counting spanning trees in self similar networks by evaluating determinants spanning trees are relevant to various aspects of networks generally the number of spanning trees in a network can be obtained by computing a related determinant of the laplacian matrix of the network however for a large generic network evaluating the relevant determinant is computationally intractable in this paper we develop a fairly generic technique for computing determinants corresponding to self similar networks thereby providing a method to determine the numbers of spanning trees in networks exhibiting self similarity we describe the computation process with a family of networks called x y flowers which display rich behavior as observed in a large variety of real systems the enumeration of spanning trees is based on the relationship between the determinants of submatrices of the laplacian matrix corresponding to the x y flowers at different generations and is devoid of the direct laborious computation of determinants using the proposed method we derive analytically the exact number of spanning trees in the x y flowers on the basis of which we also obtain the entropies of the spanning trees in these networks moreover to illustrate the universality of our technique we apply it to some other self similar networks with distinct degree distributions and obtain explicit solutions to the numbers of spanning trees and their entropies finally we compare our results for networks with the same average degree but different structural properties such as degree distribution and fractal dimension and uncover the effect of these topological features on the number of spanning trees

objective bayes model selection in probit models we describe a new variable selection procedure for categorical responses where the candidate models are all probit regression models the procedure uses objective intrinsic priors for the model parameters which do not depend on tuning parameters and ranks the models for the different subsets of covariates according to their model posterior probabilities when the number of covariates is moderate or large the number of potential models can be very large and for those cases we derive a new stochastic search algorithm that explores the potential sets of models driven by their model posterior probabilities the algorithm allows the user to control the dimension of the candidate models and thus can handle situations when the number of covariates exceed the number of observations we assess through simulations the performance of the procedure and apply the vari able selector to a gene expression data set where the response is whether a patient exhibits pneumonia software needed to run the procedures is available in the r package varselectip copyright 2011 john wiley sons ltd categorical responses often appear in the analysis of the effect of a medical treatment which is applied to a collection of patients these patients are typically characterized by a set of p potential covariates and a first important statistical problem in the presence of covariates is to reduce the dimension of the model by retaining from the original p covariates those that have some influence in the observed responses thus we have a model selection problem in the class of 2 p models in this paper we describe a new variable selection procedure applicable to dichotomous responses that are modeled with a probit regression although under the 0 1 loss function an optimal solution is the model having the highest posterior probability the 0 1 loss function might not be realistic for many applications one potential shortcoming is that submodels different from the true one are assigned the same loss regardless of how far or close to the true one they are therefore it seems appropriate to not choose a single optimal model but a subset of them having posterior probabilities over a given threshold thus the output of our model selection procedure will be a ranking of the models according to their posterior probabilities the literature on variable selection in logistic or probit models is not large researchers have car ried out variable selection in logistic regression with lasso type procedures for instance meier et al 1 used a group lasso for logistic regression whereas kyung et al 2 used the latent variable probit model for bayesian lasso variable selection the bayesian approaches tend to dominate and researchers typically solve the main difficulties in its formulation the choice of the prior for the models involved and for the model parameters by using subjective priors swartz and shete 3 did a simulation study of the five types of variable selection in case control logistic regression and found that a bayesian stochastic search based on a spike and slab prior was the best performer chen and dey 4 also used a bayesian approach and were particularly concerned with correlated binary responses in multivariate logistic regression in their study the researchers developed subjective priors and selected models on the

the variable selection problem abstract the problem of variable selection is one of the most pervasive model selection problems in statistical applications often referred to as the problem of subset selection it arises when one wants to model the relationship between a variable of interest and a subset of potential explanatory variables or predictors but there is uncertainty about which subset to use this vignette reviews some of the key developments that have led to the wide variety of approaches for this problem

new method for counting the number of spanning trees in a two tree network abstract the number of spanning trees is an important quantity characterizing the reliability of a network generally the number of spanning trees in a network can be obtained by directly calculating a related determinant corresponding to the network however for a large network evaluating the relevant determinant is intractable in this paper we investigate the number of spanning trees in two tree networks we first give a new algorithm which avoids the laborious computation of the determinant for counting the number of spanning trees using the algorithm we can obtain the number of spanning trees of any two tree network in linear time the result shows that the computation complexity is o n which is better than that of the matrix tree theorem with o n 2 where n is the number of steps we then characterize two tree networks with the maximum and minimum numbers of spanning trees denote by p t and k t respectively the two tree networks of t 2 vertices with the maximum and minimum numbers of spanning trees denote by p a and e n respectively the two tree network of t 2 vertices generated by preferential attachment and by equiprobability attachment by algorithmic analysis and through simulations we conjecture that n s t k t n s t p a n s t e n n s t p t as t tends to infinity where n s t g is the number of spanning trees of g as an application of the algorithm we give the formula of the number of spanning trees of a particular small world network

the number and degree distribution of spanning trees in the tower of hanoi graph the number of spanning trees of a graph is an important invariant related to topological and dynamic properties of the graph such as its reliability communication aspects synchronization and so on however the practical enumeration of spanning trees and the study of their properties remain a challenge particularly for large networks in this paper we study the number and degree distribution of the spanning trees in the hanoi graph we first establish recursion relations between the number of spanning trees and other spanning subgraphs of the hanoi graph from which we find an exact analytical expression for the number of spanning trees of the n disc hanoi graph this result allows the calculation of the spanning tree entropy which is then compared with those for other graphs with the same average degree then we introduce a vertex labeling which allows to find for each vertex of the graph its degree distribution among all possible spanning trees

comparing and weighting imperfect models using d probabilities abstractwe propose a new approach for assigning weights to models using a divergence based method d probabilities relying on evaluating parametric models relative to a nonparametric bayesian ref

generalization of jeffreys divergence based priors for bayesian hypothesis testing summary we introduce objective proper prior distributions for hypothesis testing and model selection based on measures of divergence between the competing models we call them diver gence based db priors db priors have simple forms and desirable properties like information finite sample consistency and are often similar to other existing proposals like intrinsic priors moreover in normal linear model scenarios they reproduce the jeffreys zellner siow priors exactly most importantly in challenging scenarios such as irregular models and mixture mod els db priors are well defined and very reasonable whereas alternative proposals are not we derive approximations to the db priors as well as markov chain monte carlo and asymptotic expressions for the associated bayes factors

