an l2 quotient algorithm for finitely presented groups abstract the paper develops algorithmic methods to enumerate all normal subgroups of a finitely presented group such that the factor groups are either isomorphic to psl 2 p n or to pgl 2 p n the case of two generators is treated in detail a range of examples starting with the free group on two generators and ending with groups having only finitely many normal subgroups of this type is discussed

a general framework for updating belief distributions we propose a framework for general bayesian inference we argue that a valid update of a prior belief distribution to a posterior can be made for parameters which are connected to observations through a loss function rather than the traditional likelihood function which is recovered as a special case modern application areas make it increasingly challenging for bayesians to attempt to model the true data generating mechanism for instance when the object of interest is low dimensional such as a mean or median it is cumbersome to have to achieve this via a complete model for the whole data distribution more importantly there are settings where the parameter of interest does not directly index a family of density functions and thus the bayesian approach to learning about such parameters is currently regarded as problematic our framework uses loss functions to connect information in the data to functionals of interest the updating of beliefs then follows from a decision theoretic approach involving cumulative loss functions importantly the procedure coincides with bayesian updating when a true likelihood is known yet provides coherent subjective inference in much more general settings connections to other inference frameworks are highlighted

on bayesian consistency we consider a sequence of posterior distributions based on a data dependent prior which we shall refer to as a pseudoposterior distribution and establish simple conditions under which the sequence is hellinger consistent it is shown how investigations into these pseudo posteriors assist with the understanding of some true posterior distributions including polya trees the infinite dimensional exponential family and mixture models

gibbs posterior for variable selection in high dimensional classification and data mining in the popular approach of bayesian variable selection bvs one uses prior and posterior distributions to select a subset of candidate variables to enter the model a completely new direction will be considered here to study bvs with a gibbs posterior originating in statistical mechanics the gibbs posterior is constructed from a risk function of practical interest such as the classification error and aims at minimizing a risk function without modeling the data probabilistically this can improve the performance over the usual bayesian approach which depends on a probability model which may be misspecified conditions will be provided to achieve good risk performance even in the presence of high dimensionality when the number of candidate variables k can be much larger than the sample size n in addition we develop a convenient markov chain monte carlo algorithm to implement bvs with the gibbs posterior

interpreting statistical evidence by using imperfect models robust adjusted likelihood functions summary the strength of statistical evidence is measured by the likelihood ratio two key performance properties of this measure are the probability of observing strong misleading evidence and the probability of observing weak evidence for the likelihood function associated with a parametric statistical model these probabilities have a simple large sample structure when the model is correct here we examine how that structure changes when the model fails this leads to criteria for determining whether a given likelihood function is robust continuing to perform satisfactorily when the model fails and to a simple technique for adjusting both likelihoods and profile likelihoods to make them robust we prove that the expected information in the robust adjusted likelihood cannot exceed the expected information in the likelihood function from a true model we note that the robust adjusted likelihood is asymptotically fully efficient when the working model is correct and we show that in some important examples this efficiency is retained even when the working model fails in such cases the bayes posterior probability distribution based on the adjusted likelihood is robust remaining correct asymptotically even when the model for the observable random variable does not include the true distribution finally we note a link to standard frequentist methodology in large samples the adjusted likelihood functions provide robust likelihood based confidence intervals

on some questions about a family of cyclically presented groups abstract we study various questions about the generalised fibonacci groups a family of cyclically presented groups which includes as special cases the fibonacci sieradski and gilbert howie groups

generalizations of cauchy s summation theorem for schur functions cauchy s summation theorem for schur functions is generalized and a number of related results are given the result is applied to a com binatorial problem involving products of pairs of permuations by appeal to properties of the group algebra of the symmetric group in 3 this summation theorem is used to derive a generalization of gessel s 3 result on counting 3 rowed latin rectangles with respect to the number of cycles in each row considered as permuations we also derive proposition 3 2 an explicit expression for a useful character sum associated with the symmetric group 4 contains a number of results for symmetric functions that are equivalent to theorem 2 1 these include a generalization of cauchy s summation theorem for

connection coefficients matchings maps and combinatorial conjectures for jack symmetric functions a power series is introduced that is an extension to three sets of variables of the cauchy sum for jack symmetric functions in the jack parameter ae we conjecture that the coefficients of this series with respect to the power sum basis are nonnegative integer polynomials in b the jack parameter shifted by 1 more strongly we make the matchings jack conjecture that the coefficients are counting series in b for matchings with respect to a parameter of nonbipartiteness evidence is presented for these conjectures and they are proved for two infinite families the coefficients of a second series essentially the logarithm of the first specialize at values 1 and 2 of the jack parameter to the numbers of hypermaps in orientable and locally orientable surfaces respectively we conjecture that these coefficients are also nonnegative integer polynomials in b and we make the hypermap jack conjecture that the coefficients are counting series in b for hypermaps in locally orientable surfaces with respect to a parameter of nonorientability

