a central limit theorem for temporally non homogenous markov chains with applications to dynamic programming we prove a central limit theorem for a class of additive processes that arise naturally in the theory of nite horizon markov decision problems the main theorem generalizes a classic result of dobrushin 1956 for tem porally non homogeneous markov chains and the principal innovation is that here the summands are permitted to depend on both the current state and a bounded number of future states of the chain we show through several examples that this added exibility gives one a direct path to asymptotic nor mality of the optimal total reward of nite horizon markov decision problems the same examples also explain why such results are not easily obtained by alternative markovian techniques such as enlargement of the state space

bayesian computation via the gibbs sampler and related markov chain monte carlo methods the use of the gibbs sampler for bayesian computation is reviewed and illustrated in the context of some canonical examples other markov chain monte carlo simulation methods are also briefly described and comments are made on the advantages of sample based approaches for bayesian inference summaries

on the geometric convergence of the gibbs sampler summary the rate of convergence of the gibbs sampler is discussed the gibbs sampler is a monte carlo simulation method with extensive application to computational issues in the bayesian paradigm conditions for the geometric rate of convergence of the algorithm for discrete and continuous parameter spaces are derived and an illustrative exponential family example is given this paper investigates conditions under which the gibbs sampler gelfand and smith 1990 tanner and wong 1987 geman and geman 1984 converges at a geometric rate the main results appear in sections 2 and 3 where geometric convergence results are established with respect to total variation and supremum norms under fairly natural conditions on the underlying distribution for ease of exposition we shall concentrate on the two most commonly encountered situations where the state space is finite or continuous all our results will establish uniform convergence a strong form of geometric convergence under appropriate regularity conditions uniform convergence is a useful property in its own right but also happens to be a sufficient condition for certain ergodic central limit theorems such results are important for estimation in markov chain simulation but will not be considered in detail here our approach is to apply the theory of markov chains to the specific gibbs sampler case in the finite state space case uniform ergodicity is automatic however the situation is more complicated for continuous state spaces where even well behaved underlying distributions can give rise to markov chains which converge slowly or have unbounded kernels we give two results in this context corollary 2 and corollary 3 which establish uniform convergence under different sets of conditions on the underlying density finally we apply these results to an example of a bayesian hierarchical model where regularity conditions for geometric convergence are naturally satisfied here the hierarchical structure of the model is crucial in permitting application of corollary 3

central limit theorem for triangular arrays of non homogeneous markov chains in this paper we obtain the central limit theorem for triangular arrays of non homogeneous markov chains under a condition imposed to the maximal coefficient of correlation the proofs are based on martingale techniques and a sharp lower bound estimate for the variance of partial sums the results complement an important central limit theorem of dobrushin based on the contraction coefficient

optimal monte carlo updating based on peskun s theorem it is shown that optimal transition matrices in markov chain monte carlo should have zero diagonal elements except for the diagonal element corresponding to the largest weight we will compare the statistical efficiency of this sampler to existing algorithms such as heat bath updating and the metropolis algorithm we provide numerical results for the potts model as an application in classical physics as an application in quantum physics we consider the spin 3 2xy model and the bose hubbard model which have been simulated by the directed loop algorithm in the stochastic series expansion framework

on random and systematic scan samplers we introduce a simple time homogeneous markov embedding of a class of time inhomogeneous markov chains widely used in the context of monte carlo sampling algorithms such as systematic scan metropolis within gibbs samplers this allows us to establish that systematic scan samplers involving two factors are always better than their random scan counterparts when asymptotic variance is the criterion of interest we also show that this embedding sheds some light on the result of maire et al 2014 and discuss the scenario involving more than two factors

rosenthal type inequalities for the maximum of partial sums of stationary processes and examples the aim of this paper is to propose new rosenthal type inequalities for moments of order higher than 2 of the maximum of partial sums of stationary sequences including martingales and their generalizations as in the recent results by peligrad et al proc amer math soc 135 2007 541 550 and rio j theoret probab 22 2009 146 163 the estimates of the moments are expressed in terms of the norms of projections of partial sums the proofs of the results are essentially based on a new maximal inequality generalizing the doob maximal inequality for martingales and dyadic induction various applications are also provided