consistency of objective bayes factors as the model dimension grows in the class of normal regression models with a finite number of regressors and for a wide class of prior distributions a bayesian model selection procedure based on the bayes factor is consistent casella and moreno j amer statist assoc 104 2009 1261 1271 however in models where the number of parameters increases as the sample size increases properties of the bayes factor are not totally understood here we study consistency of the bayes factors for nested normal linear models when the number of regressors increases with the sample size we pay attention to two successful tools for model selection schwarz ann statist 6 1978 461 464 approximation to the bayes factor and the bayes factor for intrinsic priors berger and pericchi j amer statist assoc 91 1996 109 122 moreno bertolino and racugno j amer statist assoc 93 1998 1451 1460 we find that the the schwarz approximation and the bayes factor for intrinsic priors are consistent when the rate of growth of the dimension of the bigger model is o n b for b 1 when b 1 the schwarz approximation is always inconsistent under the alternative while the bayes factor for intrinsic priors is consistent except for a small set of alternative models which is characterized

prediction via orthogonalized model mixing abstract we introduce an approach and algorithms for model mixing in large prediction problems with correlated predictors we focus on the choice of predictors in linear models and mix over possible subsets of candidate predictors our approach is based on expressing the space of models in terms of an orthogonalization of the design matrix advantages are both statistical and computational statistically orthogonalization often leads to a reduction in the number of competing models by eliminating correlations computationally large model spaces cannot be enumerated recent approaches are based on sampling models with high posterior probability via markov chains based on orthogonalization of the space of candidate predictors we can approximate the posterior probabilities of models by products of predictor specific terms this leads to an importance sampling function for sampling directly from the joint distribution over the model space without resorting to markov chains compared to the latter ortho

comparison of objective bayes factors for variable selection in parametric regression models for survival analysis this paper considers the problem of selecting a set of regressors when the response variable is distributed according to a specified parametric model and observations are censored under a bayesian perspective the most widely used tools are bayes factors bfs which are undefined when improper priors are used in order to overcome this issue fractional fbf and intrinsic ibf bfs have become common tools for model selection both depend on the size nt of a minimal training sample mts while the ibf also depends on the specific mts used in the case of regression with censored data the definition of an mts is problematic because only uncensored data allow to turn the improper prior into a proper posterior and also because full exploration of the space of the mtss which includes also censored observations is needed to avoid bias in model selection to address this concern a sequential mts was proposed but it has the drawback of an increase of the number of possible mtss as nt becomes random for this reason we explore the behaviour of the fbf contextualizing its definition to censored data we show that these are consistent providing also the corresponding fractional prior finally a large simulation study and an application to real data are used to compare ibf fbf and the well known bayesian information criterion copyright 2014 john wiley sons ltd

tutte polynomial of scale free networks the tutte polynomial of a graph or equivalently the q state potts model partition function is a two variable polynomial graph invariant of considerable importance in both statistical physics and combinatorics the computation of this invariant for a graph is np hard in general in this paper we focus on two iteratively growing scale free networks which are ubiquitous in real life systems based on their self similar structures we mainly obtain recursive formulas for the tutte polynomials of two scale free networks lattices one is fractal and large world while the other is non fractal but possess the small world property furthermore we give some exact analytical expressions of the tutte polynomial for several special points at x y plane such as the number of spanning trees the number of acyclic orientations etc

empirical bayes selection of wavelet thresholds this paper explores a class of empirical bayes methods for level dependent threshold selection in wavelet shrinkage the prior considered for each wavelet coefficient is a mixture of an atom of probability at zero and a heavy tailed density the mixing weight or sparsity parameter for each level of the transform is chosen by marginal maximum likelihood if estimation is carried out using the posterior median this is a random thresholding procedure the estimation can also be carried out using other thresholding rules with the same threshold details of the calculations needed for implementing the procedure are included in practice the estimates are quick to compute and there is software available simulations on the standard model functions show excellent performance and applications to data drawn from various fields of application are used to explore the practical performance of the approach by using a general result on the risk of the corresponding marginal maximum likelihood approach for a single sequence overall bounds on the risk of the method are found subject to membership of the unknown function in one of a wide range of besov classes covering also the case of f of bounded variation the rates obtained are optimal for any value of the parameter p in 0 simultaneously for a wide range of loss functions each dominating the lq norm of the th derivative with 0 and 0 q 2 attention is paid to the distinction between sampling the unknown function within white noise and sampling at discrete points and between placing constraints on the function itself and on the discrete wavelet transform of its sequence of values at the observation points results for all relevant combinations of these scenarios are obtained in some cases a key feature of the theory is a particular boundary corrected wavelet basis details of which are discussed overall the approach described seems so far unique in combining the properties of fast computation good theoretical properties and good performance in simulations and in practice a key feature appears to be that the estimate of sparsity adapts to three different zones of estimation first where the signal is not sparse enough for thresholding to be of benefit second where an appropriately chosen threshold results in substantially improved estimation and third where the signal is so sparse that the zero estimate gives the optimum accuracy rate

criteria for bayesian model choice with application to variable selection in objective bayesian model selection no single criterion has emerged as dominant in defining objective prior distributions indeed many criteria have been separately proposed and utilized to propose differing prior choices we first formalize the most general and compelling of the various criteria that have been suggested together with a new criterion we then illustrate the potential of these criteria in determining objective model selection priors by considering their application to the problem of variable selection in normal linear models this results in a new model selection objective prior with a number of compelling properties

theoretical explanation of observed decreasing failure rate pooled data on the times of successive failures of the air conditioning system of a fleet of jet airplanes seemed to indicate that the life distribution had a decreasing failure rate more refined analysis showed that the failure distribution for each airplane separately was exponential but with a different failure rate using the theorem that a mixture of distributions each having a non increasing failure rate e g a mixture of exponential distributions itself has a non increasing failure rate the apparent decreasing failure rate of the pooled air conditioning life distribution was satisfactorily explained the present study has implications in other areas where an observed decreasing failure rate may well be the result of mixing exponential distributions having different parameters

a bayesian information criterion for singular models summary we consider approximate bayesian model choice for model selection problems that involve models whose fisher information matrices may fail to be invertible along other competing submodels such singular models do not obey the regularity conditions underlying the derivation of schwarz s bayesian information criterion bic and the penalty structure in bic generally does not reflect the frequentist large sample behaviour of the marginal likelihood although large sample theory for the marginal likelihood of singular models has been developed recently the resulting approximations depend on the true parameter value and lead to a paradox of circular reasoning guided by examples such as determining the number of components in mixture models the number of factors in latent factor models or the rank in reduced rank regression we propose a resolution to this paradox and give a practical extension of bic for singular model selection problems

