in about 10-20% of patients with heart failure. They have substantial comorbidities as patients with preserved ejection fraction but the prevalence of ischemic heart disease in this subgroup makes it similar to heart failure with reduced ejection fraction. The response to treatment with beta-blockers and aldosterone antagonists is similar to that of heart failure with reduced ejection fraction. It is important that the mortality rates in all three groups of patients are approximately the same. This circumstance underlines the importance of further searche. Perhaps the research of «grey zone» of the syndrome will help to better understand pathophysiology of the existing heart failure phenotypes and confirm the validity of their identification based on ejection fraction

Laplacian Growth and Whitham Equations of Soliton Theory

The Laplacian growth (the Hele-Shaw problem) of multi-connected domains in the case of zero surface tension is proven to be equivalent to an integrable systems of Whitham equations known in soliton theory. The Whitham equations describe slowly modulated periodic solutions of integrable hierarchies of nonlinear differential equations. Through this connection the Laplacian growth is understood as a flow in the moduli space of Riemann surfaces.Comment: 33 pages, 7 figures, typos corrected, new references adde