conditional simulation of max stable processes summary since many environmental processes such as heat waves or precipitation are spatial in extent it is likely that a single extreme event affects several loca tions and the areal modelling of extremes is therefore essential if the spatial dependence of e xtremes has to be appropriately taken into account this paper proposes a framework for conditional simulations of max stable processes and give closed forms for brown resnick and schlather processes we test the method on simulated data and give an application to extreme rainfall around zurich and extreme temperature in switzerland results show that the proposed framework provides accurate conditional simulations and can handle real sized problems

beyond automated multilevel substructuring domain decomposition with rational filtering this paper proposes a rational filtering domain decomposition technique for the solution of large and sparse symmetric generalized eigenvalue problems the proposed technique is purely algebraic an

bayesian models for multiple local sequence alignment and gibbs sampling strategies abstract a wealth of data concerning life s basic molecules proteins and nucleic acids has emerged from the biotechnology revolution the human genome project has accelerated the growth of these data multiple observations of homologous protein or nucleic acid sequences from different organisms are often available but because mutations and sequence errors misalign these data multiple sequence alignment has become an essential and valuable tool for understanding structures and functions of these molecules a recently developed gibbs sampling algorithm has been applied with substantial advantage in this setting in this article we develop a full bayesian foundation for this algorithm and present extensions that permit relaxation of two important restrictions we also present a rank test for the assessment of the significance of multiple sequence alignment as an example we study the set of dinucleotide binding proteins and predict binding segments for dozens of its members

markov chain monte carlo methods for lattice gaussian sampling convergence analysis and enhancement sampling from lattice gaussian distribution has emerged as an important problem in coding decoding and cryptography in this paper the classic gibbs algorithm from markov chain monte carlo mcmc methods is demonstrated to be geometrically ergodic for lattice gaussian sampling which means that the markov chain arising from it converges exponentially fast to the stationary distribution meanwhile the exponential convergence rate of the markov chain is also derived through the spectral radius of the forward operator then a comprehensive analysis of the convergence rate is carried out and two sampling schemes are proposed to further enhance the convergence performance the first one referred to as a metropolis within gibbs mwg algorithm improves the convergence by refining the state space of the univariate sampling the second is a blocked strategy of the gibbs algorithm which performs sampling over multivariates at each markov move and is shown to yield a better convergence rate than the traditional univariate sampling in order to perform blocked sampling efficiently the gibbs klein gk algorithm is proposed which samples block by block using the kleins algorithm furthermore the validity of the gk algorithm is demonstrated by showing its ergodicity simulation results based on mimo detections are presented to confirm the convergence gain brought by the proposed gibbs sampling schemes

precise rates in complete moment convergence for ρ mixing sequences abstract let x 1 x 2 be a strictly stationary sequence of mixing random variables with mean zeros and positive finite variances set s n x 1 x n suppose that lim n e s n 2 n 2 0 n 1 2 q 2 n where q 2 2 we prove that if e x 1 2 log x 1 for any 0 1 then lim 0 2 n 2 log n 1 n 2 e s n 2 i s n n log n e n 2 2 where n is the standard normal random variable

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

on marcinkiewicz zygmund laws abstract marcinkiewicz zygmund laws with convergence rates are established here for a class of strictly stationary ergodic sequences

the calculation of posterior distributions by data augmentation abstract the idea of data augmentation arises naturally in missing value problems as exemplified by the standard ways of filling in missing cells in balanced two way tables thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze this device is used to great advantage by the em algorithm dempster laird and rubin 1977 in solving maximum likelihood problems in situations when the likelihood cannot be approximated closely by the normal likelihood maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements from the bayesian point of view one must now calculate the posterior distribution of parameters of interest if data augmentation can be used in the calculation of the maximum likelihood estimate then in the same cases one ought to be able to use it in the computation of the posterior distribution it is the purpose of this article to explain how this can be done the basic idea

geometric convergence and central limit theorems for multidimensional hastings and metropolis algorithms we develop results on geometric ergodicity of markov chains and apply these and other recent results in markov chain theory to multidimensional hastings and metropolis algorithms for those based on random walk candidate distributions we find sufficient conditions for moments and moment generating functions to converge at a geometric rate to a prescribed distribution by phrasing the conditions in terms of the curvature of the densities we show that the results apply to all distributions with positive densities in a large class which encompasses many commonly used statistical forms from these results we develop central limit theorems for the metropolis algorithm converse results showing non geometric convergence rates for chains where the rejection rate is not bounded away from unity are also given these show that the negative definiteness property is not redundant

