multiple window parallel adaptive boundary finding in computer vision the problem considered in this paper is the estimation of highly variable object boundaries in noisy images boundaries may be those of a tank in an ir image a spinal canal in a cat scan a cloud in a visible light image etc or they may be internal to an object such as the boundary between a spherical surface and a cylindrical surface in a manufactured object the focus of the paper is on parallel multiple window boundary estimation algorithms here the image field is parti tioned into an array of rectangular windows and boundary finders are run simultaneously within the windows the boundary segments found within the windows are then seamed together to obtain meaningful global boundaries the entire procedure is treated within a maximum likelihood estimation framework that we have developed for boundary finding although our multiple window estimation approach can be used with a number of local boundary finding algorithms we concen trate on one which is based on dynamic programming and will produce the true maximum likelihood boundary some theoretical considera tions for boundary model design and boundary finding runtime are covered included is the use of a low computational cost f test for test ing whether a window contains a boundary and an analytical treatment which shows that use of coarse pixels with a chi square test or an f test improves the probability of correctly recognizing whether a boundary is present in a window

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

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

a smoothed maximum score estimator for the binary response model this paper describes a semiparametric estimator for binary response models in which there may be arbitrary heteroskedasticity of unknown form the estimator is obtained by maximizing a smoothed version of the objective function of c manski s maximum score estimator the smoothing procedure is similar to that used in kernel nonparametric density estimation the resulting estimator s rate of convergence in probability is the fastest possible under the assumptions that are made the centered normalized estimator is asymptotically normally distributed methods are given for consistently estimating the parameters of the limiting distribution and for selecting the bandwidth required by the smoothing procedure copyright 1992 by the econometric society

the choice of variables in multivariate regression a non conjugate bayesian decision theory approach we consider the choice of explanatory variables in multivariate linear regression our approach balances prediction accuracy against costs attached to variables in a multivariate version of a decision theory approach pioneered by lindley 1968 we also employ a non conjugate proper prior distribution for the parameters of the regression model extending the standard normal inverse wishart by adding a component of error which is unexplainable by any number of predictor variables thus avoiding the determinism identified by dawid 1988 simulated annealing and fast updating algorithms are used to search for good subsets when there are very many regressors the technique is illustrated on a near infrared spectroscopy example involving 39 observations and 300 explanatory variables this demonstrates the effectiveness of multivariate regression as opposed to separate univariate regressions it also emphasises that within a bayesian framework more variables than observations can be utilised

an error probability formula for m ary dpsk in fast rician fading and gaussian noise a formula for the symbol error probability is derived which applies to differential detection of m ary phase shift keying mdpsk in fast rician fading and white gaussiau noise this formula is an extension of a result already known for mdpsk in gaussian noise interference

efficient agnostic learning of neural networks with bounded fan in we show that the class of two layer neural networks with bounded fan in is efficiently learnable in a realistic extension to the probably approximately correct pac learning model in this model a joint probability distribution is assumed to exist on the observations and the learner is required to approximate the neural network which minimizes the expected quadratic error as special cases the model allows learning real valued functions with bounded noise learning probabilistic concepts and learning the best approximation to a target function that cannot be well approximated by the neural network the networks we consider have real valued inputs and outputs an unlimited number of threshold hidden units with bounded fan in and a bound on the sum of the absolute values of the output weights the number of computation steps of the learning algorithm is bounded by a polynomial in 1 spl epsiv 1 spl delta n and b where spl epsiv is the desired accuracy spl delta is the probability that the algorithm fails n is the input dimension and b is the bound on both the absolute value of the target which may be a random variable and the sum of the absolute values of the output weights in obtaining the result we also extended some results on iterative approximation of functions in the closure of the convex hull of a function class and on the sample complexity of agnostic learning with the quadratic loss function

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

maximin performance of binary input channels with uncertain noise distributions we consider uncertainty classes of noise distributions defined by a bound on the divergence with respect to a nominal noise distribution the noise that maximizes the minimum error probability for binary input channels is found the effect of the reduction in uncertainty brought about by knowledge of the signal to noise ratio is also studied the particular class of gaussian nominal distributions provides an analysis tool for near gaussian channels the asymptotic behavior of the least favorable noise distribution and the resulting error probability are studied in a variety of scenarios namely asymptotically small divergence with and without power constraint asymptotically large divergence with and without power constraint and asymptotically large signal to noise ratio

zeta functions of discrete groups acting on trees abstract this paper generalizes bass work on zeta functions for uniform tree lattices using the theory of von neumann algebras machinery is developed to define the zeta function of a discrete group of automorphisms of a bounded degree tree the main theorems relate the zeta function to determinants of operators defined on edges or vertices of the tree a zeta function associated to a non uniform tree lattice with appropriate hilbert representation is defined zeta functions are defined for infinite graphs with a cocompact or finite covolume group action

learning without concentration we obtain sharp bounds on the estimation error of the empirical risk minimization procedure performed in a convex class and with respect to the squared loss without assuming that class members and the target are bounded functions or have rapidly decaying tails rather than resorting to a concentration based argument the method used here relies on a small ball assumption and thus holds for classes consisting of heavy tailed functions and for heavy tailed targets the resulting estimates scale correctly with the noise level of the problem and when applied to the classical bounded scenario always improve the known bounds

on the bayes risk consistency of regularized boosting methods the probability of error of classification methods based on convex combinations of simple base classifiers by boosting algorithms is investigated the main result of the paper is that certain regularized boosting algorithms provide bayes risk consistent classifiers under the sole assumption that the bayes classifier may be approximated by a convex combination of the base classifiers nonasymptotic distribution free bounds are also developed which offer interesting new insight into how boosting works and help explain its success in practical classification problems

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

digital image restoration using spatial interaction models this paper is concerned with developing fast nonrecursive algorithms for the minimum mean squared error restoration of degraded images the degradation is assumed to be due to a space invariant periodic nonseparable known point spread function and additive white noise our basic approach is to represent the images by a class of spatial interaction models namely the simultaneous autoregressve models and the conditional markov models defined on toroidal lattices and develop minimum mean squared error restoration algorithms using these models the restoration algorithms are optimal if the parameters characterizing the interaction models are exactly known however in practice the parameters are estimated from the images by using spatial interaction models we develop restoration algorithms that do not require the availability of the original image or its prototype the specific structure of the underlying lattice enables the implementation of the filters using fast fourier transform fft computations several restoration examples are given

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

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

weighted zeta functions of digraphs abstract we define a weighted zeta function of a digraph and a weighted l function of a symmetric digraph and give determinant expressions of them furthermore we give a decomposition formula for the weighted zeta function of a g cyclic cover of a symmetric digraph for any finite group and g a decomposition formula for the weighted zeta function of an oriented line graph l g of a regular covering g of a graph g is given furthermore we define a weighted l function of an oriented line graph l g of g and present a factorization formula for the weighted zeta function of l g by weighted l functions of l g as a corollay we obtain a factorization formula for the multiedge zeta function of g given by stark and terras

ruelle zeta functions for finite digraphs abstract three types of expressions for combinatorial zeta functions are considered we investigate conditions for reformulating the exponential expression to the euler product expression and the euler product expression to the determinant expression of hashimoto type the existence of the determinant expression of hashimoto type enables one to unify those zetas within a single framework that is the ruelle zeta functions for quasi finite dynamical systems constructed on finite digraphs which we call the combinatorial zeta functions

nonconcave penalized likelihood with a diverging number of parameters a class of variable selection procedures for parametric models via nonconcave penalized likelihood was proposed by fan and li to simultaneously estimate parameters and select important variables they demonstrated that this class of procedures has an oracle property when the number of parameters is finite however in most model selection problems the number of parameters should be large and grow with the sample size in this paper some asymptotic properties of the nonconcave penalized likelihood are established for situations in which the number of parameters tends to as the sample size increases under regularity conditions we have established an oracle property and the asymptotic normality of the penalized likelihood estimators furthermore the consistency of the sandwich formula of the covariance matrix is demonstrated nonconcave penalized likelihood ratio statistics are discussed and their asymptotic distributions under the null hypothesis are obtained by imposing some mild conditions on the penalty functions the asymptotic results are augmented by a simulation study and the newly developed methodology is illustrated by an analysis of a court case on the sexual discrimination of salary

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

zeta functions of digraphs abstract we define a zeta function of a digraph and an l function of a symmetric digraph and give determinant expressions of them furthermore we give a decomposition formula for the zeta function of a g cyclic cover of a symmetric digraph for any finite group and g

markov random field texture models we consider a texture to be a stochastic possibly periodic two dimensional image field a texture model is a mathematical procedure capable of producing and describing a textured image we explore the use of markov random fields as texture models the binomial model where each point in the texture has a binomial distribution with parameter controlled by its neighbors and number of tries equal to the number of gray levels was taken to be the basic model for the analysis a method of generating samples from the binomial model is given followed by a theoretical and practical analysis of the method s convergence examples show how the parameters of the markov random field control the strength and direction of the clustering in the image the power of the binomial model to produce blurry sharp line like and blob like textures is demonstrated natural texture samples were digitized and their parameters were estimated under the markov random field model a hypothesis test was used for an objective assessment of goodness of fit under the markov random field model overall microtextures fit the model well the estimated parameters of the natural textures were used as input to the generation procedure the synthetic microtextures closely resembled their real counterparts while the regular and inhomogeneous textures did not

high dimensional classification using features annealed independence rules classification using high dimensional features arises frequently in many contemporary statistical studies such as tumor classification using microarray or other high throughput data the impact of dimensionality on classifications is largely poorly understood in a seminal paper bickel and levina 2004 show that the fisher discriminant performs poorly due to diverging spectra and they propose to use the independence rule to overcome the problem we first demonstrate that even for the independence classification rule classification using all the features can be as bad as the random guessing due to noise accumulation in estimating population centroids in high dimensional feature space in fact we demonstrate further that almost all linear discriminants can perform as bad as the random guessing thus it is paramountly important to select a subset of important features for high dimensional classification resulting in features annealed independence rules fair the conditions under which all the important features can be selected by the two sample t statistic are established the choice of the optimal number of features or equivalently the threshold value of the test statistics are proposed based on an upper bound of the classification error simulation studies and real data analysis support our theoretical results and demonstrate convincingly the advantage of our new classification procedure

best subset selection persistence in high dimensional statistical learning and optimization under l1 constraint let y x 1 x m be a random vector it is desired to predict y based on x 1 x m examples of prediction methods are regression classification using logistic regression or separating hyperplanes and so on we consider the problem of best subset selection and study it in the context m n a 1 where n is the number of observations we investigate procedures that are based on empirical risk minimization it is shown that in common cases we should aim to find the best subset among those of size which is of order o n log n it is also shown that in some asymptotic sense when assuming a certain sparsity condition there is no loss in letting m be much larger than n for example m n a 1 this is in comparison to starting with the best subset of size smaller than n and regardless of the value of a we then study conditions under which empirical risk minimization subject to l 1 constraint yields nearly the best subset these results extend some recent results obtained by greenshtein and ritov finally we present a high dimensional simulation study of a boosting type classification procedure

on the foundations of relaxation labeling processes a large class of problems can be formulated in terms of the assignment of labels to objects frequently processes are needed which reduce ambiguity and noise and select the best label among several possible choices relaxation labeling processes are just such a class of algorithms they are based on the parallel use of local constraints between labels this paper develops a theory to characterize the goal of relaxation labeling the theory is founded on a definition of con sistency in labelings extending the notion of constraint satisfaction in certain restricted circumstances an explicit functional exists that can be maximized to guide the search for consistent labelings this functional is used to derive a new relaxation labeling operator when the restrictions are not satisfied the theory relies on variational cal culus it is shown that the problem of finding consistent labelings is equivalent to solving a variational inequality a procedure nearly identical to the relaxation operator derived under restricted circum stances serves in the more general setting further a local convergence result is established for this operator the standard relaxation labeling formulas are shown to approximate our new operator which leads us to conjecture that successful applications of the standard methods are explainable by the theory developed here observations about con vergence and generalizations to higher order compatibility relations are described

on the probability distribution of differential phase perturbed by tone interference and gaussian noise this paper presents a novel method for determination of the general probability distribution of a received differential phase corrupted by both continuous tone jamming and gaussian noise an alternative and yet simple expression is derived for this general probability distribution as an example of its application we analyze the error performance of a general uncoded sfh dpsk system under partial band multitone jamming

weighted zeta functions for quotients of regular coverings of graphs abstract let g be a connected graph we reformulate stark and terras galois theory for a quotient h of a regular covering k of a graph g by using voltage assignments as applications we show that the weighted bartholdi l function of h associated to the representation of the covering transformation group of h is equal to that of g associated to its induced representation in the covering transformation group of k furthermore we express the weighted bartholdi zeta function of h as a product of weighted bartholdi l functions of g associated to irreducible representations of the covering transformation group of k we generalize stark and terras galois theory to digraphs and apply to weighted bartholdi l functions of digraphs

zeta equivalent digraphs simultaneous cospectrality abstract we introduce a zeta function of digraphs that determines and is determined by the spectra of all linear combinations of the adjacency matrix its transpose the out degree matrix and the in degree matrix in particular zeta equivalence of graphs encompasses simultaneous cospectrality with respect to the adjacency the laplacian the signless laplacian and the normalized laplacian matrix in addition we express zeta equivalence in terms of markov chains and in terms of invasions where each edge is replaced by a fixed digraph we finish with a method for constructing zeta equivalent digraphs

posterior consistency of nonparametric conditional moment restricted models this paper addresses the estimation of the nonparametric conditional moment restricted model that involves an infinite dimensional parameter g0 we estimate it in a quasi bayesian way based on the limited information likelihood and investigate the impact of three types of priors on the posterior consistency i truncated prior priors supported on a bounded set ii thin tail prior a prior that has very thin tail outside a growing bounded set and iii normal prior with nonshrinking variance in addition g0 is allowed to be only partially identified in the frequentist sense and the parameter space does not need to be compact the posterior is regularized using a slowly growing sieve dimension and it is shown that the posterior converges to any small neighborhood of the identified region we then apply our results to the nonparametric instrumental regression model finally the posterior consistency using a random sieve dimension parameter is studied

robust bayesian inference via coarsening the standard approach to bayesian inference is based on the assumption that the distribution of the data belongs to the chosen model class however even a small violation of this assumption can ha

bayesian variable selection for high dimensional generalized linear models convergence rates of the fitted densities bayesian variable selection has gained much empirical success recently in a variety of applications when the number k of explanatory variables x 1 ldots x k is possibly much larger than the sample size n for generalized linear models if most of the x j s have very small effects on the response y we show that it is possible to use bayesian variable selection to reduce overfitting caused by the curse of dimensionality k gg n in this approach a suitable prior can be used to choose a few out of the many x j s to model y so that the posterior will propose probability densities p that are often close to the true density p in some sense the closeness can be described by a hellinger distance between p and p that scales at a power very close to n 1 2 which is the finite dimensional rate corresponding to a low dimensional situation these findings extend some recent work of jiang technical report 05 02 2005 dept statistics northwestern univ on consistency of bayesian variable selection for binary classification

smoothly clipped absolute deviation on high dimensions the smoothly clipped absolute deviation scad estimator proposed by fan and li has many desirable properties including continuity sparsity and unbiasedness the scad estimator also has the asymptotically oracle property when the dimension of covariates is fixed or diverges more slowly than the sample size in this article we study the scad estimator in high dimensional settings where the dimension of covariates can be much larger than the sample size first we develop an efficient optimization algorithm that is fast and always converges to a local minimum second we prove that the scad estimator still has the oracle property on high dimensional problems we perform numerical studies to compare the scad estimator with the lasso and sis scad estimators in terms of prediction accuracy and variable selectivity when the true model is sparse through the simulation we show that the variance estimator of fan and li still works well for some limited high dimensional cases where the true nonzero coeffic

properties and refinements of the fused lasso we consider estimating an unknown signal both blocky and sparse which is corrupted by additive noise we study three interrelated least squares procedures and their asymptotic properties the first procedure is the fused lasso put forward by friedman et al ann appl statist 1 2007 302 332 which we modify into a different estimator called the fused adaptive lasso with better properties the other two estimators we discuss solve least squares problems on sieves one constrains the maximal l 1 norm and the maximal total variation seminorm and the other restricts the number of blocks and the number of nonzero coordinates of the signal we derive conditions for the recovery of the true block partition and the true sparsity patterns by the fused lasso and the fused adaptive lasso and we derive convergence rates for the sieve estimators explicitly in terms of the constraining parameters

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

error probability for fh mdpsk in multitone jamming fast rician fading and gaussian noise a method is presented for the calculation of m ary dpsk error probability in partial band multitone jamming and white gaussian noise both unfaded and faded signals and jamming tones may be analyzed by the method the fading considered is frequency flat time selective rician fading the results have application in mobile slow frequency hopped military satellite systems the method uses the pawula rice roberts 1982 method for calculating the probability distribution function of the phase angle between two vectors when they are perturbed by noise the worst case error probability in partial band jamming with fading is efficiently computed using houston s 1975 approach a fundamental parameter independent of the actual jammer power being used on the system is identified the effect of the ratio of the direct signal energy to the faded indirect signal energy is shown by example as is the effect of fading bandwidth and doppler shift for practical values of the parameters jammer fading was not found to be significant results for m 2 4 and 8 are given

population theory for boosting ensembles tree ensembles are looked at in distribution space that is the limit case of infinite sample size it is shown that the simplest kind of trees is complete in d dimensional l 2 p space if the number of terminal nodes t is greater than d for such trees we show that the adaboost algorithm gives an ensemble converging to the bayes risk

zeta functions of finite graphs and coverings iii abstract a graph theoretical analog of brauer siegel theory for zeta functions of number fields is developed using the theory of artin l functions for galois coverings of graphs from parts i and ii in the process we discuss possible versions of the riemann hypothesis for the ihara zeta function of an irregular graph

estimation and choice of neighbors in spatial interaction models of images some aspects of statistical inference for a class of spatial interaction models for finite images are presented primarily the simultaneous autoregressive sar models and conditional markov cm models each of these models is characterized by a set of neighbors a set of coefficients and a noise sequence of specified characteristics we are concerned with two problems the estimation of the unknown parameters in both sar and cm models and the choice of an appropriate model from a class of such competing models assuming gaussian distributed variables we discuss maximum likelihood ml estimation methods in general the ml scheme leads to nonlinear optimization problems to avoid excessive computation an iterative scheme is given for sar models which gives approximate ml estimates in the gaussian case and reasonably good estimates in some non gaussian situations as well likewise for cm models an easily computable consistent estimate is given the asymptotic mean squared error mse of this estimate for a four neighbor cm model is shown tn be substantially less than the mse of the popular coding estimate asymptotically consistent decision rules are given for choosing an appropriate sar or cm model the usefulness of the estimation scheme and the decision rule for the choice of neighbors is illustrated by using synthetic patterns synthetic patterns obeying known sar and cm models are generated and the models corresponding to true and several competing neighbor sets are fitted the estimation scheme yields estimates close to the parameters of the true models and the decision rule for the choice of neighbors picks up the true model from the class of competing models

or forum an algorithmic approach to linear regression linear regression models are traditionally built through trial and error to balance many competing goals such as predictive power interpretability significance robustness to error in data and sparsity among others this problem lends itself naturally to a mixed integer quadratic optimization miqo approach but has not been modeled this way because of the belief in the statistics community that miqo is intractable for large scale problems however in the last 25 years 1991 2015 algorithmic advances in integer optimization combined with hardware improvements have resulted in an astonishing 450 billion factor speedup in solving mixed integer optimization problems we present an miqo based approach for designing high quality linear regression models that explicitly addresses various competing objectives and demonstrate the effectiveness of our approach on both real and synthetic data sets

bartholdi zeta and l functions of weighted digraphs their coverings and products since a zeta function of a regular graph was introduced by ihara y ihara on discrete subgroups of the two by two projective linear group over p adic fields j math soc japan 19 1966 219 235 many kinds of zeta functions and l functions of a graph or a digraph have been defined and investigated most of the works concerning zeta and l functions of a graph contain the following 1 defining a zeta function 2 defining an l function associated with a regular graph covering 3 providing their determinant expressions and 4 computing the zeta function of a graph covering and obtaining its decomposition formula as a product of l functions as a continuation of those works we introduce a zeta function of a weighted digraph and an l function associated with a weighted digraph bundle a graph bundle is a notion containing a cartesian product of graphs and a regular or irregular graph covering also we provide determinant expressions of the zeta function and the l function moreover we compute the zeta function of a weighted digraph bundle and obtain its decomposition formula as a product of the l functions