families of group presentations related to topology we study some algebraic properties of a class of group presentations depending on a finite number of integer parameters this class contains many well known groups which are interesting from a topological point of view we find arithmetic conditions on the parameters under which the considered groups cannot be fundamental groups of hyperbolic 3 manifolds of finite volume then we investigate the asphericity for many presentations contained in our family

from ɛ entropy to kl entropy analysis of minimum information complexity density estimation we consider an extension of e entropy to a kl divergence based complexity measure for randomized density estimation methods based on this extension we develop a general information theoretical inequality that measures the statistical complexity of some deterministic and randomized density estimators consequences of the new inequality will be presented in particular we show that this technique can lead to improvements of some classical results concerning the convergence of minimum description length and bayesian posterior distributions moreover we are able to derive clean finite sample convergence bounds that are not obtainable using previous approaches

universal prediction this paper consists of an overview on universal prediction from an information theoretic perspective special attention is given to the notion of probability assignment under the self information loss function which is directly related to the theory of universal data compression both the probabilistic setting and the deterministic setting of the universal prediction problem are described with emphasis on the analogy and the differences between results in the two settings

pac bayesian analysis of co clustering and beyond we derive pac bayesian generalization bounds for supervised and unsupervised learning models based on clustering such as co clustering matrix tri factorization graphical models graph clustering and pairwise clustering we begin with the analysis of co clustering which is a widely used approach to the analysis of data matrices we distinguish among two tasks in matrix data analysis discriminative prediction of the missing entries in data matrices and estimation of the joint probability distribution of row and column variables in co occurrence matrices we derive pac bayesian generalization bounds for the expected out of sample performance of co clustering based solutions for these two tasks the analysis yields regularization terms that were absent in the previous formulations of co clustering the bounds suggest that the expected performance of co clustering is governed by a trade off between its empirical performance and the mutual information preserved by the cluster variables on row and column ids we derive an iterative projection algorithm for finding a local optimum of this trade off for discriminative prediction tasks this algorithm achieved state of the art performance in the movielens collaborative filtering task our co clustering model can also be seen as matrix tri factorization and the results provide generalization bounds regularization terms and new algorithms for this form of matrix factorization the analysis of co clustering is extended to tree shaped graphical models which can be used to analyze high dimensional tensors according to the bounds the generalization abilities of tree shaped graphical models depend on a trade off between their empirical data fit and the mutual information that is propagated up the tree levels we also formulate weighted graph clustering as a prediction problem given a subset of edge weights we analyze the ability of graph clustering to predict the remaining edge weights the analysis of co clustering easily extends to this problem and suggests that graph clustering should optimize the trade off between empirical data fit and the mutual information that clusters preserve on graph nodes

counting cycles in permutations by group characters with an application to a topological problem the character theory of the symmetric group is used to derive properties of the number of permutations with k cycles which are expressible as the product of a full cycle with an element of an arbitrary but fixed conjugacy class for the conjugacy class of fixed point free involutions this problem has application to the analysis of singularities in surfaces 1 introduction for nonnegative integers k and n and a partition p of n let ef denote the number of permutations w on n symbols such that it has exactly k cycles and such that it can be expressed as a product of an arbitrary but fixed cycle of length n and a permutation in the conjugacy class indexed by p the purpose of this paper is to derive the generating function for these numbers and to obtain some of their properties the method makes direct use of combinatorial and algebraic properties of the group algebra of the symmetric group a special case of this problem is of particular interest let e p n denote the number e when p indexes the conjugacy class of permutations on pn symbols with n cycles of length p the matter of calculating ek2 n arose in connection with work by harris and morrison 4 on singularities in surfaces it has also occurred indirectly in the work of gross 2 on graph embeddings harer and zagier 3 have shown by an independent method that the sequence e k2 n for k n 1 satisfies a three term linear recurrence equation with coefficients which are polynomials in n to fix ideas note that for n 2 the permissible permutations are