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

bayesian estimation of switching arma models abstract switching arma processes have recently appeared as an efficient modelling to nonlinear time series models because they can represent multiple or heterogeneous dynamics through simple components the levels of dependence between the observations are double at a first level the parameters of the model are selected by a markovian procedure at a second level the next observation is generated according to a standard time series model when the model involves a moving average structure the complexity of the resulting likelihood function is such that simulation techniques like those proposed by shephard 1994 biometrika 81 115 131 and billio and monfort 1998 journal of statistical planning and inference 68 65 103 are necessary to derive an inference on the parameters of the model we propose in this paper a bayesian approach with a non informative prior distribution developed in mengersen and robert 1996 bayesian statistics 5 oxford university press oxford pp 255 276 and robert and titterington 1998 statistics and computing 8 2 145 158 in the setup of mixtures of distributions and hidden markov models respectively the computation of the bayes estimates relies on mcmc techniques which iteratively simulate missing states innovations and parameters until convergence the performances of the method are illustrated on several simulated examples this work also extends the papers by chib and greenberg 1994 journal of econometrics 64 183 206 and chib 1996 journal of econometrics 75 1 79 97 which deal with arma and hidden markov models respectively

multilinear maps on products of operator algebras abstract let a1 a2 ar be c algebras with second duals a1 a2 ar and let x be an arbitrary banach space let a 1 a 2 a r x be a bounded r linear map and denote by a 1 a 2 a r x the johnson kadison ringrose extension i e the separately weak to weak continuous r linear extension of the problem of characterising those for which takes its values in x was solved by villanueva when the algebras are all commutative because the dunford pettis property fails for noncommutative c algebras the obvious extension of villanueva s characterisation does not give the correct condition in this paper we solve this problem for general c algebras this result is then applied to obtaining a multilinear generalisation of the normal singular decomposition of a bounded linear operator on a von neumann algebra

the grothendieck inequality for bilinear forms on c algebras abstract the following generalization of grothendieck s inequality is proved for any bounded bilinear form v on a pair of c algebras a b there exist two states 1 2 on a and two states 1 2 on b such that v x y v 1 x x 2 xx 2 1 2 1 y y 2 yy 2 1 2 for all x a and all y b an inequality of this type was proved a few years ago by pisier in the case where one of the c algebras has the bounded approximation property it follows from the above inequality that any bounded linear map t of a c algebra into the dual of a c algebra has a factorization t r s through a hilbert space such that r s 2 t

lattice reduction aided gibbs algorithm for lattice gaussian sampling convergence enhancement and decoding optimization sampling from the lattice gaussian distribution has emerged as an important problem in coding decoding and cryptography in this paper lattice reduction technique is adopted to gibbs sampler for lattice gaussian sampling first with respect to lattice gaussian distribution we show the convergence rate of systematic scan gibbs sampling is characterized by the hirschfeld gebelein renyi maximal correlation among the multivariate of being sampled then lattice reduction aided gibbs algorithm is proposed to sample from an equivalent lattice gaussian distribution but with less correlated multivariate thus leading to a better markov mixing after that we extend the proposed lattice reduction aided gibbs sampling to lattice decoding where the choice of the standard deviation for the sampling is fully investigated a customized solution that suits for each specific decoding case by euclidean distance is given which results in a better tradeoff between markov mixing and sampler decoding based on it a startup mechanism is also proposed for gibbs sampler decoding where decoding complexity can be reduced without performance loss moreover the recycling gibbs sampling that exploits the potential of samples is also considered to improve the decoding performance in lattice decoding simulation results based on large scale uncoded multiple input multiple output detection are presented to confirm the performance gain and complexity reduction

