a characterization of markov equivalence classes for acyclic digraphs undirected graphs and acyclic digraphs adgs as well as their mutual extension to chain graphs are widely used to describe dependencies among variables in multivariate distributions in particular the likelihood functions of adg models admit convenient recursive factorizations that often allow explicit maximum likelihood estimates and that are well suited to building bayesian networks for expert systems whereas the undirected graph associated with a dependence model is uniquely determined there may however be many adgs that determine the same dependence markov model thus the family of all adgs with a given set of vertices is naturally partitioned into markov equivalence classes each class being associated with a unique statistical model statistical procedures such as model selection or model averaging that fail to take into account these equivalence classes may incur substantial computational or other inefficiencies here it is shown that each markov equivalence class is uniquely determined by a single chain graph the essential graph that is itself simultaneously markov equivalent to all adgs in the equivalence class essential graphs are characterized a polynomial time algorithm for their construction is given and their applications to model selection and other statistical questions are described

global identifiability of linear structural equation models structural equation models are multivariate statistical models that are defined by specifying noisy functional relationships among random variables we consider the classical case of linear relationships and additive gaussian noise terms we give a necessary and sufficient condition for global identifiability of the model in terms of a mixed graph encoding the linear structural equations and the correlation structure of the error terms global identifiability is understood to mean injectivity of the parametrization of the model and is fundamental in particular for applicability of standard statistical methodology

effective connectivity of fmri data using ancestral graph theory dealing with missing regions most of the current methods to assess effective connectivity from functional magnetic resonance imaging fmri rely on the assumption that all relevant brain regions are entered into the analysis if this assumption is untenable which we believe is most often the case then spurious connections between brain regions can appear in this paper we propose to use an ancestral graph to model connectivity which provides a way to avoid spurious connections the ancestral graph is determined from trial by trial variation and not from the time series a random effects model is defined for ancestral graphs which allows for individual differences in terms of graph parameters e g connection strength procedures for model selection model fit and hypothesis testing of ancestral graphs are proposed the hypothesis test can be used to find differences in connection strength between for example conditions monte carlo simulations show that the ancestral graph is appropriate to model connectivity from fmri condition specific trial data to assess the accuracy further the proposed method is applied to real fmri data to determine how brain regions interact during speech monitoring

chain graph models and their causal interpretations chain graphs are a natural generalization of directed acyclic graphs and undirected graphs however the apparent simplicity of chain graphs belies the subtlety of the conditional independence hypotheses that they represent there are many simple and apparently plausible but ultimately fallacious interpretations of chain graphs that are often invoked implicitly or explicitly these interpretations also lead to flawed methods for applying background knowledge to model selection we present a valid interpretation by showing how the distribution corresponding to a chain graph may be generated from the equilibrium distributions of dynamic models with feed back these dynamic interpretations lead to a simple theory of intervention extending the theory developed for directed acyclic graphs finally we contrast chain graph models under this interpretation with simultaneous equation models which have traditionally been used to model feed back in econometrics

structural intervention distance for evaluating causal graphs causal inference relies on the structure of a graph often a directed acyclic graph dag different graphs may result in different causal inference statements and different intervention distributions to quantify such differences we propose a pre metric between dags the structural intervention distance sid the sid is based on a graphical criterion only and quantifies the closeness between two dags in terms of their corresponding causal inference statements it is therefore well suited for evaluating graphs that are used for computing interventions instead of dags it is also possible to compare cpdags completed partially dags that represent markov equivalence classes the sid differs significantly from the widely used structural hamming distance and therefore constitutes a valuable additional measure we discuss properties of this distance and provide a reasonably efficient implementation with software code available on the first author s home page

random networks graphical models and exchangeability we study conditional independence relationships for random networks and their interplay with exchangeability we show that for finitely exchangeable network models the empirical subgraph densities are maximum likelihood estimates of their theoretical counterparts we then characterize all possible markov structures for finitely exchangeable random graphs thereby identifying a new class of markov network models corresponding to bidirected kneser graphs in particular we demonstrate that the fundamental property of dissociatedness corresponds to a markov property for exchangeable networks described by bidirected line graphs finally we study those exchangeable models that are also summarized in the sense that the probability of a network depends only on the degree distribution and we identify a class of models that is dual to the markov graphs of frank and strauss particular emphasis is placed on studying consistency properties of network models under the process of forming subnetworks and we show that the only consistent systems of markov properties correspond to the empty graph the bidirected line graph of the complete graph and the complete graph

binary models for marginal independence summary log linear models are a classical tool for the analysis of contingency tables in particular the subclass of graphical log linear models provides a general framework for modelling conditional independences however with the exception of special structures marginal independence hypotheses cannot be accommodated by these traditional models focusing on binary variables we present a model class that provides a framework for modelling marginal independences in contingency tables the approach that is taken is graphical and draws on analogies with multivariate gaussian models for marginal independence for the graphical model representation we use bidirected graphs which are in the tradition of path diagrams we show how the models can be parameterized in a simple fashion and how maximum likelihood estimation can be performed by using a version of the iterated conditional fitting algorithm finally we consider combining these models with symmetry restrictions

analysis of noise properties of a class of exact methods of inverting the 2 d exponential radon transform spect application a general approach that the authors proposed elsewhere reveals the intrinsic relationship among methods for inversion of the 2 d exponential radon transform described by bellini et al 1979 by tretiak and metz 1980 by hawkins et al 1988 and by inouye et al 1989 moreover the authors approach provides an infinite class of linear methods for inverting the 2 d exponential radon transform in the work reported here the authors systematically investigated the noise characteristics of the methods in this class obtaining analytical forms for the autocovariance and the variance of the images reconstructed by use of various methods the noise properties of a new quasi optimal method were then compared theoretically to those of other methods of the class the authors analysis demonstrates that the quasi optimal method achieves smaller global variance in the reconstructed images than do the other methods of the class extensive numerical simulation studies confirm this prediction

weighted kolmogorov smirnov test accounting for the tails accurate goodness of fit tests for the extreme tails of empirical distributions is a very important issue relevant in many contexts including geophysics insurance and finance we have derived exact asymptotic results for a generalization of the large sample kolmogorov smirnov test well suited to testing these extreme tails in passing we have rederived and made more precise the approximate limit solutions found originally in unrelated fields first in l turban j phys a 25 127 1992 and later in p l krapivsky and s redner am j phys 64 546 1996

causal inference by using invariant prediction identification and confidence intervals summary what is the difference between a prediction that is made with a causal model and that with a non causal model suppose that we intervene on the predictor variables or change the whole environment the predictions from a causal model will in general work as well under interventions as for observational data in contrast predictions from a non causal model can potentially be very wrong if we actively intervene on variables here we propose to exploit this invariance of a prediction under a causal model for causal inference given different experimental settings e g various interventions we collect all models that do show invariance in their predictive accuracy across settings and interventions the causal model will be a member of this set of models with high probability this approach yields valid confidence intervals for the causal relationships in quite general scenarios we examine the example of structural equation models in more detail and provide sufficient assumptions under which the set of causal predictors becomes identifiable we further investigate robustness properties of our approach under model misspecification and discuss possible extensions the empirical properties are studied for various data sets including large scale gene perturbation experiments

a recursive method for structural learning of directed acyclic graphs in this paper we propose a recursive method for structural learning of directed acyclic graphs dags in which a problem of structural learning for a large dag is first decomposed into two problems of structural learning for two small vertex subsets each of which is then decomposed recursively into two problems of smaller subsets until none subset can be decomposed further in our approach search for separators of a pair of variables in a large dag is localized to small subsets and thus the approach can improve the efficiency of searches and the power of statistical tests for structural learning we show how the recent advances in the learning of undirected graphical models can be employed to facilitate the decomposition simulations are given to demonstrate the performance of the proposed method

acceleration of maximum likelihood reconstruction using frequency amplification and attenuation compensation algorithms that calculate maximum likelihood ml and maximum a posteriori solutions using expectation maximization have been successfully applied to spect and pet these algorithms are appealing because of their solid theoretical basis and their guaranteed convergence a major drawback is the slow convergence which results in long processing times the authors present 2 new heuristic acceleration methods for maximum likelihood reconstruction of ect images the first method incorporates a frequency dependent amplification in the calculations to compensate for the low pass filtering of the backprojection operation in the second method an amplification factor is incorporated that suppresses the effect of attenuation on the updating factors both methods are compared to the 1 dimensional line search method proposed by lewitt all 3 methods accelerate the ml algorithm on the authors test images lewitt s method produced the strongest acceleration of the three individual methods however the combination of the frequency amplification with the line search method results in a new algorithm with still better performance under certain conditions an effective frequency amplification can be already achieved by skipping some of the calculations required for ml

marginal log linear parameters for graphical markov models marginal log linear mll models provide a flexible approach to multivariate discrete data mll parametrizations under linear constraints induce a wide variety of models including models defined by conditional independences we introduce a subclass of mll models which correspond to acyclic directed mixed graphs admgs under the usual global markov property we characterize for precisely which graphs the resulting parametrization is variation independent the mll approach provides the first description of admg models in terms of a minimal list of constraints the parametrization is also easily adapted to sparse modelling techniques which we illustrate using several examples of real data

a harmonic decomposition reconstruction algorithm for spatially varying focal length collimators spatially varying focal length fan beam collimators can be used in single photon emission computed tomography to improve detection efficiency and to reduce reconstruction artifacts resulting from the truncation of projection data it has been proven that there exists no convolution backprojection algorithm for this type of collimator so a complicated interpolation between two nonparallel projection rays is necessary for existing algorithms the interpolation may generate blurring and artifacts in the reconstructed images based on a harmonic decomposition technique and the translation property of fourier series a semifrequency resampling technique is proposed to avoid the above mentioned interpolations by this technique the harmonic decomposition of projection data for spatially varying focal length fan beam collimators has the same form as that for parallel beam collimators in the semifrequency domain fourier transform with respect to angular variables only an alternative version of the inverse cormack transform is then proposed to reconstruct the images the derived reconstruction algorithm was implemented in a pentium ii 266 pc computer numerical simulations demonstrated its efficiency 3 s for 128 spl times 128 reconstruction arrays and its robust performance compared to the existing algorithms

graphical methods for efficient likelihood inference in gaussian covariance models in graphical modelling a bi directed graph encodes marginal independences among random variables that are identified with the vertices of the graph we show how to transform a bi directed graph into a maximal ancestral graph that i represents the same independence structure as the original bi directed graph and ii minimizes the number of arrowheads among all ancestral graphs satisfying i here the number of arrowheads of an ancestral graph is the number of directed edges plus twice the number of bi directed edges in gaussian models this construction can be used for more efficient iterative maximization of the likelihood function and to determine when maximum likelihood estimates are equal to empirical counterparts

distinguishing cause from effect using observational data methods and benchmarks the discovery of causal relationships from purely observational data is a fundamental problem in science the most elementary form of such a causal discovery problem is to decide whether x causes y or alternatively y causes x given joint observations of two variables x y an example is to decide whether altitude causes temperature or vice versa given only joint measurements of both variables even under the simplifying assumptions of no confounding no feedback loops and no selection bias such bivariate causal discovery problems are challenging nevertheless several approaches for addressing those problems have been proposed in recent years we review two families of such methods methods based on additive noise models anms and information geometric causal inference igci we present the benchmark causeeffectpairs that consists of data for 100 different causee ffect pairs selected from 37 data sets from various domains e g meteorology biology medicine engineering economy etc and motivate our decisions regarding the ground truth causal directions of all pairs we evaluate the performance of several bivariate causal discovery methods on these real world benchmark data and in addition on artificially simulated data our empirical results on real world data indicate that certain methods are indeed able to distinguish cause from effect using only purely observational data although more benchmark data would be needed to obtain statistically significant conclusions one of the best performing methods overall is the method based on additive noise models that has originally been proposed by hoyer et al 2009 which obtains an accuracy of 63 10 and an auc of 0 74 0 05 on the real world benchmark as the main theoretical contribution of this work we prove the consistency of that method

a new criterion for confounder selection we propose a new criterion for confounder selection when the underlying causal structure is unknown and only limited knowledge is available we assume all covariates being considered are pretreatment variables and that for each covariate it is known i whether the covariate is a cause of treatment and ii whether the covariate is a cause of the outcome the causal relationships the covariates have with one another is assumed unknown we propose that control be made for any covariate that is either a cause of treatment or of the outcome or both we show that irrespective of the actual underlying causal structure if any subset of the observed covariates suffices to control for confounding then the set of covariates chosen by our criterion will also suffice we show that other commonly used criteria for confounding control do not have this property we use formal theory concerning causal diagrams to prove our result but the application of the result does not rely on familiarity with causal diagrams an investigator simply need ask is the covariate a cause of the treatment and is the covariate a cause of the outcome if the answer to either question is yes then the covariate is included for confounder control we discuss some additional covariate selection results that preserve unconfoundedness and that may be of interest when used with our criterion

estimation of a covariance matrix with zeros summary we consider estimation of the covariance matrix of a multivariate random vector under the constraint that certain covariances are zero we first present an algorithm which we call iterative conditional fitting for computing the maximum likelihood estimate of the constrained covariance matrix under the assumption of multivariate normality in contrast to previous approaches this algorithm has guaranteed convergence properties dropping the assumption of multivariate normality we show how to estimate the covariance matrix in an empirical likelihood approach these approaches are then compared via simulation and on an example of gene expression

the circular harmonic transform for spect reconstruction and boundary conditions on the fourier transform of the sinogram the circular harmonic transform cht solution of the exponential randon transform ert is applied to single photon emission computed tomography spect for uniform attenuation within a convex boundary an important special case also considered is the linear unattenuated radon transform lrt the solution is on the form of an orthogonal function expansion matched to projections that are in parallel ray geometry this property allows for efficient and accurate processing of the projections with fast fourier transform fft without interpolation or beam matching the algorithm is optimized by the use of boundary conditions on the 2 d fourier transform of the sinogram these boundary conditions imply that the signal energy of the sinogram is concentrated in well defined sectors in transform space the angle defining the sectors depends in a direct way on the radius of the field view these results are also obtained for fan beam geometry and the linear radon transform the fourier chebyshev transform of the sinogram to demonstrate that the boundary conditions are a more general property of the radon transform and a not a property unique to rectangular coordinates

half trek criterion for generic identifiability of linear structural equation models a linear structural equation model relates random variables of interest and corresponding gaussian noise terms via a linear equation system each such model can be represented by a mixed graph in which directed edges encode the linear equations and bidirected edges indicate possible correlations among noise terms we study parameter identifiability in these models that is we ask for conditions that ensure that the edge coefficients and correlations appearing in a linear structural equation model can be uniquely recovered from the covariance matrix of the associated distribution we treat the case of generic identifiability where unique recovery is possible for almost every choice of parameters we give a new graphical condition that is sufficient for generic identifiability and can be verified in time that is polynomial in the size of the graph it improves criteria from prior work and does not require the directed part of the graph to be acyclic we also develop a related necessary condition and examine the gap between sufficient and necessary conditions through simulations on graphs with 25 or 50 nodes as well as exhaustive algebraic computations for graphs with up to five nodes

detection of unknown signals in arbitrary noise we devise a simple method for detecting signals of unknown form buried in any noise including heavy tailed the method centers on signal noise decomposition in rank and time only stationary white noise generates data with a jointly uniform rank time probability distribution u 1 n u 1 n for n data points in a time series signals of any kind distort this uniformity such distortions are captured by rank time cumulative distributions permitting all purpose efficient detection even for single time series and noise of infinite variance

constraint based causal discovery from multiple interventions over overlapping variable sets scientific practice typically involves repeatedly studying a system each time trying to unravel a different perspective in each study the scientist may take measurements under different experimental conditions interventions manipulations perturbations and measure different sets of quantities variables the result is a collection of heterogeneous data sets coming from different data distributions in this work we present algorithm combine which accepts a collection of data sets over overlapping variable sets under different experimental conditions combine then outputs a summary of all causal models indicating the invariant and variant structural characteristics of all models that simultaneously fit all of the input data sets combine converts estimated dependencies and independencies in the data into path constraints on the data generating causal model and encodes them as a sat instance the algorithm is sound and complete in the sample limit to account for conflicting constraints arising from statistical errors we introduce a general method for sorting constraints in order of confidence computed as a function of their corresponding p values in our empirical evaluation combine outperforms in terms of efficiency the only pre existing similar algorithm the latter additionally admits feedback cycles but does not admit conflicting constraints which hinders the applicability on real data as a proof of concept combine is employed to co analyze 4 real mass cytometry data sets measuring phosphorylated protein concentrations of overlapping protein sets under 3 different interventions

on the completeness of orientation rules for causal discovery in the presence of latent confounders and selection bias causal discovery becomes especially challenging when the possibility of latent confounding and or selection bias is not assumed away for this task ancestral graph models are particularly useful in that they can represent the presence of latent confounding and selection effect without explicitly invoking unobserved variables based on the machinery of ancestral graphs there is a provably sound causal discovery algorithm known as the fci algorithm that allows the possibility of latent confounders and selection bias however the orientation rules used in the algorithm are not complete in this paper we provide additional orientation rules augmented by which the fci algorithm is shown to be complete in the sense that it can under standard assumptions discover all aspects of the causal structure that are uniquely determined by facts of probabilistic dependence and independence the result is useful for developing any causal discovery and reasoning system based on ancestral graph models

marginalization and conditioning for lwf chain graphs supported by grant fa9550 12 1 0392 from the u s air force office of scientific research afosr and the defense advanced research projects agency darpa

markov equivalence for ancestral graphs ancestral graphs can encode conditional independence relations that arise in directed acyclic graph dag models with latent and selection variables however for any ancestral graph there may be several other graphs to which it is markov equivalent we state and prove conditions under which two maximal ancestral graphs are markov equivalent to each other thereby extending analogous results for dags given by other authors these conditions lead to an algorithm for determining markov equivalence that runs in time that is polynomial in the number of vertices in the graph

a unified reconstruction framework for both parallel beam and variable focal length fan beam collimators by a cormack type inversion of exponential radon transform a variety of inversions of exponential radon transform has been derived based on the circular harmonic transform in fourier space by several research groups however these inversions cannot be directly applied to deal with the reconstruction for fan beam or variable focal length fan beam collimator geometries in single photon emission computed tomography spect in this paper the authors derived a cormack type inversion of the exponential radon transform by employing the circular harmonic transform directly in the projection space and the image space instead of the fourier space thus a unified reconstruction framework is established for parallel fan and variable focal length fan beam collimator geometries compared to many existing algorithms the presented one greatly mitigates the difficulty of image reconstruction due to the complicated collimator geometry and significantly reduces the computational burden of the special functions such as chebyshev or bessel functions by the well established fast fourier transform fft the authors algorithm is very efficient as demonstrated by several numerical simulations

causal reasoning with ancestral graphs causal reasoning is primarily concerned with what would happen to a system under external interventions in particular we are often interested in predicting the probability distribution of some random variables that would result if some other variables were forced to take certain values one prominent approach to tackling this problem is based on causal bayesian networks using directed acyclic graphs as causal diagrams to relate post intervention probabilities to pre intervention probabilities that are estimable from observational data however such causal diagrams are seldom fully testable given observational data in consequence many causal discovery algorithms based on data mining can only output an equivalence class of causal diagrams rather than a single one this paper is concerned with causal reasoning given an equivalence class of causal diagrams represented by a partial ancestral graph we present two main results the first result extends pearl 1995 s celebrated do calculus to the context of ancestral graphs in the second result we focus on a key component of pearl s calculus the property of invariance under interventions and give stronger graphical conditions for this property than those implied by the first result the second result also improves the earlier similar results due to spirtes et al 1993

a numerically stable circular harmonic reconstruction algorithm a numerically stable algorithm based on the orthogonal function solution of the 2 d radon transform is given this form of image reconstruction is often called circular harmonic transform cht reconstruction series of zernike polynomials are evaluated by solving a system of nonhomogeneous difference equations the solution is based on a recursion formula for zernike polynomials we show that the method is numerically stable for the paraxial case this algorithm overcomes the loss of spatial and contrast resolution associated with cht algorithms when compared with ramp filtered back projection it is more resistant to ringing and it is not subject to systematic errors that seem to be caused by the discretization of the back projection operator the execution time is proportional to 9n 3 32 for n 2 points on a polar grid so that it is comparable to back projection methods in execution time

linear detection of a weak signal in additive cauchy noise the detection of a weak signal in additive cauchy noise is of great importance in many applications a locally optimum detector lod exists for such a scenario however it is non linear in nature in general implementation of non linear detectors is difficult in practice and linear detectors with good properties such as high asymptotic relative efficiency are with respect to the lod are often desirable in this paper we propose a linear detector for a weak signal in additive cauchy noise the proposed test statistic is a linear combination of order statistics for the special case of a constant signal in additive cauchy noise we prove the asymptotic normality of the trimmed linear detector and show that the are of the trimmed linear detector with respect to the lod is unity extensive simulation results are provided to demonstrate that the loss in the performance of the linear detector is very small compared with the non linear lod we also discuss the hardware complexities of the lod and the linear detector and demonstrate the advantages of the linear detector over the lod in terms of hardware implementation

discussion latent variable graphical model selection via convex optimization we want to congratulate the authors for a thought provoking and very interesting paper sparse modeling of the concentration matrix has enjoyed popularity in recent years it has been framed as a computationally convenient convex 1constrained estimation problem in yuan and lin 2007 and can be applied readily to higher dimensional problems the authors argue we think correctly that the sparsity of the concentration matrix is for many applications more plausible after the effects of a few latent variables have been removed the most attractive point about their method is surely that it is formulated as a convex optimization problem latent variable fitting and sparse graphical modeling of the conditional distribution of the observed variables can then be obtained through a single fitting procedure practical aspects the method deserves wide adoption but this will only be realistic if software is made available for example as an r package not many users will go to the trouble of implementing the method on their own so we will strongly urge the authors to do so an imputation method in the absence of readily available software it is worth thinking whether the proposed fitting procedure can be approximated by methods involving known and well tested computational techniques the concentration matrix of observed and hidden variables is k ko koh k ho kh where we have deviated from the notation in the paper by omitting the asterisk the proposed estimator sn ko of ko was defined as

