Eaton (1992) considered a general parametric statistical model paired with an improper prior distribution for the parameter and proved that if a certain Markov chain, constructed using the model and the prior, is recurrent, then the improper prior is strongly admissible, which (roughly speaking) means that the generalized Bayes estimators derived from the corresponding posterior distribution are admissible. Hobert and Robert (1999) proved that Eaton’s Markov chain is recurrent if and only if its so-called conjugate Markov chain is recurrent. The focus of this paper is a family of Markov chains that contains all of the conjugate chains that arise in the context of a Poisson model paired with an arbitrary improper prior for the mean parameter. Sufficient conditions for recurrence and transience are developed and these are used to establish new results concerning the strong admissibility of non-conjugate improper priors for the Poisson mean.

We construct a class of non-reversible Metropolis kernels as a multivariate extension of the guided-walk kernel proposed by Gustafson (Statist. Comput. 8, 1998). The main idea of our method is to introduce a projection that maps a state space to a totally ordered group. By using Haar measure, we construct a novel Markov kernel termed the Haar mixture kernel, which is of interest in its own right. This is achieved by inducing a topological structure to the totally ordered group. Our proposed method, the $\Delta$-guided Metropolis–Haar kernel, is constructed by using the Haar mixture kernel as a proposal kernel. The proposed non-reversible kernel is at least 10 times better than the random-walk Metropolis kernel and Hamiltonian Monte Carlo kernel for the logistic regression and a discretely observed stochastic process in terms of effective sample size per second.

Reversible contraceptives, such as melengestrol acetate (MGA) implants, have become an invaluable tool that captive-animal managers use to maintain various species in social groupings while avoiding unwanted pregnancies. The American Zoo and Aquarium Contraception Advisory Group (AZA CAG) monitors the efficacy, reversibility, and safety of contraceptives used in captive exotic mammals worldwide. Because so few data exist on contraceptive efficacy and safety in exotic species, it is critical that evaluations reflect an understanding of the mechanism of action of the active ingredient as well as of the delivery system. The following discussion addresses the concerns of the CAG over the manner in which De Vleeschouwer et al (Animal Welfare 2000, 9: 251-271) analysed MGA implant reversibility data in golden-headed lion tamarins.

We consider a stochastic matching model with a general compatibility graph, as introduced by Mairesse and Moyal (2016). We show that the natural necessary condition of stability of the system is also sufficient for the natural ‘first-come, first-matched’ matching policy. To do so, we derive the stationary distribution under a remarkable product form, by using an original dynamic reversibility property related to that of Adan, Bušić, Mairesse, and Weiss (2018) for the bipartite matching model.

The structural details of chromosomes have been of interest to researchers for many years, but how the metaphase chromosome is constructed remains unsolved. Divalent cations have been suggested to be required for the organization of chromosomes. However, detailed information about the role of these cations in chromosome organization is still limited. In the current study, we investigated the effects of Ca2+ and Mg2+ depletion and the reversibility upon re-addition of one of the two ions. Human chromosomes were treated with different concentrations of Ca2+and Mg2+. Depletion of Ca2+ and both Ca2+ and Mg2+ were carried out using 1, 2-bis(2-aminophenoxy)ethane-N,N,N',N'-tetraacetic acid and ethylenediaminetetraacetic acid (EDTA), respectively. Chromosome structure was examined by fluorescence microscopy and scanning electron microscopy. The results indicated that chromosome structures after treatment with a buffer without Mg2+, after Ca2+ depletion, as well as after depletion of both Mg2+, and Ca2+, yielded fewer compact structures with fibrous chromatin than those without cation depletion. Interestingly, the chromatin of EDTA-treated chromosomes reversed to their original granular diameters after re-addition of either Mg2+ or Ca2+ only. These findings signify the importance of divalent cations on the chromosome structure and suggest the interchangeable role of Ca2+ and Mg2+.