polynomially filtered exact diagonalization approach to many body localization polynomially filtered exact diagonalization method polfed for large sparse matrices is introduced the algorithm finds an optimal basis of a subspace spanned by eigenvectors with eigenvalues close to a specified energy target by a spectral transformation using a high order polynomial of the matrix the memory requirements scale better with system size than in the state of the art shift invert approach the potential of polfed is demonstrated examining many body localization transition in 1d interacting quantum spin 1 2 chains we investigate the disorder strength and system size scaling of thouless time system size dependence of bipartite entanglement entropy and of the gap ratio highlights the importance of finite size effects we discuss possible scenarios regarding the many body localization transition obtaining estimates for the critical disorder strength

convergence rates in the law of large numbers for arrays of martingale differences abstract we study the convergence rates in the law of large numbers for arrays of martingale differences for n 1 let x n 1 x n 2 be a sequence of real valued martingale differences with respect to a filtration f n 0 f n 1 f n 2 and set s n n x n 1 x n n under a simple moment condition on j 1 n e x n j f n j 1 for some 1 2 we show necessary and sufficient conditions for the convergence of the series n 1 n p s n n e n where e 0 and is a positive function we also give a criterion for n p s n n e n 0 the most interesting case where is a regularly varying function is considered with attention in the special case where x n j j 1 is the same sequence x j j 1 of independent and identically distributed random variables our result on the series n 1 n p s n n e n corresponds to the theorems of hsu robbins and erdos 1947 1949 if 1 and n 1 of spitzer 1956 if 1 and n 1 n and of baum and katz 1965 if 1 2 and n n b 1 with b 0 in the single martingale case where x n j x j for all n and j it generalizes the results of alsmeyer 1990 the consideration of martingale arrays rather than a single martingale makes the results very adapted in the study of weighted sums of identically distributed random variables for which we prove new theorems about the rates of convergence in the law of large numbers the results are established in a more general setting for sums of infinitely many martingale differences say s n j 1 x n j instead of s n n the obtained results improve and extend those of ghosal and chandra 1998 the one sided cases and the supermartingale case are also considered

grothendieck s inequalities for real and complex jbw triples we prove that if m 4 1 2 3 and e 0 if v and w are complex jbw triples with preduals v andw respectively and if u is a separately weak continuous bilinear form on v w then there exist norm one functionals 1 2 v and 1 2 w satisfying u x y m u x 2 2 e 2 x 2 1 1 2 y 2 2 e 2 y 2 1 1 2 for all x y v w here for a norm one functional on a complex jb triple v stands for the prehilertian seminorm on v associated to in bf1 we arrive in this grothendieck s inequality through results of c h chu b iochum and g loupias cil and a corrected version of the little grothendieck s inequality for complex jb triples due to t barton and y friedman bf1 we also obtain extensions of these results to the setting of real jb triples 2000 mathematics subject classification 17c65 46k70 46l05 46l10 and 46l70

quasi completely continuous multilinear operators we introduce the concept of quasi completely continuous multilinear operators and use this concept to characterize for a wide class of banach spaces x1 xk the multilinear operators t x1 xk x with an x valued aron berner extension

convergence rate of markov chain methods for genomic motif discovery we analyze the convergence rate of a popular gibbs sampling method used for statistical discovery of gene regulatory binding motifs in dna sequences this sampler satisfies a very strong form of ergodicity uniform however we show that due to multimodality of the posterior distribution the rate of convergence often decreases exponentially as a function of the length of the dna sequence specifically we show that this occurs whenever there is more than one true repeating pattern in the data in practice there are typically multiple even numerous such patterns in biological data the goal being to detect the most well conserved and frequently occurring of these our findings match empirical results in which the motif discovery gibbs sampler has exhibited such poor convergence that it is used only for finding modes of the posterior distribution candidate motifs rather than for obtaining samples from that distribution ours appear to be the first meaningful bounds on the convergence rate of a markov chain method for sampling from a multimodal posterior distribution as a function of statistical quantities like the number of observations

