a riemannian optimization approach to the matrix singular value decomposition the problem of the singular value decomposition of a matrix can be brought into an optimization problem on the product of two stiefel manifolds of different sizes the steepest descent the conjugate gradient and newton s methods for the problem are developed and applied with several numerical experiments these algorithms do not need the preconditioning that is inevitable in the usual singular value decomposition algorithm the present newton s method can serve to make more accurate the singular value decomposition obtained by other existing algorithms in addition degenerate optimal solutions are studied together with numerical experiments to show that those solutions form a disconnected submanifold

worst case evaluation complexity for unconstrained nonlinear optimization using high order regularized models the worst case evaluation complexity for smooth possibly nonconvex unconstrained optimization is considered it is shown that if one is willing to use derivatives of the objective function up to order p for p ge 1 p 1 and to assume lipschitz continuity of the p th derivative then an epsilon approximate first order critical point can be computed in at most o epsilon p 1 p o p 1 p evaluations of the problem s objective function and its derivatives this generalizes and subsumes results known for p 1 p 1 and p 2 p 2

plates with regular stiffening in acoustic media vibration and radiation flat plates and cylindrical shells with regular and identical stiffening constitute spatially periodic structures specially convenient methods of vibration analysis are available for these some of which are suitable for the inclusion of the effects of fluid loading from adjacent acoustic media this paper outlines the nature of free wave motion in periodic structures stiffened either in one direction or in two orthogonal directions the structural response to distributed harmonic pressure fields is analyzed by using displacement functions consisting of a series of space harmonics the sound radiated or transmitted by the vibrating structure is also found the method of space harmonics is next combined with the method of phased array receptance functions to yield equations for the propagation constants of fluid loaded periodic plates stiffened in just one direction some calculated propagation constants are presented recent developments in the application of the hierarchical finite element method to periodic structures are described computed results are presented for the propagation constants of an orthogonally stiffened plate without fluid loading consideration is given to those frequency ranges and wave motions that could radiate into an adjacent medium

sound radiation from fluid loaded orthogonally stiffened plates abstract the radiation of sound from infinite fluid loaded plates is examined when the plates are reinforced with two sets of orthogonal line stiffeners the stiffeners are assumed to be equally spaced and exert only forces on the plate the response to a convected harmonic pressure is found by using fourier transforms and is given in terms of the harmonic amplitudes of the stiffener forces these forces satisfy an infinite set of simultaneous equations to which a numerical solution must be found an expression for the response to a general excitation is derived and from this the acoustic pressure in the far field is determined with particular reference to point force excitation

formulation of mindlin engesser model for stiffened plate vibration in this paper a mindlin engesser model is developed for the vibration analysis of moderately thick plates with arbitrarily oriented stiffeners the theoretical derivation incorporates the mindlin theory to account for the effects of transverse shear deformation and rotary inertia of plates and the engesser theory to account for the shear deformation of stiffeners with the inclusion of torsion effect in the method of solution the resulting energy functionals are minimized using the ritz procedure with a set of admissible two dimensional functions expressed in the form of simple polynomials the key kinematic feature of these shape functions is that they are boundary oriented and no boundary losses are introduced as in discretization methods with the aim of demonstrating the applicability and versatility of the method numerical examples including plates of various shapes with arbitrarily oriented stiffeners are presented several findings and conclusions regarding the method have been highlighted and discussed

free vibration of an orthogonally stiffened flat plate abstract a flat plate reinforced by a regular orthogonal array of uniform beams is analyzed by using techniques developed for studying wave propagation in two dimensional periodic structures a plane wave type of motion is considered which may be characterized by different propagation phase constants in the x and y directions the hierarchical finite element method is used to set up the governing equations of free wave motion and these are then solved as an eigenvalue problem for the frequencies at which particular waves will propagate plots of phase constant surfaces vs frequency are presented for a number of different plate beam configurations excellent agreement is found between some of these and the results of earlier investigators when the plate is supported by flexible beams in both directions wave propagation is found to commerce at zero frequency at higher frequencies alternating and overlapping attenuation and propagation bands occur the nature and explanations of these are discussed wave speed surfaces vs frequency are also presented and these give insight into the critical coincidence frequencies of the plate under acoustic excitation

parameter estimation for differential equations a generalized smoothing approach summary we propose a new method for estimating parameters in models that are defined by a system of non linear differential equations such equations represent changes in system outputs by linking the behaviour of derivatives of a process to the behaviour of the process itself current methods for estimating parameters in differential equations from noisy data are computationally intensive and often poorly suited to the realization of statistical objectives such as inference and interval estimation the paper describes a new method that uses noisy measurements on a subset of variables to estimate the parameters defining a system of non linear differential equations the approach is based on a modification of data smoothing methods along with a generalization of profiled estimation we derive estimates and confidence intervals and show that these have low bias and good coverage properties respectively for data that are simulated from models in chemical engineering and neurobiology the performance of the method is demonstrated by using real world data from chemistry and from the progress of the autoimmune disease lupus

the asymptotic efficiency of simulation estimators a decision theoretic framework is proposed for evaluating the efficiency of simulation estimators the framework includes the cost of obtaining the estimate as well as the cost of acting based on the estimate the cost of obtaining the estimate and the estimate itself are represented as realizations of jointly distributed stochastic processes in this context the efficiency of a simulation estimator based on a given computational budget is defined as the reciprocal of the risk the overall expected cost this framework is appealing philosophically but it is often difficult to apply in practice e g to compare the efficiency of two different estimators because only rarely can the efficiency associated with a given computational budget be calculated however a useful practical framework emerges in a large sample context when we consider the limiting behavior as the computational budget increases a limit theorem established for this model supports and extends a fairly well known efficiency principle

adaptive cubic regularisation methods for unconstrained optimization part ii worst case function and derivative evaluation complexity an adaptive regularisation framework using cubics arc was proposed for unconstrained optimization and analysed in cartis gould and toint part i math program doi 10 1007 s10107 009 0286 5 2009 generalizing at the same time an unpublished method due to griewank technical report na 12 1981 damtp university of cambridge an algorithm by nesterov and polyak math program 108 1 177 205 2006 and a proposal by weiser deuflhard and erdmann optim methods softw 22 3 413 431 2007 in this companion paper we further the analysis by providing worst case global iteration complexity bounds for arc and a second order variant to achieve approximate first order and for the latter second order criticality of the iterates in particular the second order arc algorithm requires at most mathcal o epsilon 3 2 iterations or equivalently function and gradient evaluations to drive the norm of the gradient of the objective below the desired accuracy epsilon and mathcal o epsilon 3 iterations to reach approximate nonnegative curvature in a subspace the orders of these bounds match those proved for algorithm 3 3 of nesterov and polyak which minimizes the cubic model globally on each iteration our approach is more general in that it allows the cubic model to be solved only approximately and may employ approximate hessians

a coupling regeneration scheme for diagnosing convergence in markov chain monte carlo algorithms abstract here i propose a convergence diagnostic for markov chain monte carlo mcmc algorithms based on couplings of a markov chain with an auxiliary chain that is periodically restarted from a fixed parameter value the diagnostic provides a mechanism for estimating the specific constants governing the rate of convergence of geometrically and uniformly ergodic chains and provides a lower bound on the effective sample size of a mcmc run it also provides a simple procedure for obtaining what is with high probability an independent sample from the stationary distribution

monte carlo approximation of bayes factors via mixing with surrogate distributions by mixing the target posterior distribution with a surrogate distribution of which the normalizing constant is tractable we propose a method for estimating the marginal likelihood using the wang

particle markov chain monte carlo methods summary markov chain monte carlo and sequential monte carlo methods have emerged as the two main tools to sample from high dimensional probability distributions although asymptotic convergence of markov chain monte carlo algorithms is ensured under weak assumptions the performance of these algorithms is unreliable when the proposal distributions that are used to explore the space are poorly chosen and or if highly correlated variables are updated independently we show here how it is possible to build efficient high dimensional proposal distributions by using sequential monte carlo methods this allows us not only to improve over standard markov chain monte carlo schemes but also to make bayesian inference feasible for a large class of statistical models where this was not previously so we demonstrate these algorithms on a non linear state space model and a levy driven stochastic volatility model

approximate riemannian conjugate gradient learning for fixed form variational bayes variational bayesian vb methods are typically only applied to models in the conjugate exponential family using the variational bayesian expectation maximisation vb em algorithm or one of its variants in this paper we present an efficient algorithm for applying vb to more general models the method is based on specifying the functional form of the approximation such as multivariate gaussian the parameters of the approximation are optimised using a conjugate gradient algorithm that utilises the riemannian geometry of the space of the approximations this leads to a very efficient algorithm for suitably structured approximations it is shown empirically that the proposed method is comparable or superior in efficiency to the vb em in a case where both are applicable we also apply the algorithm to learning a nonlinear state space model and a nonlinear factor analysis model for which the vb em is not applicable for these models the proposed algorithm outperforms alternative gradient based methods by a significant margin

