order based dependent dirichlet processes in this article we propose a new framework for bayesian nonparametric modeling with continuous covariates in particular we allow the nonparametric distribution to depend on covariates through ordering the random variables building the weights in the stick breaking representation we focus mostly on the class of random distributions that induces a dirichlet process at each covariate value we derive the correlation between distributions at different covariate values and use a point process to implement a practically useful type of ordering two main constructions with analytically known correlation structures are proposed practical and efficient computational methods are introduced we apply our framework through mixtures of these processes to regression modeling the modeling of stochastic volatility in time series data and spatial geostatistical modeling

kernel stick breaking processes we propose a class of kernel stick breaking processes for uncountable collections of dependent random probability measures the process is constructed by first introducing an infinite sequence of random locations independent random probability measures and beta distributed random weights are assigned to each location predictor dependent random probability measures are then constructed by mixing over the locations with stick breaking probabilities expressed as a kernel multiplied by the beta weights some theoretical properties of the process are described including a covariate dependent prediction rule a retrospective markov chain monte carlo algorithm is developed for posterior computation and the methods are illustrated using a simulated example and an epidemiological application

subclass discriminant nonnegative matrix factorization for facial image analysis nonnegative matrix factorization nmf is among the most popular subspace methods widely used in a variety of image processing problems recently a discriminant nmf method that incorporates linear discriminant analysis inspired criteria has been proposed which achieves an efficient decomposition of the provided data to its discriminant parts thus enhancing classification performance however this approach possesses certain limitations since it assumes that the underlying data distribution is unimodal which is often unrealistic to remedy this limitation we regard that data inside each class have a multimodal distribution thus forming clusters and use criteria inspired by clustering based discriminant analysis the proposed method incorporates appropriate discriminant constraints in the nmf decomposition cost function in order to address the problem of finding discriminant projections that enhance class separability in the reduced dimensional projection space while taking into account subclass information the developed algorithm has been applied for both facial expression and face recognition on three popular databases experimental results verified that it successfully identified discriminant facial parts thus enhancing recognition performance

on slicing invariants of knots the slicing number of a knot u s k is the minimum number of crossing changes required to convert k to a slice knot this invariant is bounded above by the unknotting number and below by the slice genus g s k we show that for many knots previous bounds on the unknotting number obtained by ozsvath and szabo and by the author in fact give bounds on the slicing number livingston defined another invariant u s k which takes into account signs of crossings changed to get a slice knot and which is bounded above by the slicing number and below by the slice genus we exhibit an infinite family of knots k n with slice genus n and livingston invariant greater than n our bounds are based on restrictions using donaldson s diagonalisation theorem or heegaard floer homology on the intersection forms of four manifolds bounded by the double branched cover of a knot

bayesian density regression summary the paper considers bayesian methods for density regression allowing a random probability distribution to change flexibly with multiple predictors the conditional response distribution is expressed as a non parametric mixture of regression models with the mixture distribution changing with predictors a class of weighted mixture of dirichlet process priors is proposed for the uncountable collection of mixture distributions it is shown that this specification results in a generalized polya urn scheme which incorporates weights that are dependent on the distance between subjects predictor values to allow local dependence in the mixture distributions we propose a kernel based weighting scheme a gibbs sampling algorithm is developed for posterior computation the methods are illustrated by using simulated data examples and an epidemiologic application

some applications of tristram levine signatures and relation to vassiliev invariants abstract tristram and levine introduced a continuous family of signature invariants for knots we show that any possible value of such an invariant is realized by a knot with given vassiliev invariants of bounded degree we also show that one can make a knot prime preserving alexander polynomial and vassiliev invariants of bounded degree finally the tristram levine signatures are applied to obtain a condition on signed unknotting number

generalized spatial dirichlet process models summary many models for the study of point referenced data explicitly introduce spatial random effects to capture residual spatial association these spatial effects are customarily modelled as a zeromean stationary gaussian process the spatial dirichlet process introduced by gelfand et al 2005 produces a random spatial process which is neither gaussian nor stationary rather it varies about a process that is assumed to be stationary and gaussian the spatial dirichlet process arises as a probability weighted collection of random surfaces this can be limiting for modelling and inferential purposes since it insists that a process realization must be one of these surfaces we introduce a random distribution for the spatial effects that allows different surface selection at different sites moreover we can specify the model so that the marginal distribution of the effect at each site still comes from a dirichlet process the development is offered constructively providing a multivariate extension of the stick breaking representation of the weights we then introduce mixing using this generalized spatial dirichlet process we illustrate with a simulated dataset of independent replications and note that we can embed the generalized process within a dynamic model specification to eliminate the independence assumption

gibbs sampling methods for stick breaking priors a rich and flexible class of random probability measures which we call stick breaking priors can be constructed using a sequence of independent beta random variables examples of random measures that have this characterization include the dirichlet process its two parameter extension the two parameter poisson dirichlet process finite dimensional dirichlet priors and beta two parameter processes the rich nature of stick breaking priors offers bayesians a useful class of priors for nonparametric problems while the similar construction used in each prior can be exploited to develop a general computational procedure for fitting them in this article we present two general types of gibbs samplers that can be used to fit posteriors of bayesian hierarchical models based on stick breaking priors the first type of gibbs sampler referred to as a polya urn gibbs sampler is a generalized version of a widely used gibbs sampling method currently employed for dirichlet process computing this method applies t

bayesian nonparametric spatial modeling with dirichlet process mixing customary modeling for continuous point referenced data assumes a gaussian process that is often taken to be stationary when such models are fitted within a bayesian framework the unknown parameters of the process are assumed to be random so a random gaussian process results here we propose a novel spatial dirichlet process mixture model to produce a random spatial process that is neither gaussian nor stationary we first develop a spatial dirichlet process model for spatial data and discuss its properties because of familiar limitations associated with direct use of dirichlet process models we introduce mixing by convolving this process with a pure error process we then examine properties of models created through such dirichlet process mixing in the bayesian framework we implement posterior inference using gibbs sampling spatial prediction raises interesting questions but these can be handled finally we illustrate the approach using simulated data as well as a dataset involving precipitati

product partition models for change point problems product partition models assume that observations in different components of a random partition of the data are independent if the probability distribution of random partitions is in a certain product form prior to making the observations it is also in product form given the observations the product model thus provides a convenient machinery for allowing the data to weight the partitions likely to hold and inference about particular future observations may then be made by first conditioning on the partition and then averaging over all partitions these models apply with special computational simplicity to change point problems where the partitions divide the sequence of observations into components within which different regimes hold we show with appropriate selection of prior product models that the observations can eventually determine approximately the true partition

instantons concordance and whitehead doubling we use moduli spaces of instantons and chern simons invariants of flat connections to prove that the whitehead doubles of 2 2 n 1 torus knots are independent in the smooth knot concordance group that is they freely generate a subgroup of infinite rank call two knots in the 3 sphere concordant if they arise as the boundary of a smooth and properly embedded cylinder in the 3 sphere times an interval concordance is clearly an equivalence relation modulo this relation the set of knots forms an abelian group c with the role of addition played by connected sum and inverse given by considering the mirror image with reversed orientation this concordance group is a much studied object well motivated by its role as a gateway into the mysterious world of 4 dimensional topology indeed even in this relative situation of studying 3 dimensional manifolds in relation to the 4 dimensional manifolds they bound one can observe the distinction between the smooth and topological categories in dimension four moreover the group structure afforded by passing to concordance paves a clearer path through the often intractable field of knot theory our results focus on a particular satellite operation positive untwisted whitehead doubling and its effect on the concordance group see figure 1 and section 3 1 for a definition our motivation comes from the following conjecture to state it recall that a knot is slice if it is concordant to the unknot or equivalently if it bounds a smooth and properly embedded disk in the 4 ball