the collapsed gibbs sampler in bayesian computations with applications to a gene regulation problem abstract this article describes a method of grouping and collapsing in using the gibbs sampler and proves from an operator theory viewpoint that the method is in general beneficial the norms of the forward operators associated with the corresponding nonreversible markov chains are used to discriminate among different simulation schemes when applied to bayesian missing data problems the idea of collapsing suggests skipping the steps of sampling parameter s values in standard data augmentation by doing this we obtain a predictive update version of the gibbs sampler a procedure of calculating the posterior odds ratio via the collapsed gibbs sampler when incomplete observations are involved is presented as an illustration of possible applications three examples along with a bayesian treatment for identifying common protein binding sites in unaligned dna sequences are provided

antithetic coupling of two gibbs sampler chains two coupled gibbs sampler chains both with invariant probability density are run in parallel so that the chains are negatively correlated we define an asymptotically unbiased estimator of the expectation e f x which achieves significant variance reduction with respect to the usual gibbs sampler at comparable computational cost the variance of the estimator based on the new algorithm is always smaller than the variance of a single gibbs sampler chain if is attractive and f is monotone nondecreasing in all components of x for nonattractive targets our results are not complete the new antithetic algorithm outperforms the standard gibbs sampler when is a multivariate normal density or the ising model more generally nonrigorous arguments and numerical experiments support the usefulness of the antithetically coupled gibbs samplers also for other nonattractive models in our experiments the variance is reduced to at least a third and the efficiency also improves significantly

weakly compact operators and the strong topology for a banach space the strong topology s x of a banach space x is defined as the locally convex topology generated by the seminorms x 7 ksxk for bounded linear maps s from x into hilbert spaces the w right topology for x x is a stronger locally convex topology which may be analogously characterised by taking reflexive banach spaces in place of hilbert spaces for any banach space y a linear map t x y is known to be weakly compact precisely when t is continuous from the w right topology to the norm topology of y the main results deal with conditions for and consequences of the coincidence of these two topologies on norm bounded sets a large class of banach spaces including all c algebras and more generally all jb triples exhibit this behaviour

a thick restart lanczos algorithm with polynomial filtering for hermitian eigenvalue problems polynomial filtering can provide a highly effective means of computing all eigenvalues of a real symmetric or complex hermitian matrix that are located in a given interval anywhere in the spectrum this paper describes a technique for tackling this problem by combining a thick restart version of the lanczos algorithm with deflation locking and a new type of polynomial filter obtained from a least squares technique the resulting algorithm can be utilized in a spectrum slicing approach whereby a very large number of eigenvalues and associated eigenvectors of the matrix are computed by extracting eigenpairs located in different subintervals independently from one another

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

asymptotic behavior of the gibbs sampler mild sufficient conditions for the ergodicity of the gibbs sampler are obtained the geometric ergodicity of the gibbs sampler is then discussed where the main tool is the drift criterion this criterion requires the construction of a generalized energy function so that the chain is dissipating energy at a suitable rate whenever it lies outside the center of the state space for the two variable case a useful recipe is provided for constructing appropriate energy functions these results are illustrated with four examples

topological characterisation of weakly compact operators let x be a banach space then there is a locally convex topology for x the right topology such that a linear map t from x into a banach space y is weakly compact precisely when t is a continuous map from x equipped with the right topology into y equipped with the norm topology when t is only sequentially continuous with respect to the right topology it is said to be pseudo weakly compact this notion is related to pelczynski s property v

central limit theorems for the shrinking target problem suppose b i b p r i are nested balls of radius r i about a point p in a dynamical system t x the question of whether t i x b i infinitely often i o for a e x is often called the shrinking target problem in many dynamical settings it has been shown that if e n sum i 1 n mu b i diverges then there is a quantitative rate of entry and lim n to infty frac 1 e n sum j 1 n 1 b i t i x to1 for a e x x this is a self norming type of strong law of large numbers we establish self norming central limit theorems clt of the form lim n to infty frac 1 a n sum i 1 n 1 b i t i x mu b i to n 0 1 in distribution for a variety of hyperbolic and non uniformly hyperbolic dynamical systems the normalization constants are a 2 n sim e sum i 1 n 1 b i t i x mu b i 2 dynamical systems to which our results apply include smooth expanding maps of the interval rychlik type maps gibbs markov maps rational maps and in higher dimensions piecewise expanding maps for such central limit theorems the main difficulty is to prove that the non stationary variance has a limit in probability