cubic regularization of newton method and its global performance in this paper we provide theoretical analysis for a cubic regularization of newton method as applied to unconstrained minimization problem for this scheme we prove general local convergence results however the main contribution of the paper is related to global worst case complexity bounds for different problem classes including some nonconvex cases it is shown that the search direction can be computed by standard linear algebra technique

optimal scaling of discrete approximations to langevin diffusions we consider the optimal scaling problem for proposal distributions in hastings metropolis algorithms derived from langevin diffusions we prove an asymptotic diffusion limit theorem and show that the relative efficiency of the algorithm can be characterized by its overall acceptance rate independently of the target distribution the asymptotically optimal acceptance rate is 0 574 we show that as a function of dimension n the complexity of the algorithm is o n1 3 which compares favourably with the o n complexity of random walk metropolis algorithms we illustrate this comparison with some example simulations

geometry of stochastic state vector reduction the state space of a quantum mechanical system is a complex projective space the space of rays in the associated hilbert space the state space comes equipped with a natural riemannian metric the fubini study metric and a compatible symplectic structure the operations of ordinary quantum mechanics can thus be reinterpreted in the language of differential geometry it is interesting in this spirit to scrutinize the probabilistic assumptions that are brought in at various stages in the analy sis of quantum dynamics particularly in connection with state vector reduction a promising approach to understanding reduction studied recently by a number of authors involves the use of nonlinear stochastic dynamics to modify the ordinary linear schrodinger evolution here we use stochastic differential geometry to give a systematic geometric formulation for such stochastic models of state vector collapse in this picture the conventional schrodinger evolution which corresponds to the unitary flow associated with a killing vector of the fubini study metric is replaced by a more general stochastic flow on the state manifold in the simplest example of such a flow the volatility term in the stochastic differential equation for the state trajectory is proportional to the gradient of the expectation of the hamiltonian the conservation of energy is represented by the requirement that the actual process followed by the expectation of the hamiltonian as the state evolves should be a martingale this requirement implies the existence of a nonlinear term in the drift vector of the state process which is always oriented opposite to the direction of in creasing energy uncertainty as a consequence the state vector necessarily collapses to an energy eigenstate and a martingale argument can be used to show that the probability of collapse to a given eigenstate from any particular initial state is in fact given by precisely the usual quantum mechanical probability

projection like retractions on matrix manifolds this paper deals with constructing retractions a key step when applying optimization algorithms on matrix manifolds for submanifolds of euclidean spaces we show that the operation consisting of taking a tangent step in the embedding euclidean space followed by a projection onto the submanifold is a retraction we also show that the operation remains a retraction if the projection is generalized to a projection like procedure that consists of coming back to the submanifold along admissible directions and we give a sufficient condition on the admissible directions for the generated retraction to be second order this theory offers a framework in which previously proposed retractions can be analyzed as well as a toolbox for constructing new ones illustrations are given for projection like procedures on some specific manifolds for which we have an explicit easy to compute expression

vibration of rectangular plates with edge restraints and intermediate stiffeners vibrations of stiffened plates with elastically restrained edges are analyzed by the rayleigh ritz method the first lower four frequencies for restrained plates with up to six intermediate stiffeners are calculated the results of this study are compared with existing data and the agreement is excellent

local minima and convergence in low rank semidefinite programming the low rank semidefinite programming problem lrsdpr is a restriction of the semidefinite programming problem sdp in which a bound r is imposed on the rank of x and it is well known that lrsdpr is equivalent to sdp if r is not too small in this paper we classify the local minima of lrsdpr and prove the optimal convergence of a slight variant of the successful yet experimental algorithm of burer and monteiro 5 which handles lrsdpr via the nonconvex change of variables x rrt in addition for particular problem classes we describe a practical technique for obtaining lower bounds on the optimal solution value during the execution of the algorithm computational results are presented on a set of combinatorial optimization relaxations including some of the largest quadratic assignment sdps solved to date

optimization methods on riemannian manifolds and their application to shape space we extend the scope of analysis for linesearch optimization algorithms on possibly infinite dimensional riemannian manifolds to the convergence analysis of the bfgs quasi newton scheme and the fletcher reeves conjugate gradient iteration numerical implementations for exemplary problems in shape spaces show the practical applicability of these methods

a hierarchical functions set for predicting very high order plate bending modes with any boundary conditions in this paper a new hierarchical functions set is proposed to predict flexural motion of plate like structures in the medium frequency range this functions set is built from trigonometric functions instead of polynomials as classically encountered in the literature it is shown that such a trigonometric set presents all the advantages of a classical hierarchical polynomials set and additional ones which are of interest if very high order functions are intended to be used it is stated that this new trigonometric set can be used at very high orders up to 2048 without taking care of computer round off errors while the polynomials set fail at order 46 because of the limited numerical dynamics of computers this trigonometric set can be easily implemented on a computer it does not require quadruple precision pre computed arrays only a very low number which does not depend on the function order of basic operations is needed when calling such functions moreover it is shown that this trigonometric set presents a better convergence rate than polynomials when predicting high order natural flexural modes of rectangular plates with any boundary conditions

shapefit exact location recovery from corrupted pairwise directions let t1 tn r and consider the location recovery problem given a subset of pairwise direction observations ti tj ti tj 2 i j n n where a constant fraction of these observations are arbitrarily corrupted find ti i 1 up to a global translation and scale we propose a novel algorithm for the location recovery problem which consists of a simple convex program over dn real variables we prove that this program recovers a set of n i i d gaussian locations exactly and with high probability if the observations are given by an erdos renyi graph d is large enough and provided that at most a constant fraction of observations involving any particular location are adversarially corrupted

adaptive regularization with cubics on manifolds adaptive regularization with cubics arc is an algorithm for unconstrained non convex optimization akin to the trust region method its iterations can be thought of as approximate safe guarded newton steps for cost functions with lipschitz continuous hessian arc has optimal iteration complexity in the sense that it produces an iterate with gradient smaller than varepsilon in o 1 varepsilon 1 5 iterations for the same price it can also guarantee a hessian with smallest eigenvalue larger than sqrt varepsilon in this paper we study a generalization of arc to optimization on riemannian manifolds in particular we generalize the iteration complexity results to this richer framework our central contribution lies in the identification of appropriate manifold specific assumptions that allow us to secure these complexity guarantees both when using the exponential map and when using a general retraction a substantial part of the paper is devoted to studying these assumptions relevant beyond arc and providing user friendly sufficient conditions for them numerical experiments are encouraging

rates of convergence of the hastings and metropolis algorithms we apply recent results in markov chain theory to hastings and metropolis algorithms with either independent or symmetric candidate distributions and provide necessary and sufficient conditions for the algorithms to converge at a geometric rate to a prescribed distribution pi in the independence case in mathbb r k these indicate that geometric convergence essentially occurs if and only if the candidate density is bounded below by a multiple of pi in the symmetric case in mathbb r only we show geometric convergence essentially occurs if and only if pi has geometric tails we also evaluate recently developed computable bounds on the rates of convergence in this context examples show that these theoretical bounds can be inherently extremely conservative although when the chain is stochastically monotone the bounds may well be effective

on the use of three dimensional h and p version finite elements in solving vibration response problems abstract the potential for reliably solving vibration response problems by using h and p versions of the finite element fe method is investigated each fe model is based on full three dimensional 3 d displacement based continuum theory special attention is given to the ability to handle thin and or nearly incompressible elastic and viscoelastic materials steady state time harmonic problems in the low and mid frequency ranges are treated convergence studies are performed on plate like model problems with simply supported boundary conditions using a series elasto dynamic navier solution extended to the viscoelasto dynamic case by means of the elasto viscoelastic correspondence principle the importance of eliminating the discretisation error when using numerical solutions to estimate the frequency dependent viscoelastic material parameters from experiments is stressed the superior efficiency of p enrichments compared to h refinements for resolving regular elastostatic solutions is observed in elasto viscoelasto dynamics as well requirements on p enrichments to avoid shear and volumetric locking are given the frequently used polynomial interpolation functions employed are shown to produce a mass contribution to the dynamic stiffness whose condition number deteriorates with increasing polynomial order the practical implications and limitations of this observation are outlined the recommended approach is corroborated using the measured frequency response on a laminate consisting of two aluminium plates and a constrained viscoelastic polymer damping treatment

bayesian prediction of spatial count data using generalized linear mixed models summary spatial weed count data are modeled and predicted using a generalized linear mixed model combined with a bayesian approach and markov chain monte carlo informative priors for a data set with sparse sampling are elicited using a previously collected data set with extensive sampling furthermore we demonstrate that so called langevin hastings updates are useful for efficient simulation of the posterior distributions and we discuss computational issues concerning prediction

low rank optimization on the cone of positive semidefinite matrices we propose an algorithm for solving optimization problems defined on a subset of the cone of symmetric positive semidefinite matrices this algorithm relies on the factorization x yy t where the number of columns of y fixes an upper bound on the rank of the positive semidefinite matrix x it is thus very effective for solving problems that have a low rank solution the factorization x yy t leads to a reformulation of the original problem as an optimization on a particular quotient manifold the present paper discusses the geometry of that manifold and derives a second order optimization method with guaranteed quadratic convergence it furthermore provides some conditions on the rank of the factorization to ensure equivalence with the original problem in contrast to existing methods the proposed algorithm converges monotonically to the sought solution its numerical efficiency is evaluated on two applications the maximal cut of a graph and the problem of sparse principal component analysis

vibration of stiffened plates using hierarchical trigonometric functions the vibration analysis of stiffened plates using hierarchical finite elements with a set of local trigonometric interpolation functions is presented the local functions extend on the plate domain comprised between consecutive stiffeners thereby allowing a coarse discretization of the global structure convergence studies as well as comparison of the present approach with the literature and experimental results are presented the great numerical stability of the trigonometric functions and their readiness for symbolic manipulations make them potentially attractive for vibration and sound radiation analysis in the mid frequency range

stochastic gradient descent on riemannian manifolds stochastic gradient descent is a simple approach to find the local minima of a cost function whose evaluations are corrupted by noise in this paper we develop a procedure extending stochastic gradient descent algorithms to the case where the function is defined on a riemannian manifold we prove that as in the euclidian case the gradient descent algorithm converges to a critical point of the cost function the algorithm has numerous potential applications and is illustrated here by four examples in particular a novel gossip algorithm on the set of covariance matrices is derived and tested numerically

equivariant adaptive source separation source separation consists of recovering a set of independent signals when only mixtures with unknown coefficients are observed this paper introduces a class of adaptive algorithms for source separation that implements an adaptive version of equivariant estimation and is henceforth called equivariant adaptive separation via independence easi the easi algorithms are based on the idea of serial updating this specific form of matrix updates systematically yields algorithms with a simple structure for both real and complex mixtures most importantly the performance of an easi algorithm does not depend on the mixing matrix in particular convergence rates stability conditions and interference rejection levels depend only on the normalized distributions of the source signals closed form expressions of these quantities are given via an asymptotic performance analysis the theme of equivariance is stressed throughout the paper the source separation problem has an underlying multiplicative structure the parameter space forms a matrix multiplicative group we explore the favorable consequences of this fact on implementation performance and optimization of easi algorithms

solving the trust region subproblem using the lanczos method the approximate minimization of a quadratic function within an ellipsoidal trust region is an important subproblem for many nonlinear programming methods when the number of variables is large the most widely used strategy is to trace the path of conjugate gradient iterates either to convergence or until it reaches the trust region boundary in this paper we investigate ways of continuing the process once the boundary has been encountered the key is to observe that the trust region problem within the currently generated krylov subspace has a very special structure which enables it to be solved very efficiently we compare the new strategy with existing methods the resulting software package is available as hsl vf05 within the harwell subroutine library

riemann manifold langevin and hamiltonian monte carlo methods the paper proposes metropolis adjusted langevin and hamiltonian monte carlo sampling methods defined on the riemann manifold to resolve the shortcomings of existing monte carlo algorithms when sampling from target densities that may be high dimensional and exhibit strong correlations the methods provide fully automated adaptation mechanisms that circumvent the costly pilot runs that are required to tune proposal densities for metropolis hastings or indeed hamiltonian monte carlo and metropolis adjusted langevin algorithms this allows for highly efficient sampling even in very high dimensions where different scalings may be required for the transient and stationary phases of the markov chain the methodology proposed exploits the riemann geometry of the parameter space of statistical models and thus automatically adapts to the local structure when simulating paths across this manifold providing highly efficient convergence and exploration of the target density the performance of these riemann manifold monte carlo methods is rigorously assessed by performing inference on logistic regression models log gaussian cox point processes stochastic volatility models and bayesian estimation of dynamic systems described by non linear differential equations substantial improvements in the time normalized effective sample size are reported when compared with alternative sampling approaches matlab code that is available from http www ucl ac uk statistics research rmhmc allows replication of all the results reported

bayesian calibration of computer models we consider prediction and uncertainty analysis for systems which are approximated using complex mathematical models such models implemented as computer codes are often generic in the sense that by a suitable choice of some of the model s input parameters the code can be used to predict the behaviour of the system in a variety of specific applications however in any specific application the values of necessary parameters may be unknown in this case physical observations of the system in the specific context are used to learn about the unknown parameters the process of fitting the model to the observed data by adjusting the parameters is known as calibration calibration is typically effected by ad hoc fitting and after calibration the model is used with the fitted input values to predict the future behaviour of the system we present a bayesian calibration technique which improves on this traditional approach in two respects first the predictions allow for all sources of uncertainty including the remaining uncertainty over the fitted parameters second they attempt to correct for any inadequacy of the model which is revealed by a discrepancy between the observed data and the model predictions from even the best fitting parameter values the method is illustrated by using data from a nuclear radiation release at tomsk and from a more complex simulated nuclear accident exercise

manopt a matlab toolbox for optimization on manifolds optimization on manifolds is a rapidly developing branch of nonlinear optimization its focus is on problems where the smooth geometry of the search space can be leveraged to design efficient numerical algorithms in particular optimization on manifolds is well suited to deal with rank and orthogonality constraints such structured constraints appear pervasively in machine learning applications including low rank matrix completion sensor network localization camera network registration independent component analysis metric learning dimensionality reduction and so on the manopt toolbox available at www manopt org is a user friendly documented piece of software dedicated to simplify experimenting with state of the art riemannian optimization algorithms by dealing internally with most of the differential geometry the package aims particularly at lowering the entrance barrier

ricci curvature for metric measure spaces via optimal transport we dene a notion of a measured length space x having nonnegative n ricci curvature for n 2 1 1 or having1 ricci curvature bounded below byk fork2 r the denitions are in terms of the displacement convexity of certain functions on the associated wasserstein metric space p2 x of probability measures we show that these properties are preserved under measured gromov hausdor limits we give geometric and analytic consequences this paper has dual goals one goal is to extend results about optimal transport from the setting of smooth riemannian manifolds to the setting of length spaces a second goal is to use optimal transport to give a notion for a measured length space to have ricci curvature bounded below we refer to 11 and 44 for background material on length spaces and optimal transport respectively further bibliographic notes on optimal transport are in appendix f in the present introduction we motivate the questions that we address and we state the main results to start on the geometric side there are various reasons to try to extend notions of curvature from smooth riemannian manifolds to more general spaces a fairly general setting is that of length spaces meaning metric spaces x d in which the distance between two points equals the inmum of the lengths of curves joining the points in the rest of this introduction we assume that x is a compact length space alexandrov gave a good notion of a length space having curvature bounded below by k with k a real number in terms of the geodesic triangles in x in the case of a riemannian manifold m with the induced length structure one recovers the riemannian notion of having sectional curvature bounded below by k length spaces with alexandrov curvature bounded below by k behave nicely with respect to the gromovhausdor topology on compact metric spaces modulo isometries they form

scaling limits for the transient phase of local metropolis hastings algorithms summary the paper considers high dimensional metropolis and langevin algorithms in their initial transient phase in stationarity these algorithms are well understood and it is now well known how to scale their proposal distribution variances for the random walk metropolis algorithm convergence during the transient phase is extremely regular to the extent that the algo rithm s sample path actually resembles a deterministic trajectory in contrast the langevin algorithm with variance scaled to be optimal for stationarity performs rather erratically we give weak convergence results which explain both of these types of behaviour and practical guidance on implementation based on our theory

regeneration in markov chain samplers abstract markov chain sampling has recently received considerable attention in particular in the context of bayesian computation and maximum likelihood estimation this article discusses the use of markov chain splitting originally developed for the theoretical analysis of general state space markov chains to introduce regeneration into markov chain samplers this allows the use of regenerative methods for analyzing the output of these samplers and can provide a useful diagnostic of sampler performance the approach is applied to several samplers including certain metropolis samplers that can be used on their own or in hybrid samplers and is illustrated in several examples