Ternary systems of 152Eu(III), bulk bentonite and ethylenediaminetetraacetic acid (EDTA) ([Eu] = 7.9 × 10–10 M; pH = 6.0–7.0) have been studied. Without EDTA, there was slow uptake in a two-stage process, with initial rapid sorption of Eu(III) (96%), followed by slower uptake of a much smaller fraction (3.0% over a period of one month). The reversibility of Eu(III) binding was tested by allowing Eu(III) to sorb to bentonite for 1–322 days. EDTA was added to the pre-equilibrated Eu bentonite systems at 0.01 M, a concentration that was sufficient to suppress sorption in a system where EDTA was present prior to the contact of Eu(III) with bentonite. A fraction of the Eu was released instantaneously (30–50%), but a significant amount remained bound. With time, the amount of Eu(III) retained by the bentonite reduced, with a slow fraction dissociation rate constant of approximately 4.3 × 10–8 s–1 (values in the range 2.2 × 10–8 – 1.0 × 10–7 s–1) for pre-equilibration times ≥7 days. Eventually, the amount of Eu(III) remaining bound to the bentonite was within error of that when EDTA was present prior to contact (4.5% ± 0.6), although in systems with pre-equilibration times >100 days, full release took up to 500 days. Europium interactions with colloidal bentonite were also studied, and the dissociation rate constant measured by a resin competition method. For the colloids, more Eu was found in the slowly dissociating fraction (60–70%), but the first-order dissociation rate constant was faster, with an average rate constant of 8.8 × 10–7 s–1 and a range of 7.7 × 10–7 –9.5 × 10–7 s–1. For both bulk and colloidal bentonite, although slow dissociation was observed for Eu(III), there was no convincing evidence for 'irreversible' binding.

We propose a class of models of random walks in a random environment where an exact solution can be given for a stationary distribution. The environment is cast in terms of a Jackson/Gordon–Newell network although alternative interpretations are possible. The main tool is the detailed balance equations. The difference compared to earlier works is that the position of the random walk influences the transition intensities of the network environment and vice versa, creating strong correlations. The form of the stationary distribution is closely related to the well-known product formula.

For a stationary Markov process the detailed balance condition is equivalent to thetime-reversibility of the process. For stochastic differential equations (SDE’s), the timediscretization of numerical schemes usually destroys the time-reversibility property.Despite an extensive literature on the numerical analysis for SDE’s, their stabilityproperties, strong and/or weak error estimates, large deviations and infinite-timeestimates, no quantitative results are known on the lack of reversibility of discrete-timeapproximation processes. In this paper we provide such quantitative estimates by using theconcept of entropy production rate, inspired by ideas from non-equilibrium statisticalmechanics. The entropy production rate for a stochastic process is defined as the relativeentropy (per unit time) of the path measure of the process with respect to the pathmeasure of the time-reversed process. By construction the entropy production rate isnonnegative and it vanishes if and only if the process is reversible. Crucially, from anumerical point of view, the entropy production rate is an a posterioriquantity, hence it can be computed in the course of a simulation as the ergodicaverage of a certain functional of the process (the so-called Gallavotti−Cohen (GC) action functional). We computethe entropy production for various numerical schemes such as explicit Euler−Maruyama and explicit Milstein’s forreversible SDEs with additive or multiplicative noise. In addition we analyze the entropyproduction for the BBK integrator for the Langevin equation. The order (in thetime-discretization step Δt) of the entropy production rate provides a tool toclassify numerical schemes in terms of their (discretization-induced) irreversibility. Ourresults show that the type of the noise critically affects the behavior of the entropyproduction rate. As a striking example of our results we show that the Euler scheme formultiplicative noise is not an adequate scheme from a reversibilitypoint of view since its entropy production rate does not decrease withΔt.

The microstructural analysis of the dehydrogenation products of the Ca(BH4)2–MgH2 composite was performed using transmission electron microscopy. It was found that nanocrystalline CaB6 crystallites formed as a dehydrogenation product throughout the areas where the signals of Ca and Mg were simultaneously detected, in addition to relatively coarse Mg crystallites. The uniform distribution of the nanocrystalline CaB6 crystallites appears to play a key role in the rehydrogenation of the dehydrogenation products, which implies that microstructure is a crucial factor determining the reversibility of reactive hydride composites.