evaluating non analytic functions of matrices abstract the paper revisits the classical problem of evaluating f a for a real function f and a matrix a with real spectrum the evaluation is based on expanding f in chebyshev polynomials and the focus of the paper is to study the convergence rates of these expansions in particular we derive bounds on the convergence rates which reveal the relation between the smoothness of f and the diagonalizability of the matrix a we present several numerical examples to illustrate our analysis

covariance structure and convergence rate of the gibbs sampler with various scans this paper presents results on covariance structure and convergence for the gibbs sampler with both systematic and random scans it is shown that under conditions that guarantee the compactness of the markov forward operator and irreducibility of the corresponding chain the gibbs sampling scheme converges geometrically in terms of pearson 2 distance in particular for the random scan the autocovariance can be expressed as variances of iterative conditional expectations as a consequence the autocorrelations are all positive and decrease monotonically

solution of the k th eigenvalue problem in large scale electronic structure calculations abstract we consider computing the k th eigenvalue and its corresponding eigenvector of a generalized hermitian eigenvalue problem of n n large sparse matrices in electronic structure calculations several properties of materials such as those of optoelectronic device materials are governed by the eigenpair with a material specific index k we present a three stage algorithm for computing the k th eigenpair with validation of its index in the first stage of the algorithm we propose an efficient way of finding an interval containing the k th eigenvalue 1 k n with a non standard application of the lanczos method in the second stage spectral bisection for large scale problems is realized using a sparse direct linear solver to narrow down the interval of the k th eigenvalue in the third stage we switch to a modified shift and invert lanczos method to reduce bisection iterations and compute the k th eigenpair with validation numerical results with problem sizes up to 1 5 million are reported and the results demonstrate the accuracy and efficiency of the three stage algorithm

rao blackwellisation of sampling schemes summary this paper proposes a post simulation improvement for two common monte carlo methods the accept reject and metropolis algorithms the improvement is based on a rao blackwellisation method that integrates over the uniform random variables involved in the algorithms and thus post processes the standard estimators we show how the rao blackwellised versions of these algorithms can be implemented and through examples illustrate the improvement in variance brought by these new procedures we also compare the improved version of the metropolis algorithm with ordinary and rao blackwellised importance sampling procedures for independent and general metropolis set ups

covariance structure of the gibbs sampler with applications to the comparisons of estimators and augmentation schemes summary we study the covariance structure of a markov chain generated by the gibbs sampler with emphasis on data augmentation when applied to a bayesian missing data problem the gibbs sampler produces two natural approximations for the posterior distribution of the parameter vector the empirical distribution based on the sampled values of the parameter vector and a mixture of complete data posteriors we prove that rao blackwellization causes a one lag delay for the autocovariances among dependent samples obtained from data augmentation and consequently the mixture approximation produces estimates with smaller variances than the empirical approximation the covariance structure results are used to compare different augmentation schemes it is shown that collapsing and grouping random components in a gibbs sampler with two or three components usually result in more efficient sampling schemes

bayesian methods for hidden markov models markov chain monte carlo mcmc sampling strategies can be used to simulate hidden markov model hmm parameters from their posterior distribution given observed data some mcmc methods used in practice for computing likelihood conditional probabilities of hidden states and the most likely sequence of states can be improved by incorporating established recursive algorithms the most important of these is a set of forward backward recursions calculating conditional distributions of the hidden states given observed data and model parameters i show how to use the recursive algorithms in an mcmc context and demonstrate mathematical and empirical results showing a gibbs sampler using the forward backward recursions mixes more rapidly than another sampler often used for hmms iintroduce an augmented variables technique for obtaining unique state labels in hmms and finite mixture models i show how recursive computing allows the statistically efficient use of mcmc output when estimating the hidden states i

maximal inequalities for partial sums of rho mixing sequences a rosenthal type inequality for maximum partial sums of p mixing sequences is obtained applications to the complete convergence and almost sure summability of partial sums are also discussed

peskun s theorem and a modified discrete state gibbs sampler attention is drawn to the use of peskun s theorem in improving statistical efficiency of discretestate gibbs sampling