variable selection for cox s proportional hazards model and frailty model a class of variable selection procedures for parametric models via nonconcave penalized likelihood was proposed in fan and li 2001a it has been shown there that the resulting procedures perform as well as if the subset of significant variables were known in advance such a property is called an oracle property the proposed procedures were illustrated in the context of linear regression robust linear regression and generalized linear models in this paper the nonconcave penalized likelihood approach is extended further to the cox proportional hazards model and the cox proportional hazards frailty model two commonly used semi parametric models in survival analysis as a result new variable selection procedures for these two commonly used models are proposed it is demonstrated how the rates of convergence depend on the regularization parameter in the penalty function further with a proper choice of the regularization parameter and the penalty function the proposed estimators possess an oracle property standard error formulae are derived and their accuracies are empirically tested simulation studies show that the proposed procedures are more stable in prediction and more effective in computation than the best subset variable selection and they reduce model complexity as effectively as the best subset variable selection compared with the lasso which is the penalized likelihood method with the l 1 penalty proposed by tibshirani the newly proposed approaches have better theoretic properties and finite sample performance

misspecification in infinite dimensional bayesian statistics we consider the asymptotic behavior of posterior distributions if the model is misspecified given a prior distribution and a random sample from a distribution p 0 which may not be in the support of the prior we show that the posterior concentrates its mass near the points in the support of the prior that minimize the kullback leibler divergence with respect to p 0 an entropy condition and a prior mass condition determine the rate of convergence the method is applied to several examples with special interest for infinite dimensional models these include gaussian mixtures nonparametric regression and parametric models

the lasso method for variable selection in the cox model summary i propose a new method for variable selection and shrinkage in cox s proportional hazards model my proposal minimizes the log partial likelihood subject to the sum of the absolute values of the parameters being bounded by a constant because of the nature of this constraint it shrinks coe cients and produces some coe cients that are exactly zero as a result it reduces the estimation variance while providing an interpretable final model the method is a variation of the lasso proposal of tibshirani designed for the linear regression context simulations indicate that the lasso can be more accurate than stepwise selection in this setting

fast calibrated additive quantile regression we propose a novel framework for fitting additive quantile regression models which provides well calibrated inference about the conditional quantiles and fast automatic estimation of the smoothing

on a certain class of cyclically presented groups abstract we continue the study of the cyclically presented groups g g n w where w x 0 x i x j x i x j we prove that under certain conditions on the parameters n i j the group g is infinite these results support our conjecture which states for n 4 exactly when g is trivial and that if g is non trivial then g is infinite

assigning a value to a power likelihood in a general bayesian model summarybayesian robustness under model misspecification is a current area of active research among recent ideas is that of raising the likelihood function to a power in this paper we discuss the choice of appropriate power and provide examples

revising previsions a geometric interpretation summary this paper considers the way in which an individual s previsions are changed by new information firstly reasons are discussed for preferring prevision to probability as the fundamental element in the subjective formulation of statistical procedures then following de finetti an inner product space is constructed such that the vectors are random quantities of interest and the inner product corresponds to the previsions of the individual the expected effect of new data is expressed by a bounded self adjoint operator on this space the information expected from the data is represented in terms of the eigenstructure of the operator if the operator is compact this representation has particularly simple form new data impose orthogonal co ordinate axes on the original inner product space the information expected about any random quantity is represented by a specified shrinkage along each axis to any order of accuracy all the information may be represented by a finite dimensional subspace finally the geometric representation is related to more conventional totally specified models

bayesian variable selection method for censored survival data a bayesian variable selection method for censored data is proposed in this paper based on the sufficiency and asymptotic normality of the maximum partial likelihood estimator we approximate the posterior distribution of the parameters in a proportional hazards model we consider a parsimonious model as the full model with some covariates unobserved and replaced by their conditional expected values a loss function based on the posterior expected estimation error of the log risk for the proportional hazards model is used to select a parsimonious model we derive computational expressions for this loss function for both continuous and binary covariates this approach provides an extension of lindley s 1968 journal of the royal statistical society series b 30 31 66 variable selection criterion for the linear case data from a randomized clinical trial of patients with primary biliary cirrhosis of the liver pbc fleming and harrington 1991 counting processes and survival analysis is used to illustrate the proposed method and a simulation study compares it with the backward elimination procedure

tadpole labelled oriented graph groups and cyclically presented groups abstract we study a class of labelled oriented graph log group where the underlying graph is a tadpole graph we show that such a group is the natural hnn extension of a cyclically presented group and investigate the relationship between the log group and the cyclically presented group we relate the second homotopy groups of their presentations and show that hyperbolicity of the cyclically presented group implies solvability of the conjugacy problem for the log group in the case where the label on the tail of the log spells a positive word in the vertices in the circuit we show that the logs and groups coincide with those considered by szczepanski and vesnin we obtain new presentations for these cyclically presented groups and show that the groups of fibonacci type introduced by johnson and mawdesley are of this form these groups generalise the fibonacci groups and the sieradski groups and have been studied by various authors we continue these investigations using small cancellation and curvature methods to obtain results on hyperbolicity automaticity sq universality and solvability of decision problems