objective bayesian comparison of constrained analysis of variance models in the social sciences we are often interested in comparing models specified by parametric equality or inequality constraints for instance when examining three group means mu 1 mu 2 mu 3 through an analysis of variance anova a model may specify that mu 1 mu 2 mu 3 while another one may state that mu 1 mu 3 mu 2 and finally a third model may instead suggest that all means are unrestricted this is a challenging problem because it involves a combination of nonnested models as well as nested models having the same dimension we adopt an objective bayesian approach requiring no prior specification from the user and derive the posterior probability of each model under consideration our method is based on the intrinsic prior methodology suitably modified to accommodate equality and inequality constraints focussing on normal anova models a comparative assessment is carried out through simulation studies we also present an application to real data collected in a psychological experiment

bayes model averaging with selection of regressors when a number of distinct models contend for use in prediction the choice of a single model can offer rather unstable predictions in regression stochastic search variable selection with bayesian model averaging offers a cure for this robustness issue but at the expense of requiring very many predictors here we look at bayes model averaging incorporating variable selection for prediction this offers similar mean square errors of prediction but with a vastly reduced predictor space this can greatly aid the interpretation of the model it also reduces the cost if measured variables have costs the development here uses decision theory in the context of the multivariate general linear model in passing this reduced predictor space bayes model averaging is contrasted with single model approximations a fast algorithm for updating regressions in the markov chain monte carlo searches for posterior inference is developed allowing many more variables than observations to be contemplated we discuss the merits of absolute rather than proportionate shrinkage in regression especially when there are more variables than observations the methodology is illustrated on a set of spectroscopic data used for measuring the amounts of different sugars in an aqueous solution

the intrinsic bayes factor for model selection and prediction abstract in the bayesian approach to model selection or hypothesis testing with models or hypotheses of differing dimensions it is typically not possible to utilize standard noninformative or default prior distributions this has led bayesians to use conventional proper prior distributions or crude approximations to bayes factors in this article we introduce a new criterion called the intrinsic bayes factor which is fully automatic in the sense of requiring only standard noninformative priors for its computation and yet seems to correspond to very reasonable actual bayes factors the criterion can be used for nested or nonnested models and for multiple model comparison and prediction from another perspective the development suggests a general definition of a reference prior for model comparison

model selection and the principle of minimum description length this article reviews the principle of minimum description length mdl for problems of model selection by viewing statistical modeling as a means of generating descriptions of observed data the mdl framework discriminates between competing models based on the complexity of each description this approach began with kolmogorov s theory of algorithmic complexity matured in the literature on information theory and has recently received renewed attention within the statistics community here we review both the practical and the theoretical aspects of mdl as a tool for model selection emphasizing the rich connections between information theory and statistics at the boundary between these two disciplines we find many interesting interpretations of popular frequentist and bayesian procedures as we show mdl provides an objective umbrella under which rather disparate approaches to statistical modeling can coexist and be compared we illustrate the mdl principle by considering problems in regression nonpar

potts model partition functions on two families of fractal lattices abstract the partition function of q state potts model or equivalently the tutte polynomial is computationally intractable for regular lattices the purpose of this paper is to compute partition functions of q state potts model on two families of fractal lattices based on their self similar structures and by applying the subgraph decomposition method we divide their tutte polynomials into two summands and for each summand we obtain a recursive formula involving the other summand as a result the number of spanning trees and their asymptotic growth constants and a lower bound of the number of connected spanning subgraphs or acyclic root connected orientations for each of such two lattices are obtained

bayesian variable selection in linear regression abstract this article is concerned with the selection of subsets of predictor variables in a linear regression model for the prediction of a dependent variable it is based on a bayesian approach intended to be as objective as possible a probability distribution is first assigned to the dependent variable through the specification of a family of prior distributions for the unknown parameters in the regression model the method is not fully bayesian however because the ultimate choice of prior distribution from this family is affected by the data it is assumed that the predictors represent distinct observables the corresponding regression coefficients are assigned independent prior distributions for each regression coefficient subject to deletion from the model the prior distribution is a mixture of a point mass at 0 and a diffuse uniform distribution elsewhere that is a spike and slab distribution the random error component is assigned a normal distribution with mean 0 and standard deviation

computing bayes factors by combining simulation and asymptotic approximations abstract the bayes factor is a ratio of two posterior normalizing constants which may be difficult to compute we compare several methods of estimating bayes factors when it is possible to simulate observations from the posterior distributions via markov chain monte carlo or other techniques the methods that we study are all easily applied without consideration of special features of the problem provided that each posterior distribution is well behaved in the sense of having a single dominant mode we consider a simulated version of laplace s method a simulated version of bartlett correction importance sampling and a reciprocal importance sampling technique we also introduce local volume corrections for each of these in addition we apply the bridge sampling method of meng and wong we find that a simulated version of laplace s method with local volume correction furnishes an accurate approximation that is especially useful when likelihood function evaluations are costly a simple bridge sampli

the number of spanning trees of an infinite family of outerplanar small world and self similar graphs abstract in this paper we give an exact analytical expression for the number of spanning trees of an infinite family of outerplanar small world and self similar graphs this number is an important graph invariant related to different topological and dynamic properties of the graph such as its reliability synchronization capability and diffusion properties the calculation of the number of spanning trees is a demanding and difficult task in particular for large graphs and thus there is much interest in obtaining closed expressions for relevant infinite graph families we have also calculated the spanning tree entropy of the graphs which we have compared with those for graphs with the same average degree

tests based on intrinsic priors for the equality of two correlated proportions correlated proportions arise in longitudinal panel studies a typical example is the opinion swing problem has the proportion of people favoring a politician changed after his recent speech to the nation on tv because the same group of individuals is interviewed before and after the speech the two proportions are correlated a natural null hypothesis to be tested is whether the corresponding population proportions are equal a standard bayesian approach to this problem has already been considered in the literature based on a dirichlet prior for the cell probabilities of the underlying 2 2 table under the alternative hypothesis together with an induced prior under the null with a lack of specific prior information a diffuse e g uniform distribution may be used we claim that this approach is not satisfactory because in a testing problem one should make sure that the prior under the alternative is adequately centered around the region specified by the null in order to obtain a fairer co

enumeration of spanning trees in planar unclustered networks abstract among a variety of subgraphs spanning trees are one of the most important and fundamental categories they are relevant to diverse aspects of networks including reliability transport self organized criticality loop erased random walks and so on in this paper we introduce a family of modular self similar planar networks with zero clustering relevant properties of this family are comparable to those networks associated with technological systems having low clustering like power grids some electronic circuits the internet and some biological systems so it is very significant to research on spanning trees of planar networks however for a large network evaluating the relevant determinant is intractable in this paper we propose a fairly generic linear algorithm for counting the number of spanning trees of a planar network using the algorithm we derive analytically the exact numbers of spanning trees in planar networks our result shows that the computational complexity is o t which is better than that of the matrix tree theorem with o m 2 t 2 where t is the number of steps and m is the girth of the planar network we also obtain the entropy for the spanning trees of a given planar network we find that the entropy of spanning trees in the studied network is small which is in sharp contrast to the previous result for planar networks with the same average degree we also determine an upper bound and a lower bound for the numbers of spanning trees in the family of planar networks by the algorithm as another application of the algorithm we give a formula for the number of spanning trees in an outerplanar network with small world features

nonparametric regression using bayesian variable selection abstract this paper estimates an additive model semiparametrically while automatically selecting the significant independent variables and the appropriate power transformation of the dependent variable the nonlinear variables are modeled as regression splines with significant knots selected from a large number of candidate knots the estimation is made robust by modeling the errors as a mixture of normals a bayesian approach is used to select the significant knots the power transformation and to identify outliers using the gibbs sampler to carry out the computation empirical evidence is given that the sampler works well on both simulated and real examples and that in the univariate case it compares favorably with a kernel weighted local linear smoother the variable selection algorithm in the paper is substantially faster than previous bayesian variable selection algorithms

pfaffian orientations and perfect matchings of scale free networks counting perfect matchings is of great interest in theoretical computer science mathematics among other disciplines however explicitly determining the number of perfect matchings of general graphs is a challenge in this paper we present a first analytical study of the enumeration problem of perfect matchings on scale free networks which are ubiquitous in real life systems we focus on two iteratively growing scale free networks one is fractal while the other is non fractal for each network we construct an orientation and prove that it is pfaffian then based on the connection between the number of perfect matchings and determinant of the skew adjacency matrix corresponding to a pfaffian orientation we derive the recursive expressions for the number of perfect matchings of the networks at two successive iterations furthermore using the recursive relations we show that for both scale free networks their entropy is zero which is in sharp contrast to the previously obtained results for other networks without scale free behavior but having the same average degree as the studied networks such as square lattice and honeycomb lattice whose entropies are larger than zero our results indicate from another angle that scale free networks have a distinct architecture in comparison with those lattices lacking the power law property

bayesian model comparison with the hyvarinen score computation and consistency the bayes factor is a widely used criterion in model comparison and its logarithm is a difference of out of sample predictive scores under the logarithmic scoring rule however when some of the ca

the deviance information criterion 12 years on summary the essentials of our paper of 2002 are briefly summarized and compared with other criteria for model comparison after some comments on the paper s reception and influence we consider criticisms and proposals forimprovement made by us and others

counting spanning trees of a type of generalized farey graphs abstract the farey graph f n is derived from the famous farey sequence and it is a small world network with a connectivity distribution decaying exponentially by using the matrix tree theorem zhang et al 2012 obtained the exact formula of the number of spanning trees of f n in this paper by using the electrical network method we consider a type of generalized farey graphs and give the exact solution for the number of spanning trees of these generalized farey graphs which generalizes some previous results about the farey graphs

an iteration method for computing the total number of spanning trees and its applications in graph theory abstract calculating and analyzing the number of spanning trees of graphs network models is an important and interesting research project in wide variety of fields such as mathematics physics theoretical computer science chemistry and so on the number of spanning trees of graphs models displays amounts of information on its structural features and also on some relevant dynamical properties in particular network security random walks and percolation in this paper firstly due to lots of graphs models are built on the basis of various simple and small elements components we provide primarily some helpful network operation such as link operation and merge operation to generate more realistic and complicated graphs models secondly considering reliability of fault tolerance to random faults and of intrusion tolerance to selectively remove attacks synchronization capability and diffusion properties of networks we present an iterative method algorithm for computing the total number of spanning trees as a pellucid example we apply our method to tree space and cycle space notice that it is proved to be indeed a better tool in order to reflect more widely practical meanings we study its applications in graph theory including ladder graph with zero clustering coefficient wheel graph having nonzero clustering coefficient as constituent ingredients of maximal planar graphs in the rest of this paper we make a brief summary that the method described by us can be designed a program algorithm for obtaining easily the exact number of spanning trees of some models

bayesian model averaging for linear regression models abstract we consider the problem of accounting for model uncertainty in linear regression models conditioning on a single selected model ignores model uncertainty and thus leads to the underestimation of uncertainty when making inferences about quantities of interest a bayesian solution to this problem involves averaging over all possible models i e combinations of predictors when making inferences about quantities of interest this approach is often not practical in this article we offer two alternative approaches first we describe an ad hoc procedure occam s window which indicates a small set of models over which a model average can be computed second we describe a markov chain monte carlo approach that directly approximates the exact solution in the presence of model uncertainty both of these model averaging procedures provide better predictive performance than any single model that might reasonably have been selected in the extreme case where there are many candidate predictors but

multiple bayes factors for testing hypotheses abstract partial and multiple bayes factors are introduced to obtain pairwise comparisons of hypotheses in a statistical experiment with a partition on the parameter space robust bayesian analyses are performed by introducing suitable classes of priors and by calculating lower and upper bounds of bayes factors and posterior probabilities classes of intuitively meaningful priors are introduced including unimodal densities without the constraint of symmetry for the case of precise hypotheses procedures for the corresponding optimizations are specified and examples are given

spanning trees in a fractal scale free lattice spanning trees provide crucial insight into the origin of fractality in fractal scale free networks in this paper we present the number of spanning trees in a particular fractal scale free lattice network we first study analytically the topological characteristics of the lattice and show that it is simultaneously scale free highly clustered large world fractal and disassortative any previous model does not have all the properties as the studied one then by using the renormalization group technique we derive analytically the number of spanning trees in the network under consideration based on which we also determine the entropy for the spanning trees of the network these results shed light on understanding the structural characteristics of and dynamical processes on scale free networks with fractality moreover our method and process for employing the decimation technique to enumerate spanning trees are general and can be easily extended to other deterministic media with self similarity

properties and applications of laplacian spectra for koch networks for a network knowledge of its laplacian eigenvalues is central to understanding its structure and dynamics in this paper we study the laplacian spectra of a family of koch networks with scale free and small world properties we derive some recursive relations between the laplacian characteristic polynomials of koch networks and their subgraphs at different iterations based on the obtained recurrence relations we determine explicitly the product of all nonzero laplacian eigenvalues as well as the sum of the reciprocals of these eigenvalues then using these results we further evaluate the number of spanning trees kirchhoff index global mean first passage time and average path length of the family of koch networks finally we determine the number of spanning forests under certain conditions we expect that our method can be adapted to other types of self similar networks

an objective bayesian analysis of a crossover design via model selection and model averaging inference about the treatment effect in a crossover design has received much attention over time owing to the uncertainty in the existence of the carryover effect and its impact on the estimation of the treatment effect adding to this uncertainty is that the existence of the carryover effect and its size may depend on the presence of the treatment effect and its size we consider estimation and testing hypothesis about the treatment effect in a two period crossover design assuming normally distributed response variable and use an objective bayesian approach to test the hypothesis about the treatment effect and to estimate its size when it exists while accounting for the uncertainty about the presence of the carryover effect as well as the treatment and period effects we evaluate and compare the performance of the proposed approach with a standard frequentist approach using simulated data and real data copyright 2016 john wiley sons ltd

inference from intrinsic bayes procedures under model selection and uncertainty in this article we present a fully coherent and consistent objective bayesian analysis of the linear regression model using intrinsic priors the intrinsic prior is a scaled mixture of g priors and promotes shrinkage toward the subspace defined by a base or null model while it has been established that the intrinsic prior provides consistent model selectors across a range of models the posterior distribution of the model parameters has not previously been investigated we prove that the posterior distribution of the model parameters is consistent under both model selection and model averaging when the number of regressors is fixed further we derive tractable expressions for the intrinsic posterior distribution as well as sampling algorithms for both a selected model and model averaging we compare the intrinsic prior to other mixtures of g priors and provide details on the consistency properties of modified versions of the zellner siow prior and hyper g priors supplementary materials for this arti

bayes variable selection in semiparametric linear models there is a rich literature on bayesian variable selection for parametric models our focus is on generalizing methods and asymptotic theory established for mixtures of g priors to semiparametric linear regression models having unknown residual densities using a dirichlet process location mixture for the residual density we propose a semiparametric g prior which incorporates an unknown matrix of cluster allocation indicators for this class of priors posterior computation can proceed via a straightforward stochastic search variable selection algorithm in addition bayes factor and variable selection consistency is shown to result under a class of proper priors on g even when the number of candidate predictors p is allowed to increase much faster than sample size n while making sparsity assumptions on the true model size

vertex operators for non simply laced algebras a vertex operator construction is given for the level one representations of the affine kac moody algebras associated with non simply laced finite dimensional lie algebras using free boson and interacting fermion fields the fermion fields are constructed explicitly and a detailed discussion is given of the theory of the cocycles necessary for this and other vertex operator constructions the construction is related in detail to the folding of dynkin diagrams and a generalisation of it yields freudenthal s magic square

bayesian inference for the common location parameter of several shifted exponential populations abstract consider k random samples which are independently drawn from k shifted exponential distributions with respective scale parameters 1 2 k and common location parameter on the basis of the given samples and in a bayesian framework we address the problem of point and interval estimation of the location parameter under the conjugate priors which are usually proper priors moreover we also address the problem of testing the equality of the location parameters we propose bayesian hypothesis testing procedures for the equality of the location parameters under the noninformative prior the noninformative prior is usually improper which yields a calibration problem that makes the bayes factor to be defined up to a multiplicative constant so we propose the default bayesian hypothesis testing procedures based on the fractional bayes factor and the intrinsic bayes factors under the reference priors our proposed bayesian procedures are compared and contrasted via a comparison study a simulation study and a real world data analysis to the existing classical exact procedures proposed by tippett tippett s method tippett 1931 fisher fisher s method fisher 1932 stouffer inverse normal method stouffer et al 1949 and george logit method george 1977 and to the generalized variable procedures proposed by tsui and weerahandi generalized p value method tsui and weerahandi 1989

enumeration of spanning trees on contact graphs of disk packings abstract obtaining the number of spanning trees of complex networks is an outstanding challenge since traditional approaches such as calculating the eigenvalues of the matrix and decomposing of spanning subgraphs are awkward or even infeasible for a large scale network the foundation and importance of this quantity relating to some topological and dynamic properties prompt us to explore the role of determinant identities for laplace matrices we introduce the basic electrically equivalent technique to determine an exact analytical expression for the quantity on the contact graph of disk packings which is proposed by zhang et al 2009 our theoretical results shed light on the relationship between the microscopic change of the quantity and topological iteration of the network in particular we compare the entropy of spanning trees on the network with the other two dimensional and three dimensional lattices we show that the new model is a small world scale free network with the maximum entropy so far found in addition our method for employing the electrically equivalent technique to enumerate spanning trees is general and can be easily extended to other complex networks

a reference bayesian test for nested hypotheses and its relationship to the schwarz criterion abstract to compute a bayes factor for testing h 0 0 in the presence of a nuisance parameter priors under the null and alternative hypotheses must be chosen as in bayesian estimation an important problem has been to define automatic or reference methods for determining priors based only on the structure of the model in this article we apply the heuristic device of taking the amount of information in the prior on equal to the amount of information in a single observation then after transforming to be null orthogonal to we take the marginal priors on to be equal under the null and alternative hypotheses doing so and taking the prior on to be normal we find that the log of the bayes factor may be approximated by the schwarz criterion with an error of order o p n rather than the usual error of order o p 1 this result suggests the schwarz criterion should provide sensible approximate solutions to bayesian testing problems at least when the hypotheses are nested when

q analogues of clifford and weyl algebras spinor and oscillator representations of quantum enveloping algebras we introduceq analogues of clifford and weyl algebras using these we construct spinor and oscillator representations of quantum enveloping algebras of typean 1 bn cn dn andan 1 1 also we discuss the irreducibility and the unitarity of these representations

reconciling bayesian and frequentist evidence in the one sided testing problem abstract for the one sided hypothesis testing problem it is shown that it is possible to reconcile bayesian evidence against h 0 expressed in terms of the posterior probability that h 0 is true with frequentist evidence against h 0 expressed in terms of the p value in fact for many classes of prior distributions it is shown that the infimum of the bayesian posterior probability of h 0 is equal to the p value in other cases the infimum is less than the p value the results are in contrast to recent work of berger and sellke 1987 in the two sided point null case where it was found that the p value is much smaller than the bayesian infimum some comments on the point null problem are also given

testing a point null hypothesis the irreconcilability of p values and evidence abstract the problem of testing a point null hypothesis or a small interval null hypothesis is considered of interest is the relationship between the p value or observed significance level and conditional and bayesian measures of evidence against the null hypothesis although one might presume that a small p value indicates the presence of strong evidence against the null such is not necessarily the case expanding on earlier work especially edwards lindman and savage 1963 and dickey 1977 it is shown that actual evidence against a null as measured say by posterior probability or comparative likelihood can differ by an order of magnitude from the p value for instance data that yield a p value of 05 when testing a normal mean result in a posterior probability of the null of at least 30 for any objective prior distribution objective here means that equal prior weight is given the two hypotheses and that the prior is symmetric and nonincreasing away from the null other definiti