The theory of L2-spectral gaps for reversible Markov chains has been studied by many authors. In this paper we consider positive recurrent general state space Markov chains with stationary transition probabilities. Replacing the assumption of reversibility with a weaker assumption, we still obtain a simple necessary and sufficient condition for the spectral gap property of the associated Markov operator in terms of the isoperimetric constant. We show that this result can be applied to a large class of Markov chains, including those that are related to positive recurrent finite-range random walks on Z.

In a homogeneous continuous-time Markov chain on a finite state space, two states that jump to every other state with the same rate are called similar. By partitioning states into similarity classes, the algebraic derivation of the transition matrix can be simplified, using hidden holding times and lumped Markov chains. When the rate matrix is reversible, the transition matrix is explicitly related in an intuitive way to that of the lumped chain. The theory provides a unified derivation for a whole range of useful DNA base substitution models, and a number of amino acid substitution models.

We show that, for reversible continuous-time Markov chains, the closeness of the nonzero eigenvalues of the generator to zero provides complete information about the sensitivity of the distribution vector to perturbations of the generator. Our results hold for both the transient and the stationary states.

A new structure with the special property that instantaneous resurrection and mass disaster are imposed on an ordinary birth-death process is considered. Under the condition that the underlying birth-death process is exit or bilateral, we are able to give easily checked existence criteria for such Markov processes. A very simple uniqueness criterion is also established. All honest processes are explicitly constructed. Ergodicity properties for these processes are investigated. Surprisingly, it can be proved that all the honest processes are not only recurrent but also ergodic without imposing any extra conditions. Equilibrium distributions are then established. Symmetry and reversibility of such processes are also investigated. Several examples are provided to illustrate our results.

We introduce a new model of cellular automaton called a one-dimensional number-conserving partitioned cellular automaton (NC-PCA). An NC-PCA is a system such that a state of a cell is represented by a triple of non-negative integers, and the total (i.e., sum) of integers over the configuration is conserved throughout its evolving (computing) process. It can be thought as a kind of modelization of the physical conservation law of mass (particles) or energy. We also define a reversible version of NC-PCA, and prove that a reversible NC-PCA is computation-universal. It is proved by showing that a reversible two-counter machine, which has been known to be universal, can be simulated by a reversible NC-PCA.

The paper characterizes matrices which have a given system of vectors orthogonal with respect to a given probability distribution as its right eigenvectors. Results of Hoare and Rahman are unified in this context, then all matrices with a given orthogonal polynomial system as right eigenvectors under the constraint a0j = 0 for j ≥ 2 are specified. The only stochastic matrices P = {pij} satisfying p00 + p01 = 1 with the Hahn polynomials as right eigenvectors have the form of the Moran mutation model.

We define a kind of cellular automaton called a hexagonal partitioned cellular automaton (HPCA), and study logical universality of a reversible HPCA. We give a specific 64-state reversible HPCA H1, and show that a Fredkin gate can be embedded in this cellular space. Since a Fredkin gate is known to be a universal logic element, logical universality of H1 is concluded. Although the number of states of H1 is greater than those of the previous models of reversible CAs having universality, the size of the configuration realizing a Fredkin gate is greatly reduced, and its local transition function is still simple. Comparison with the previous models, and open problems related to these model are also discussed.

Consider a system of interacting finite Markov chains in continuous time, where each subsystem is aggregated by a common partitioning of the state space. The interaction is assumed to arise from dependence of some of the transition rates for a given subsystem at a specified time on the states of the other subsystems at that time. With two subsystem classes, labelled 0 and 1, the superposition process arising from a system counts the number of subsystems in the latter class. Key structure and results from the theory of aggregated Markov processes are summarized. These are then applied also to superposition processes. In particular, we consider invariant distributions for the level m entry process, marginal and joint distributions for sojourn-times of the superposition process at its various levels, and moments and correlation functions associated with these distributions. The distributions are obtained mainly by using matrix methods, though an approach based on point process methods and conditional probability arguments is outlined. Conditions under which an interacting aggregated Markov chain is reversible are established. The ideas are illustrated with simple examples for which numerical results are obtained using Matlab. Motivation for this study has come from stochastic modelling of the behaviour of ion channels; another application is in reliability modelling.