the factorial schur function the application of the divided difference of a function to the inhomogeneous symmetric functions factorial schur functions of biedenharn and louck is shown to lead to new relations and simplified proofs of their properties these results include determinantal definitions and the factorial jacobi trudi identities with extensions to skew versions similar properties of a second class of symmetric functions depending on an arbitrary parameter and of importance for generalized hypergeometric functions and series are shown also to be derivable from the divided difference notion slightly extended

updating subjective probability abstract jeffrey s rule for revising a probability p to a new probability p based on new probabilities p ei on a partition ei i 1 n is p a p a ei p ei jeffrey s rule is applicable if it is judged that p a ei p a ei for all a and i this article discusses some of the mathematical properties of this rule connecting it with sufficient partitions and maximum entropy updating of contingency tables the main results concern simultaneous revision on two partitions

the selection of prior distributions by formal rules abstract subjectivism has become the dominant philosophical foundation for bayesian inference yet in practice most bayesian analyses are performed with so called noninformative priors that is priors constructed by some formal rule we review the plethora of techniques for constructing such priors and discuss some of the practical and philosophical issues that arise when they are used we give special emphasis to jeffreys s rules and discuss the evolution of his viewpoint about the interpretation of priors away from unique representation of ignorance toward the notion that they should be chosen by convention we conclude that the problems raised by the research on priors chosen by formal rules are serious and may not be dismissed lightly when sample sizes are small relative to the number of parameters being estimated it is dangerous to put faith in any default solution but when asymptotics take over jeffreys s rules and their variants remain reasonable choices we also provide an annotated b

information theoretic upper and lower bounds for statistical estimation in this paper we establish upper and lower bounds for some statistical estimation problems through concise information theoretic arguments our upper bound analysis is based on a simple yet general inequality which we call the information exponential inequality we show that this inequality naturally leads to a general randomized estimation method for which performance upper bounds can be obtained the lower bounds applicable for all statistical estimators are obtained by original applications of some well known information theoretic inequalities and approximately match the obtained upper bounds for various important problems moreover our framework can be regarded as a natural generalization of the standard minimax framework in that we allow the performance of the estimator to vary for different possible underlying distributions according to a predefined prior

the consistency of posterior distributions in nonparametric problems we give conditions that guarantee that the posterior probability of every hellinger neighborhood of the true distribution tends to 1 almost surely the conditions are 1 a requirement that the prior not put high mass near distributions with very rough densities and 2 a requirement that the prior put positive mass in kullback leibler neighborhoods of the true distribution the results are based on the idea of approximating the set of distributions with a finite dimensional set of distributions with sufficiently small hellinger bracketing metric entropy we apply the results to some examples

on the properties of variational approximations of gibbs posteriors the pac bayesian approach is a powerful set of techniques to derive nonasymptotic risk bounds for random estimators the corresponding optimal distribution of estimators usually called the gibbs posterior is unfortunately often intractable one may sample from it using markov chain monte carlo but this is usually too slow for big datasets we consider instead variational approximations of the gibbs posterior which are fast to compute we undertake a general study of the properties of such approximations our main finding is that such a variational approximation has often the same rate of convergence as the original pac bayesian procedure it approximates in addition we show that when the risk function is convex a variational approximation can be obtained in polynomial time using a convex solver we give finite sample oracle inequalities for the corresponding estimator we specialize our results to several learning tasks classification ranking matrix completion discuss how to implement a variational approximation in each case and illustrate the good properties of said approximation on real datasets

discontinuous hamiltonian monte carlo for discrete parameters and discontinuous likelihoods hamiltonian monte carlo has emerged as a standard tool for posterior computation in this article we present an extension that can efficiently explore target distributions with discontinuous densities our extension in particular enables efficient sampling from ordinal parameters though embedding of probability mass functions into continuous spaces we motivate our approach through a theory of discontinuous hamiltonian dynamics and develop a corresponding numerical solver the proposed solver is the first of its kind with a remarkable ability to exactly preserve the hamiltonian we apply our algorithm to challenging posterior inference problems to demonstrate its wide applicability and competitive performance

new approaches to bayesian consistency we use martingales to study bayesian consistency we derive sufficient conditions for both hellinger and kullback leibler consistency which do not rely on the use of a sieve alternative sufficient conditions for hellinger consistency are also found and demonstrated on examples

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures