a bayesian approach to multiscale inverse problems with on the fly scale determination abstract a bayesian computational approach is presented to provide a multi resolution estimate of an unknown spatially varying parameter from indirect measurement data in particular we are interested in spatially varying parameters with multiscale characteristics in our work we consider the challenge of not knowing the characteristic length scale s of the unknown a priori and present an algorithm for on the fly scale determination our approach is based on representing the spatial field with a wavelet expansion wavelet basis functions are hierarchically structured localized in both spatial and frequency domains and tend to provide sparse representations in that a large number of wavelet coefficients are approximately zero for these reasons wavelet bases are suitable for representing permeability fields with non trivial correlation structures moreover the intra scale correlations between wavelet coefficients form a quadtree and this structure is exploited to identify additional basis functions to refine the model bayesian inference is performed using a sequential monte carlo smc sampler with a markov chain monte carlo mcmc transition kernel the smc sampler is used to move between posterior densities defined on different scales thereby providing a computationally efficient method for adaptive refinement of the wavelet representation we gain insight from the marginal likelihoods by computing bayes factors for model comparison and model selection the marginal likelihoods provide a termination criterion for our scale determination algorithm the bayesian computational approach is rather general and applicable to several inverse problems concerning the estimation of a spatially varying parameter the approach is demonstrated with permeability estimation for groundwater flow using pressure sensor measurements

correlated random measures abstractwe develop correlated random measures random measures where the atom weights can exhibit a flexible pattern of dependence and use them to develop powerful hierarchical bayesian nonparametric models hierarchical bayesian nonparametric models are usually built from completely random measures a poisson process based construction in which the atom weights are independent completely random measures imply strong independence assumptions in the corresponding hierarchical model and these assumptions are often misplaced in real world settings correlated random measures address this limitation they model correlation within the measure by using a gaussian process in concert with the poisson process with correlated random measures for example we can develop a latent feature model for which we can infer both the properties of the latent features and their dependency pattern we develop several other examples as well we study a correlated random measure model of pairwise count data we derive an eff

the discrete fourier transform method of solving differential integral equations in scattering theory an accurate and efficient numerical method is presented for solving many differential integral equations arising from electromagnetic scattering theory it uses the discrete fourier transform technique to treat both the derivatives and the convolution integrals which often appear in these equations as a consequence this method is extremely simple to implement uses less computer memory than comparable methods and yields accurate predictions the differential integral equation is recast into a periodic form conducive to application of the discrete fourier convolution theorem the differential operators are approximated by appropriate finite difference and discrete convolution operators all these quantities are computed by using the fast fourier transform an approximate solution is obtained by using the conjugate gradient method results are compared to experimental data or analytical solutions for a 3 lambda 3 lambda metal plate where lambda is the wavelength a homogeneous and a layered infinite circular dielectric cylinder and a dielectric sphere the accuracy of the method is further illustrated by comparing predictions with independent measurements by r a ross 1966 on a 2 lambda 1 lambda metal plate at grazing incidence in all cases agreement is excellent

multistability and stable asynchronous periodic oscillations in a multiple delayed neural system abstract we consider a network of three identical neurons with multiple discrete signal transmission delays the model for such a network is a system of nonlinear delay differential equations after some consideration of the absolute synchronization of the system and the global attractivity of the zero solution we present a detailed discussion about the boundaries of the stability region of the trivial solution this allows us to determine the possible codimension one bifurcations which occur in the system in particular we show the existence of standard hopf bifurcations giving rise to synchronized periodic solutions and of d 3 equivariant hopf bifurcations giving rise to three types of periodic solutions phase locked mirror reflecting and standing waves hopf hopf and hopf steady state bifurcations interactions are shown to exist and give rise to coexistence of stable synchronized and desynchronized solutions perturbation techniques coupled with the floquet theory are used to determine the stability of the phase locked oscillations

stochastic variational inference we develop stochastic variational inference a scalable algorithm for approximating posterior distributions we develop this technique for a large class of probabilistic models and we demonstrate it with two probabilistic topic models latent dirichlet allocation and the hierarchical dirichlet process topic model using stochastic variational inference we analyze several large collections of documents 300k articles from nature 1 8m articles from the new york times and 3 8m articles from wikipedia stochastic inference can easily handle data sets of this size and outperforms traditional variational inference which can only handle a smaller subset we also show that the bayesian nonparametric topic model outperforms its parametric counterpart stochastic variational inference lets us apply complex bayesian models to massive data sets

variational message passing bayesian inference is now widely established as one of the principal foundations for machine learning in practice exact inference is rarely possible and so a variety of approximation techniques have been developed one of the most widely used being a deterministic framework called variational inference in this paper we introduce variational message passing vmp a general purpose algorithm for applying variational inference to bayesian networks like belief propagation vmp proceeds by sending messages between nodes in the network and updating posterior beliefs using local operations at each node each such update increases a lower bound on the log evidence unless already at a local maximum in contrast to belief propagation vmp can be applied to a very general class of conjugate exponential models because it uses a factorised variational approximation furthermore by introducing additional variational parameters vmp can be applied to models containing non conjugate distributions the vmp framework also allows the lower bound to be evaluated and this can be used both for model comparison and for detection of convergence variational message passing has been implemented in the form of a general purpose inference engine called vibes variational inference for bayesian networks which allows models to be specified graphically and then solved variationally without recourse to coding

iterative computational techniques in scattering based upon the integrated square error criterion an iterative technique is developed to rigorously compute the electromagnetic time and frequency domain scattering problems the method is based upon a wave function expansion technique this also includes the integral representation techniques in which the electromagnetic field equations and causality conditions are satisfied analytically while the boundary conditions or the constitutive relations have to be satisfied in a computational manner the latter is accomplished by an iterative minimization of the integrated square error for the solution of an integral equation it is shown how to obtain optimum convergence some numerical results pertaining to a number of representative problems illustrate the numerical advantages and disadvantages of the iterative method

variational bayesian approach to canonical correlation analysis as a dimension reduction algorithm canonical correlation analysis cca encounters the issue of selecting the number of canonical correlations in this letter we present a bayesian model selection algorithm for cca based on a probabilistic interpretation a hierarchical bayesian model is applied to probabilistic cca and learned by variational approximation this method not only estimates the model parameters but also automatically determines the number of canonical correlations and avoids overfitting experiments show that it performs better compared with maximum likelihood and some other model selection methods

an online expectation maximization algorithm for exploring general structure in massive networks abstract mixture model and stochastic block model sbm for structure discovery employ a broad and flexible definition of vertex classes such that they are able to explore a wide variety of structure compared to the existing algorithms based on the sbm their time complexities are o m c 2 where m and c are the number of edges and clusters the algorithms of mixture model are capable of dealing with networks with a large number of communities more efficiently due to their o m c time complexity however the algorithms of mixture model using expectation maximization em technique are still too slow to deal with real million node networks since they compute hidden variables on the entire network in each iteration in this paper an online variational em algorithm is designed to improve the efficiency of the em algorithms in each iteration our online algorithm samples a node and estimates its cluster memberships only by its adjacency links and model parameters are then estimated by the memberships of the sampled node and old model parameters obtained in the previous iteration the provided online algorithm updates model parameters subsequently by the links of a new sampled node and explores the general structure of massive and growing networks with millions of nodes and hundreds of clusters in hours compared to the relevant algorithms on synthetic and real networks the proposed online algorithm costs less with little or no degradation of accuracy results illustrate that the presented algorithm offers a good trade off between precision and efficiency

semiparametric mean field variational bayes general principles and numerical issues we introduce the term semiparametric mean field variational bayes to describe the relaxation of mean field variational bayes in which some density functions in the product density restriction are pre specified to be members of convenient parametric families this notion has appeared in various guises in the mean field variational bayes literature during its history and we endeavor to unify this important topic we lay down a general framework and explain how previous relevant methodologies fall within this framework a major contribution is elucidation of numerical issues that impact semiparametric mean field variational bayes in practice

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

path space variational inference for non equilibrium coarse grained systems in this paper we discuss information theoretic tools for obtaining optimized coarse grained molecular models for both equilibrium and non equilibrium molecular simulations the latter are ubiquitous in physicochemical and biological applications where they are typically associated with coupling mechanisms multi physics and or boundary conditions in general the non equilibrium steady states are not known explicitly as they do not necessarily have a gibbs structure the presented approach can compare microscopic behavior of molecular systems to parametric and non parametric coarse grained models using the relative entropy between distributions on the path space and setting up a corresponding path space variational inference problem the methods can become entirely data driven when the microscopic dynamics are replaced with corresponding correlated data in the form of time series furthermore we present connections and generalizations of force matching methods in coarse graining with path space information methods we demonstrate the enhanced transferability of information based parameterizations to different observables at a specific thermodynamic point due to information inequalities we discuss methodological connections between information based coarse graining of molecular systems and variational inference methods primarily developed in the machine learning community however we note that the work presented here addresses variational inference for correlated time series due to the focus on dynamics the applicability of the proposed methods is demonstrated on high dimensional stochastic processes given by overdamped and driven langevin dynamics of interacting particles

variational inference a review for statisticians abstractone of the core problems of modern statistics is to approximate difficult to compute probability densities this problem is especially important in bayesian statistics which frames all inference about unknown quantities as a calculation involving the posterior density in this article we review variational inference vi a method from machine learning that approximates probability densities through optimization vi has been used in many applications and tends to be faster than classical methods such as markov chain monte carlo sampling the idea behind vi is to first posit a family of densities and then to find a member of that family which is close to the target density closeness is measured by kullback leibler divergence we review the ideas behind mean field variational inference discuss the special case of vi applied to exponential family models present a full example with a bayesian mixture of gaussians and derive a variant that uses stochastic optimization to scale up to massive data

stability and bifurcations of equilibria in a multiple delayed differential equation the influence of multiple negative delayed feedback loops on the stability of a single action mechanism are considered a characteristic equation for the linearized stability of the equilibrium is completely analyzed as a function of two parameters describing a delay in one loop and a ratio of the gains in the two feedback loops the bifurcations occurring as the linear stability is lost are analyzed by the construction of a centre manifold in particular the nature of hopf and more degenerate higher codimension bifurcations are explicitly determined

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

a double variational bayesian framework in random fourier features for indefinite kernels random fourier features rffs have been successfully employed to kernel approximation in large scale situations the rationale behind rff relies on bochner s theorem but the condition is too strict and excludes many widely used kernels e g dot product kernels violates the shift invariant condition and indefinite kernels violates the positive definite pd condition in this article we present a unified rff framework for indefinite kernel approximation in the reproducing kernel kre n spaces rkkss besides our model is also suited to approximate a dot product kernel on the unit sphere as it can be transformed into a shift invariant but indefinite kernel by the kolmogorov decomposition scheme an indefinite kernel in rkks can be decomposed into the difference of two unknown pd kernels the spectral distribution of each underlying pd kernel can be formulated as a nonparametric bayesian gaussian mixtures model based on this we propose a double infinite gaussian mixture model in rff by placing the dirichlet process prior it takes full advantage of high flexibility on the number of components and has the capability of approximating indefinite kernels on a wide scale in model inference we develop a non conjugate variational algorithm with a sub sampling scheme for the posterior inference it allows for the non conjugate case in our model and is quite efficient due to the sub sampling strategy experimental results on several large classification data sets demonstrate the effectiveness of our nonparametric bayesian model for indefinite kernel approximation when compared to other representative random feature based methods

sequential tests for large scale learning we argue that when faced with big data sets learning and inference algorithms should compute updates using only subsets of data items we introduce algorithms that use sequential hypothesis tests to adaptively select such a subset of data points the statistical properties of this subsampling process can be used to control the efficiency and accuracy of learning or inference in the context of learning by optimization we test for the probability that the update direction is no more than 90 degrees in the wrong direction in the context of posterior inference using markov chain monte carlo we test for the probability that our decision to accept or reject a sample is wrong we experimentally evaluate our algorithms on a number of models and data sets

hierarchical dirichlet processes we consider problems involving groups of data where each observation within a group is a draw from a mixture model and where it is desirable to share mixture components between groups we assume that the number of mixture components is unknown a priori and is to be inferred from the data in this setting it is natural to consider sets of dirichlet processes one for each group where the well known clustering property of the dirichlet process provides a nonparametric prior for the number of mixture components within each group given our desire to tie the mixture models in the various groups we consider a hierarchical model specifically one in which the base measure for the child dirichlet processes is itself distributed according to a dirichlet process such a base measure being discrete the child dirichlet processes necessarily share atoms thus as desired the mixture models in the different groups necessarily share mixture components we discuss representations of hierarchical dirichlet processes

variational bayesian multinomial probit regression with gaussian process priors it is well known in the statistics literature that augmenting binary and polychotomous response models with gaussian latent variables enables exact bayesian analysis via gibbs sampling from the parameter posterior by adopting such a data augmentation strategy dispensing with priors over regression coefficients in favor of gaussian process gp priors over functions and employing variational approximations to the full posterior we obtain efficient computational methods for gp classification in the multiclass setting the model augmentation with additional latent variables ensures full a posteriori class coupling while retaining the simple a priori independent gp covariance structure from which sparse approximations such as multiclass informative vector machines ivm emerge in a natural and straightforward manner this is the first time that a fully variational bayesian treatment for multiclass gp classification has been developed without having to resort to additional explicit approximations to the nongaussian likelihood term empirical comparisons with exact analysis use markov chain monte carlo mcmc and laplace approximations illustrate the utility of the variational approximation as a computationally economic alternative to full mcmc and it is shown to be more accurate than the laplace approximation

normal forms for retarded functional differential equations with parameters and applications to hopf bifurcation abstract the paper addresses for retarded functional differential equations fdes with parameters the computation of normal forms associated with the flow on a finite dimensional invariant manifold tangent to an invariant space for the infinitesimal generator of the linearized equation at a singularity the analysis is based on the theory previously developed for fdes without parameters by considering an extension of a phase space appropriate to the computation of these normal forms as an application the generic hopf bifurcation for scalar retarded fdes is considered

existence and stability of a synchronous oscillation in a neural system with delayed coupling abstract the article deals with a particular case of a two neuron network with two discrete time delays we analyse the linear stability for the particular equilibrium 0 0 and address the hopf bifurcation which may occur as one of the delays crosses some critical values the network under consideration has no self connections and thus only one time delay appears in the characteristic equation

natural gradient works efficiently in learning when a parameter space has a certain underlying structure the ordinary gradient of a function does not represent its steepest direction but the natural gradient does information geometry is used for calculating the natural gradients in the parameter space of perceptrons the space of matrices for blind source separation and the space of linear dynamical systems for blind source deconvolution the dynamical behavior of natural gradient online learning is analyzed and is proved to be fisher efficient implying that it has asymptotically the same performance as the optimal batch estimation of parameters this suggests that the plateau phenomenon which appears in the backpropagation learning algorithm of multilayer perceptrons might disappear or might not be so serious when the natural gradient is used an adaptive method of updating the learning rate is proposed and analyzed

normal forms for retarded functional differential equations and applications to bogdanov takens singularity abstract the paper addresses for retarded functional differential equations fdes the computation of normal forms associated with the flow on a finite dimensional invariant manifold tangent to invariant spaces for the infinitesimal generator of the linearized equation at a singularity a phase space appropriate to the computation of these normal forms is introduced and adequate nonresonance conditions for the computation of the normal forms are derived as an application the general situation of bogdanov takens singularity and its versal unfolding for scalar retarded fdes with nondegeneracy at second order is considered both in the general case and in the case of differential delay equations of the form t x t x t 1

stability bifurcation and multistability in a system of two coupled neurons with multiple time delays a system of delay differential equations representing a model for a pair of neurons with time delayed connections between the neurons and time delayed feedback from each neuron to itself is studied conditions for the linear stability of the trivial solution of this system are represented in a parameter space consisting of the sum of the time delays between the elements and the product of the strengths of the connections between the elements it is shown that the trivial fixed point may lose stability via a pitchfork bifurcation a hopf bifurcation or one of three types of codimension two bifurcations multistability near these latter bifurcations is predicted using center manifold analysis and confirmed using numerical simulations

application of a conjugate gradient fft method to scattering from thin planar material plates the backscatter cross section is calculated for thin material plates with finite electric permittivity conductivity and magnetic permeability illuminated by a plane wave the plates are assumed to be planar with an arbitrary perimeter the integral equations are formed and solved by a combined conjugate gradient fast fourier transform cg fft method the cg fft method was tested for several geometrics and materials measured and computed backscatter results are compared for a perfectly conducting equilateral triangle plate a square dielectric and magnetic plate and a circular dielectric plate the agreement between measured and computed data is generally good except toward edge on incidence where several factors cause discrepancies accurate approximations to the geometry and far field integrals become critical near edge on incidence and it is postulated that as the incidence angle approaches edge on the sampling interval and tolerance should be decreased

scalable information inequalities for uncertainty quantification abstract in this paper we demonstrate the only available scalable information bounds for quantities of interest of high dimensional probabilistic models scalability of inequalities allows us to a obtain uncertainty quantification bounds for quantities of interest in the large degree of freedom limit and or at long time regimes b assess the impact of large model perturbations as in nonlinear response regimes in statistical mechanics c address model form uncertainty i e compare different extended models and corresponding quantities of interest we demonstrate some of these properties by deriving robust uncertainty quantification bounds for phase diagrams in statistical mechanics models

bayesian density estimation and inference using mixtures abstract we describe and illustrate bayesian inference in models for density estimation using mixtures of dirichlet processes these models provide natural settings for density estimation and are exemplified by special cases where data are modeled as a sample from mixtures of normal distributions efficient simulation methods are used to approximate various prior posterior and predictive distributions this allows for direct inference on a variety of practical issues including problems of local versus global smoothing uncertainty about density estimates assessment of modality and the inference on the numbers of components also convergence results are established for a general class of normal mixture models

a combined finite element boundary integral formulation for solution of two dimensional scattering problems via cgfft a novel technique is presented for computing the scattering by two dimensional structures of arbitrary inhomogeneity the proposed approach combines the usual finite element fe method with the boundary integral equation to formulate a discrete system this is subsequently solved via the conjugate gradient cg algorithm a particular characteristic of the method is the use of rectangular boundaries to enclose the scatterer several of the resulting boundary integrals are then convolutions and can be evaluated via the fast fourier transform fft in the implementation of the cg algorithm the solution approach presented here offers the principal advantage of having o n memory demand and employs a one dimensional fft as against the two dimensional fft required in a traditional implementation of the proposed cg fft algorithm the speed of the proposed solution method is compared with that of the traditional cg fft algorithm results are presented for several rectangular composite cylinders and one perfectly conducting cylinder these are shown to be in excellent agreement with the moment method

differential geometry of curved exponential families curvatures and information loss the differential geometrical framework is given for analyzing statistical problems related to multi parameter families of distributions the dualistic structures of the exponential families and curved exponential families are elucidated from the geometrical viewpoint the duality connected by the legendre transformation is thus extended to include two kinds of affine connections and two kinds of curvatures the second order information loss is calculated for fisher efficient estimators and is decomposed into the sum of two non negative terms one is related to the exponential curvature of the statistical model and the other is related to the mixture curvature of the estimator only the latter term depends on the estimator and vanishes for the maximum likelihood estimator a set of statistics which recover the second order information loss are given the second order efficiency also is obtained the differential geometry of the function space of distributions is discussed

application of the discrete fourier transform method to thin dielectric structures em scattering the discrete fourier transform method is a numerical technique designed to take advantage of the convolution structure that often appears in the differential integral equations arising from electromagnetic scattering it used the fast fourier transform fft to compute both the derivatives and the convolution integrals as a consequence this method is easy to program uses less computer memory than comparable methods yields accurate predictions and in general offers a better rate of convergence this technique which is particularly suited for solving problems where the scatterer is made of a dielectric material and has a shape which can be approximated accurately by a rectangular grid is applied to thin dielectric slabs with both electric and magnetic properties it is shown that by choosing the conductivity sufficiently large a thin dielectric slab behaves like a metallic plate on the other hand with a suitable choice of conductivity a particular thin dielectric slab will act like a resistive plate

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

on line expectation maximization algorithm for latent data models in this contribution we propose a generic online also sometimes called adaptive or recursive version of the expectation maximisation em algorithm applicable to latent variable models of independent observations compared to the algorithm of titterington 1984 this approach is more directly connected to the usual em algorithm and does not rely on integration with respect to the complete data distribution the resulting algorithm is usually simpler and is shown to achieve convergence to the stationary points of the kullback leibler divergence between the marginal distribution of the observation and the model distribution at the optimal rate i e that of the maximum likelihood estimator in addition the proposed approach is also suitable for conditional or regression models as illustrated in the case of the mixture of linear regressions model

bayesian nonparametric inference of switching dynamic linear models many complex dynamical phenomena can be effectively modeled by a system that switches among a set of conditionally linear dynamical modes we consider two such models the switching linear dynamical system slds and the switching vector autoregressive var process our bayesian nonparametric approach utilizes a hierarchical dirichlet process prior to learn an unknown number of persistent smooth dynamical modes we additionally employ automatic relevance determination to infer a sparse set of dynamic dependencies allowing us to learn slds with varying state dimension or switching var processes with varying autoregressive order we develop a sampling algorithm that combines a truncated approximation to the dirichlet process with efficient joint sampling of the mode and state sequences the utility and flexibility of our model are demonstrated on synthetic data sequences of dancing honey bees the ibovespa stock index and a maneuvering target tracking application

simple scalable and accurate posterior interval estimation summarystandard posterior sampling algorithms such as markov chain monte carlo procedures face major challenges in scaling up to massive datasets we propose a simple and general posterior interval estimation algorithm to rapidly and accurately estimate quantiles of the posterior distributions for one dimensional functionals our algorithm runs markov chain monte carlo in parallel for subsets of the data and then averages quantiles estimated from each subset we provide strong theoretical guarantees and show that the credible intervals from our algorithm asymptotically approximate those from the full posterior in the leading parametric order our algorithm has a better balance of accuracy and efficiency than its competitors across a variety of simulations and a real data example

probabilistic principal component analysis principal component analysis pca is a ubiquitous technique for data analysis and processing but one which is not based upon a probability model in this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximum likelihood estimation of parameters in a latent variable model closely related to factor analysis we consider the properties of the associated likelihood function giving an em algorithm for estimating the principal subspace iteratively and discuss the advantages conveyed by the definition of a probability density function for pca