an anova model for dependent random measures we consider dependent nonparametric models for related random probability distributions for example the random distributions might be indexed by a categorical covariate indicating the treatment levels in a clinical trial and might represent random effects distributions under the respective treatment combinations we propose a model that describes dependence across random distributions in an analysis of variance anova type fashion we define a probability model in such a way that marginally each random measure follows a dirichlet process dp and use the dependent dirichlet process to define the desired dependence across the related random measures the resulting probability model can alternatively be described as a mixture of anova models with a dp prior on the unknown mixing measure the main features of the proposed approach are ease of interpretation and computational simplicity because the model follows the standard anovastructure interpretation and inference parallels conventions for anova mode

bayesian clustering and product partition models summary we present a decision theoretic formulation of product partition models ppms that allows a formal treatment of different decision problems such as estimation or hypothesis testing and clustering methods simultaneously a key observation in our construction is the fact that ppms can be formulated in the context of model selection the underlying partition structure in these models is closely related to that arising in connection with dirichlet processes this allows a straightforward adaptation of some computational strategies originally devised for nonparametric bayesian problems to our framework the resulting algorithms are more flexible than other competing alternatives that are used for problems involving ppms we propose an algorithm that yields bayes estimates of the quantities of interest and the groups of experimental units we explore the application of our methods to the detection of outliers in normal and student t regression models with clustering structure equivalent to that induced by a dirichlet process prior we also discuss the sensitivity of the results considering different prior distributions for the partitions

bayesian semiparametric dynamic frailty models for multiple event time data many biomedical studies collect data on times of occurrence for a health event that can occur repeatedly such as infection hospitalization recurrence of disease or tumor onset to analyze such data it is necessary to account for within subject dependency in the multiple event times motivated by data from studies of palpable tumors this article proposes a dynamic frailty model and bayesian semiparametric approach to inference the widely used shared frailty proportional hazards model is generalized to allow subject specific frailties to change dynamically with age while also accommodating nonproportional hazards parametric assumptions on the frailty distribution are avoided by using dirichlet process priors for a shared frailty and for multiplicative innovations on this frailty by centering the semiparametric model on a conditionally conjugate dynamic gamma model we facilitate posterior computation and lack of fit assessments of the parametric model our proposed method is demonstrated using data from a cancer chemoprevention study

alternating knots with unknotting number one abstract we show that an alternating knot with unknotting number one has an unknotting crossing in any alternating diagram we also prove that an alternating knot has unknotting number one if and only if its branched double cover arises as half integer surgery on a knot in s 3 thus establishing a converse to the montesinos trick along the way we reprove a characterisation of almost alternating diagrams of the unknot originally due to tsukamoto these results are established using the obstruction to unknotting number one developed by greene

donaldson s theorem heegaard floer homology and knots with unknotting number one abstract we establish an obstruction to unknotting an alternating knot by a single crossing change the obstruction is lattice theoretic in nature and combines donaldson s diagonalization theorem with an obstruction developed by ozsvath and szabo using heegaard floer homology as an application we enumerate the alternating 3 braid knots with unknotting number one and show that each has an unknotting crossing in its standard alternating diagram

unknotting information from heegaard floer homology we use heegaard floer homology to obtain bounds on unknotting numbers this is a generalisation of ozsvath and szabo s obstruction to unknotting number one we determine the unknotting numbers of 910 913 935 938 1053 10101 and 10120 this completes the table of unknotting numbers for prime knots with crossing number nine or less our obstruction uses a refined version of montesinos theorem which gives a dehn surgery description of the branched double cover of a knot

an application of gauge theory to four dimensional topology soit x une u variete compacte reguliere simplement connexe orientee avec la propriete que la forme associee q est definie positive alors cette forme est equivalente sur les entiers a la forme diagonale standard soit dans une base q u 1 u 2 u 2 u 2 1 u 2 2 u 2 2