optimal treatments in cost effectiveness analysis in the presence of covariates improving patient subgroup definition abstract in the presence of covariates the cost effectiveness analysis of medical treatments shows that the optimal treatment varies across the patient population subgroups and hence to accurately define the subgroups is a crucial step in the analysis a patient subgroup definition using only influential covariates within the potential set of patients covariates established by the expert has recently been proposed and the influential covariates were chosen from the univariate distributions of the effectiveness and the cost conditional on the effectiveness in this paper we argue that the bayesian variable selection procedure should be developed using the bivariate distribution of the cost and the effectiveness which is not the usual practice this new approach provides results with wider applicability and more understandable without a significative increase in the complexity of the procedure for real and simulated data sets optimal treatments for subgroups are found and compared with that from previous methods

calibration and empirical bayes variable selection for the problem of variable selection for the normal linear model selection criteria such as aic c p bic and ric have fixed dimensionality penalties such criteria are shown to correspond to selection of maximum posterior models under implicit hyperparameter choices for a particular hierarchical bayes formulation based on this calibration we propose empirical bayes selection criteria that use hyperparameter estimates instead of fixed choices for obtaining these estimates both marginal and conditional maximum likelihood methods are considered as opposed to traditional fixed penalty criteria these empirical bayes criteria have dimensionality penalties that depend on the data their performance is seen to approximate adaptively the performance of the best fixed penalty criterion across a variety of orthogonal and nonorthogonal set ups including wavelet regression empirical bayes shrinkage estimators of the selected coefficients are also proposed

objective bayesian meta analysis for sparse discrete data this paper presents a bayesian model for meta analysis of sparse discrete binomial data which are out of the scope of the usual hierarchical normal random effect models treatment effectiveness data are often of this type the crucial linking distribution between the effectiveness conditional on the healthcare center and the unconditional effectiveness is constructed from specific bivariate classes of distributions with given marginals this assures coherency between the marginal and conditional prior distributions utilized in the analysis further we impose a bivariate class of priors that is able to accommodate a wide range of heterogeneity degrees between the multicenter clinical trials involved applications to real multicenter data are given and compared with previous meta analysis copyright 2014 john wiley sons ltd

mixtures of g priors for bayesian variable selection zellner s g prior remains a popular conventional prior for use in bayesian variable selection despite several undesirable consistency issues in this article we study mixtures of g priors as an alternative to default g priors that resolve many of the problems with the original formulation while maintaining the computational tractability that has made the g prior so popular we present theoretical properties of the mixture g priors and provide real and simulated examples to compare the mixture formulation with fixed g priors empirical bayes approaches and other default procedures please see arnold zellner s letter and the author s response

objective bayesian variable selection a novel fully automatic bayesian procedure for variable selection in normal regression models is proposed the procedure uses the posterior probabilities of the models to drive a stochastic search the posterior probabilities are computed using intrinsic priors which can be considered default priors for model selection problems that is they are derived from the model structure and are free from tuning parameters thus they can be seen as objective priors for variable selection the stochastic search is based on a metropolis hastings algorithm with a stationary distribution proportional to the model posterior probabilities the procedure is illustrated on both simulated and real examples

the risk inflation criterion for multiple regression a new criterion is proposed for the evaluation of variable selection procedures in multiple regression this criterion which we call the risk inflation is based on an adjustment to the risk essentially the risk inflation is the maximum increase in risk due to selecting rather than knowing the correct predictors a new variable selection procedure is obtained which in the case of orthogonal predictors substantially improves on aic c p and bic and is close to optimal in contrast to aic c p and bic which use dimensionality penalties of 2 2 and log n respectively this new procedure uses a penalty 2 log p where p is the number of available predictors for the case of nonorthogonal predictors bounds for the optimal penalty are obtained

testing hardy weinberg equilibrium an objective bayesian analysis we analyze the general multiallelic hardy weinberg equilibrium problem from an objective bayesian testing standpoint we argue that for small or moderate sample sizes the answer is rather sensitive to the prior chosen and this suggests to carry out a sensitivity analysis with respect to the prior this goal is achieved through the identification of a class of priors specifically designed for this testing problem in this paper we consider the class of intrinsic priors under the full model indexed by a tuning quantity the training sample size these priors are objective satisfy savage s continuity condition and have proved to behave extremely well for many statistical testing problems we compute the posterior probability of the hardy weinberg equilibrium model for the class of intrinsic priors assess robustness over the range of plausible answers as well as stability of the decision in favor of either hypothesis copyright 2010 john wiley sons ltd

fractional bayes factors for model comparison bayesian comparison of models is achieved simply by calculation of posterior probabilities of the models themselves however there are difficulties with this approach when prior information about the parameters of the various models is weak partial bayes factors offer a resoiution of the problem by setting aside part of the data as a training sampie the training sampie is used to obtain an initiai informative posterior distribution of the parameters in each model model comparison is then based on a bayes factor calculated from the remaining data properties of partial bayes factors are discussed particularly in the context of weak prior information and they are found to have advantages over other proposed methods of model comparison a new variant of the partial bayes factor the fractional bayes factor is advocated on grounds of consistency simplicity robustness and coherence

on the use of non local prior densities in bayesian hypothesis tests we examine philosophical problems and sampling deficiencies that are associated with current bayesian hypothesis testing methodology paying particular attention to objective bayes methodology because the prior densities that are used to define alternative hypotheses in many bayesian tests assign non negligible probability to regions of the parameter space that are consistent with null hypotheses resulting tests provide exponential accumulation of evidence in favour of true alternative hypotheses but only sublinear accumulation of evidence in favour of true null hypotheses thus it is often impossible for such tests to provide strong evidence in favour of a true null hypothesis even when moderately large sample sizes have been obtained we review asymptotic convergence rates of bayes factors in testing precise null hypotheses and propose two new classes of prior densities that ameliorate the imbalance in convergence rates that is inherited by most bayesian tests using members of these classes we obtain analytic expressions for bayes factors in linear models and derive approximations to bayes factors in large sample settings

lower bounds on bayes factors for multinomial distributions with application to chi squared tests of fit lower bounds on bayes factors in favor of the null hypothesis in multinomial tests of point null hypotheses are developed these are then applied to derive lower bounds on bayes factors in both exact and asymptotic chi squared testing situations the general conclusion is that the lower bounds tend to be substantially larger than p values raising serious questions concerning the routine use of moderately small p values to represent significant evidence against the null hypothesis

training samples in objective bayesian model selection central to several objective approaches to bayesian model selection is the use of training samples subsets of the data so as to allow utilization of improper objective priors the most common prescription for choosing training samples is to choose them to be as small as possible subject to yielding proper posteriors these are called minimal training samples when data can vary widely in terms of either information content or impact on the improper priors use of minimal training samples can be inadequate important examples include certain cases of discrete data the presence of censored observations and certain situations involving linear models and explanatory variables such situations require more sophisticated methods of choosing training samples a variety of such methods are developed in this paper and successfully applied in challenging situations

extending conventional priors for testing general hypotheses in linear models summary we consider that observations come from a general normal linear model and that it is desirable to test a simplifying null hypothesis about the parameters we approach this problem from an objective bayesian model selection perspective crucial ingredients for this approach are proper objective priors to be used for deriving the bayes factors jeffreys zellner siow priors have good properties for testing null hypotheses defined by specific values of the parameters in full rank linear models we extend these priors to deal with general hypotheses in general linear models not necessarily of full rank the resulting priors which we call conventional priors are expressed as a generalization of recently introduced partially informative distributions the corresponding bayes factors are fully automatic easily computed and very reasonable the methodology is illustrated for the change point problem and the equality of treatments effects problem we compare the conventional priors derived for these problems with other objective bayesian proposals like the intrinsic priors it is concluded that both priors behave similarly although interesting subtle differences arise we adapt the conventional priors to deal with nonnested model selection as well as multiple model comparison finally we briefly address a generalization of conventional priors to nonnormal scenarios

finding the observed information matrix when using the em algorithm a procedure is derived for extracting the observed information matrix when the em algorithm is used to find maximum likelihood estimates in incomplete data problems the technique requires computation of a complete data gradient vector or second derivative matrix but not those associated with the incomplete data likelihood in addition a method useful in speeding up the convergence of the em algorithm is developed two examples are presented

an intrinsic limiting procedure for model selection and hypotheses testing abstract improper priors typically arise in default bayesian estimation problems in the bayesian approach to model selection or hypothesis testing the main tool is the bayes factor when improper priors for the parameters appearing in the models are used the bayes factor is not well defined the intrinsic bayes factor introduced by berger and pericchi is an interesting method for overcoming that difficulty that method is of particular interest as a means for generating proper prior distributions intrinsic priors for model comparison from the improper priors typically used in estimation the goal of this article is to develop a limiting procedure that provides a solid justification for the use of bayes factor with intrinsic priors the procedure is formalized and discussed for nested and nonnested models illustrations and comparisons with other approximations to bayes factors such as the bayesian information criterion of schwarz and the fractional bayes factor of o hagan are provided

eigenvalue spectrum of transition matrix of dual sierpinski gaskets and its applications the eigenvalue spectrum of the transition matrix of a network contains much information about its structural properties and is related to the behavior of various dynamical processes performed on it in this paper we study the eigenvalues of the transition matrix of the dual sierpinski gaskets embedded in d dimensional euclidean spaces we obtain all the eigenvalues as well as their corresponding degeneracies by making use of the spectral decimation technique we then apply the obtained eigenvalues to determine the exact number of spanning trees in the studied fractals and derive an explicit formula of the eigentime identity for random walks taking place on them

accurate approximations for posterior moments and marginal densities abstract this article describes approximations to the posterior means and variances of positive functions of a real or vector valued parameter and to the marginal posterior densities of arbitrary i e not necessarily positive parameters these approximations can also be used to compute approximate predictive densities to apply the proposed method one only needs to be able to maximize slightly modified likelihood functions and to evaluate the observed information at the maxima nevertheless the resulting approximations are generally as accurate and in some cases more accurate than approximations based on third order expansions of the likelihood and requiring the evaluation of third derivatives the approximate marginal posterior densities behave very much like saddle point approximations for sampling distributions the principal regularity condition required is that the likelihood times prior be unimodal

mixtures of g priors in generalized linear models abstractmixtures of zellner s g priors have been studied extensively in linear models and have been shown to have numerous desirable properties for bayesian variable selection and model averaging

laplace approximations for hypergeometric functions with matrix argument in this paper we present laplace approximations for two functions of matrix argument the type i confluent hypergeometric function and the gauss hypergeometric function both of these functions play an important role in distribution theory in multivariate analysis but from a practical point of view they have proved challenging and they have acquired a reputation for being difficult to approximate appealing features of the approximations we present are i they are fully explicit and simple to evaluate in practice and ii typically they have excellent numerical accuracy the excellent numerical accuracy is demonstrated in the calculation of noncentral moments of wilks a and the likelihood ratio statistic for testing block independence and in the calculation of the cdf of the noncentral distribution of wilks a via a sequential saddlepoint approximation relative error properties of these approximations are also studied and it is noted that the approximations have uniformly bounded relative errors in important cases

consistency of bayesian procedures for variable selection it has long been known that for the comparison of pairwise nested models a decision based on the bayes factor produces a consistent model selector in the frequentist sense here we go beyond the usual consistency for nested pairwise models and show that for a wide class of prior distributions including intrinsic priors the corresponding bayesian procedure for variable selection in normal regression is consistent in the entire class of normal linear models we find that the asymptotics of the bayes factors for intrinsic priors are equivalent to those of the schwarz bic criterion also recall that the jeffreys lindley paradox refers to the well known fact that a point null hypothesis on the normal mean parameter is always accepted when the variance of the conjugate prior goes to infinity this implies that some limiting forms of proper prior distributions are not necessarily suitable for testing problems intrinsic priors are limits of proper prior distributions and for finite sample sizes they have been proved to behave extremely well for variable selection in regression a consequence of our results is that for intrinsic priors lindley s paradox does not arise

assessing robustness of intrinsic tests of independence in two way contingency tables for testing nested hypotheses from a bayesian standpoint a desirable condition is that the prior for the alternative model concentrates mass around the smaller or null model for testing independence in contingency tables the intrinsic priors satisfy this requirement furthermore the degree of concentration of the priors is controlled by a discrete parameter t the training sample size which plays an important role in the resulting answer in this article we report on the robustness of the tests of independence for small or moderate sample sizes in contingency tables with respect to intrinsic priors with different degrees of concentration around the null we compare these tests to frequentist tests and other robust bayes tests for large sample sizes robustness is achieved because the intrinsic bayesian tests are consistent examples using real and simulated data are given supplemental materials technical details and data sets are available online

counting spanning trees in a small world farey graph the problem of spanning trees is closely related to various interesting problems in the area of statistical physics but determining the number of spanning trees in general networks is computationally intractable in this paper we perform a study on the enumeration of spanning trees in a specific small world network with an exponential distribution of vertex degrees which is called a farey graph since it is associated with the famous farey sequence according to the particular network structure we provide some recursive relations governing the laplacian characteristic polynomials of a farey graph and its subgraphs then making use of these relations obtained here we derive the exact number of spanning trees in the farey graph as well as an approximate numerical solution for the asymptotic growth constant characterizing the network finally we compare our results with those of different types of networks previously investigated

optimal predictive model selection often the goal of model selection is to choose a model for future prediction and it is natural to measure the accuracy of a future prediction by squared error loss under the bayesian approach it is commonly perceived that the optimal predictive model is the model with highest posterior probability but this is not necessarily the case in this paper we show that for selection among normal linear models the optimal predictive model is often the median probability model which is defined as the model consisting of those variables which have overall posterior probability greater than or equal to 1 2 of being in a model the median probability model often differs from the highest probability model

non local priors for high dimensional estimation abstractjointly achieving parsimony and good predictive power in high dimensions is a main challenge in statistics nonlocal priors nlps possess appealing properties for model choice but their use for estimation has not been studied in detail we show that for regular models nlp based bayesian model averaging bma shrink spurious parameters either at fast polynomial or quasi exponential rates as the sample size n increases while nonspurious parameter estimates are not shrunk we extend some results to linear models with dimension p growing with n coupled with our theoretical investigations we outline the constructive representation of nlps as mixtures of truncated distributions that enables simple posterior sampling and extending nlps beyond previous proposals our results show notable high dimensional estimation for linear models with p n at low computational cost nlps provided lower estimation error than benchmark and hyper g priors scad and lasso in simulations and in gene expression data

variable selection via gibbs sampling abstract a crucial problem in building a multiple regression model is the selection of predictors to include the main thrust of this article is to propose and develop a procedure that uses probabilistic considerations for selecting promising subsets this procedure entails embedding the regression setup in a hierarchical normal mixture model where latent variables are used to identify subset choices in this framework the promising subsets of predictors can be identified as those with higher posterior probability the computational burden is then alleviated by using the gibbs sampler to indirectly sample from this multinomial posterior distribution on the set of possible subset choices those subsets with higher probability the promising ones can then be identified by their more frequent appearance in the gibbs sample

on bayes factors for the linear model summarywe show that the bayes factor for testing whether a subset of coefficients are zero in the normal linear regression model gives the uniformly most powerful test amongst the class of invariant tests discussed in lehmann romano 2005 if the prior distributions for the regression coefficients are in a specific class of distributions the priors in this class can have any elliptical distribution with a specific scale matrix for the subset of coefficients that are being tested we also show under mild conditions that the bayes factor gives the uniformly most powerful invariant test only if the prior for the coefficients being tested is an elliptical distribution with this scale matrix the implications are discussed

on sampling strategies in bayesian variable selection problems with large model spaces one important aspect of bayesian model selection is how to deal with huge model spaces since the exhaustive enumeration of all the models entertained is not feasible and inferences have to be based on the very small proportion of models visited this is the case for the variable selection problem with a moderately large number of possible explanatory variables considered in this article we review some of the strategies proposed in the literature from a theoretical point of view using arguments of sampling theory and in practical terms using several examples with a known answer all our results seem to indicate that sampling methods with frequency based estimators outperform searching methods with renormalized estimators supplementary materials for this article are available online

bayesian subset modeling for high dimensional generalized linear models this article presents a new prior setting for high dimensional generalized linear models which leads to a bayesian subset regression bsr with the maximum a posteriori model approximately equivalent to the minimum extended bayesian information criterion model the consistency of the resulting posterior is established under mild conditions further a variable screening procedure is proposed based on the marginal inclusion probability which shares the same properties of sure screening and consistency with the existing sure independence screening sis and iterative sure independence screening isis procedures however since the proposed procedure makes use of joint information from all predictors it generally outperforms sis and isis in real applications this article also makes extensive comparisons of bsr with the popular penalized likelihood methods including lasso elastic net sis and isis the numerical results indicate that bsr can generally outperform the penalized likelihood methods the m

some new spinor representations of quantum groups bq n cq n gq 2 using the known realizations of the quantum algebras aq n dq n in terms of quantum clifford algebra and folding procedure of lie algebras we construct some new spinor representations of bq n cq n gq 2

an alternative to the standard bayesian procedure for discrimination between normal linear models summary we consider the standard bayesian procedure for discrimination focusing on its tendency to give low posterior probabilities to some correct models even in conditions of moderate to large sampling sizes furthermore in some situations that could well occur in practice the standard procedure is even asymptotically inconsistent we claim that the explanation is that the standard procedure inflates the posterior probability of the model that has the smallest expected increase in information about its parameters we then propose to modify it by an amount related to the expected gain in information and show that the lindley paradox does not occur finally we obtain the limiting alternative criterion that results by letting the prior densities become noninformative which enables us to measure how sensitive the discrimation is to the informative prior densities used

flexible empirical bayes estimation for wavelets summary wavelet shrinkage estimation is an increasingly popular method for signal denoising and compression although bayes estimators can provide excellent mean squared error mse properties the selection of an effective prior is a difficult task to address this problem we propose empirical bayes eb prior selection methods for various error distributions including the normal and the heavier tailed student t distributions under such eb prior distributions we obtain threshold shrinkage estimators based on model selection and multiple shrinkage estimators based on model averaging these eb estimators are seen to be computationally competitive with standard classical thresholding methods and to be robust to outliers in both the data and wavelet domains simulated and real examples are used to illustrate the flexibility and improved mse performance of these methods in a wide variety of settings

annealing markov chain monte carlo with applications to ancestral inference abstract markov chain monte carlo mcmc the metropolis hastings algorithm has been used for many statistical problems including bayesian inference likelihood inference and tests of significance though the method generally works well doubts about convergence often remain here we propose mcmc methods distantly related to simulated annealing our samplers mix rapidly enough to be usable for problems in which other methods would require eons of computing time they simulate realizations from a sequence of distributions allowing the distribution being simulated to vary randomly over time if the sequence of distributions is well chosen then the sampler will mix well and produce accurate answers for all the distributions even when there is only one distribution of interest these annealing like samplers may be the only known way to get a rapidly mixing sampler these methods are essential for attacking very hard problems which arise in areas such as statistical genetics we illustrate the methods wi