online learning of hierarchical pitman yor process mixture of generalized dirichlet distributions with feature selection in this paper a novel statistical generative model based on hierarchical pitman yor process and generalized dirichlet distributions gds is presented the proposed model allows us to perform joint clustering and feature selection thanks to the interesting properties of the gd distribution we develop an online variational inference algorithm formulated in terms of the minimization of a kullback leibler divergence of our resulting model that tackles the problem of learning from high dimensional examples this variational bayes formulation allows simultaneously estimating the parameters determining the model s complexity and selecting the appropriate relevant features for the clustering structure moreover the proposed online learning algorithm allows data instances to be processed in a sequential manner which is critical for large scale and real time applications experiments conducted using challenging applications namely scene recognition and video segmentation where our approach is viewed as an unsupervised technique for visual learning in high dimensional spaces showed that the proposed approach is suitable and promising

dynamic infinite mixed membership stochastic blockmodel directional and pairwise measurements are often used to model interactions in a social network setting the mixed membership stochastic blockmodel mmsb was a seminal work in this area and its ability has been extended however models such as mmsb face particular challenges in modeling dynamic networks for example with the unknown number of communities accordingly this paper proposes a dynamic infinite mixed membership stochastic blockmodel a generalized framework that extends the existing work to potentially infinite communities inside a network in dynamic settings i e networks are observed over time additional model parameters are introduced to reflect the degree of persistence among one s memberships at consecutive time stamps under this framework two specific models namely mixture time variant and mixture time invariant models are proposed to depict two different time correlation structures two effective posterior sampling strategies and their results are presented respectively using synthetic and real world data

variational inference in nonconjugate models mean field variational methods are widely used for approximate posterior inference in many probabilistic models in a typical application mean field methods approximately compute the posterior with a coordinate ascent optimization algorithm when the model is conditionally conjugate the coordinate updates are easily derived and in closed form however many models of interest like the correlated topic model and bayesian logistic regression are nonconjugate in these models mean field methods cannot be directly applied and practitioners have had to develop variational algorithms on a case by case basis in this paper we develop two generic methods for nonconjugate models laplace variational inference and delta method variational inference our methods have several advantages they allow for easily derived variational algorithms with a wide class of nonconjugate models they extend and unify some of the existing algorithms that have been derived for specific models and they work well on real world data sets we studied our methods on the correlated topic model bayesian logistic regression and hierarchical bayesian logistic regression

online model selection based on the variational bayes the bayesian framework provides a principled way of model selection this framework estimates a probability distribution over an ensemble of models and the prediction is done by averaging over the ensemble of models accordingly the uncertainty of the models is taken into account and complex models with more degrees of freedom are penalized however integration over model parameters is often intractable and some approximation scheme is needed recently a powerful approximation scheme called the variational bayes vb method has been proposed this approach defines the free energy for a trial probability distribution which approximates a joint posterior probability distribution over model parameters and hidden variables the exact maximization of the free energy gives the true posterior distribution the vb method uses factorized trial distributions the integration over model parameters can be done analytically and an iterative expectation maximization like algorithm whose convergence is guaranteed is derived in this article we derive an online version of the vb algorithm and prove its convergence by showing that it is a stochastic approximation for finding the maximum of the free energy by combining sequential model selection procedures the online vb method provides a fully online learning method with a model selection mechanism in preliminary experiments using synthetic data the online vb method was able to adapt the model structure to dynamic environments

consistency and fluctuations for stochastic gradient langevin dynamics applying standard markov chain monte carlo mcmc algorithms to large data sets is computationally expensive both the calculation of the acceptance probability and the creation of informed proposals usually require an iteration through the whole data set the recently proposed stochastic gradient langevin dynamics sgld method circumvents this problem by generating proposals which are only based on a subset of the data by skipping the accept reject step and by using decreasing step sizes sequence m m 0 we provide in this article a rigorous mathematical framework for analysing this algorithm we prove that under verifiable assumptions the algorithm is consistent satisfies a central limit theorem clt and its asymptotic bias variance decomposition can be characterized by an explicit functional of the step sizes sequence m m 0 we leverage this analysis to give practical recommendations for the notoriously difficult tuning of this algorithm it is asymptotically optimal to use a step size sequence of the type m m 1 3 leading to an algorithm whose mean squared error mse decreases at rate o m 1 3