stratified exponential families graphical models and model selection we describe a hierarchy of exponential families which is useful for distinguishing types of graphical models undirected graphical models with no hidden variables are linear exponential families lefs directed acyclic graphical dag models and chain graphs with no hidden variables including dag models with several families of local distributions are curved exponential families cefs graphical models with hidden variables are what we term stratified exponential families sefs a sef is a finite union of cefs of various dimensions satisfying some regularity conditions we also show that this hierarchy of exponential families is noncollapsing with respect to graphical models by providing a graphical model which is a cef but not a lef and a graphical model that is a sef but not a cef finally we show how to compute the dimension of a stratified exponential family these results are discussed in the context of model selection of graphical models

towards integrative causal analysis of heterogeneous data sets and studies we present methods able to predict the presence and strength of conditional and unconditional dependencies correlations between two variables y and z never jointly measured on the same samples based on multiple data sets measuring a set of common variables the algorithms are specializations of prior work on learning causal structures from overlapping variable sets this problem has also been addressed in the field of statistical matching the proposed methods are applied to a wide range of domains and are shown to accurately predict the presence of thousands of dependencies compared against prototypical statistical matching algorithms and within the scope of our experiments the proposed algorithms make predictions that are better correlated with the sample estimates of the unknown parameters on test data this is particularly the case when the number of commonly measured variables is low the enabling idea behind the methods is to induce one or all causal models that are simultaneously consistent with fit all available data sets and prior knowledge and reason with them this allows constraints stemming from causal assumptions e g causal markov condition faithfulness to propagate several methods have been developed based on this idea for which we propose the unifying name integrative causal analysis inca a contrived example is presented demonstrating the theoretical potential to develop more general methods for co analyzing heterogeneous data sets the computational experiments with the novel methods provide evidence that causally inspired assumptions such as faithfulness often hold to a good degree of approximation in many real systems and could be exploited for statistical inference code scripts and data are available at www mensxmachina org

quasi bandlimited properties of radon transforms and their implications for increasing angular sampling densities the n dimensional n d radon transform which forms the mathematical basis for a broad variety of tomographic imaging applications can be viewed as an n d function in n d sinogram space accurate reconstruction of continuous or discrete tomographic images requires full knowledge of the radon transform in the corresponding n d sinogram space in practice however one can have only a finite set of discrete samples of the radon transform in the sinogram space one often derives the desired full knowledge of the radon transform from its discrete samples by invoking various interpolation algorithms according to the wittaker shannon sampling theorem a necessary condition for a full and unique recovery of the radon transform from its discrete samples is that the radon transform itself be bandlimited therefore it is necessary to analyze the bandlimited properties of the radon transform in this work the authors analyze explicitly the bandlimited properties of the radon transform and show that the radon transform is mathematically quasi bandlimited or essentially bandlimited in two quantitative senses and can essentially be treated as bandlimited in practice the quasi bandlimited properties can be used for increasing the angular sampling density of the radon transform

graphical gaussian modelling of multivariate time series with latent variable in time series analysis inference about causeeffect relationships among multiple times series is commonly based on the concept of granger causality which exploits temporal structure to achieve causal ordering of dependent variables one major problem in the application of granger causality for the identification of causal relationships is the possible presence of latent variables that affect the measured components and thus lead to so called spurious causalities in this paper we describe a new graphical approach for modelling the dependence structure of multivariate stationary time series that are affected by latent variables to this end we introduce dynamic maximal ancestral graphs dmags in which each time series is represented by a single vertex for gaussian processes this approach leads to vector autoregressive models with errors that are not independent but correlated according to the dashed edges in the graph we discuss identifiability of the parameters and show that these models can be viewed as graphical arma models that satisfy the granger causality restrictions encoded by the associated dynamic maximal ancestral graph

joint response graphs and separation induced by triangular systems summary we consider joint probability distributions generated recursively in terms of univariate conditional distributions satisfying conditional independence restrictions the independences are captured by missing edges in a directed graph a matrix form of such a graph called the generating edge matrix is triangular so the distributions that are generated over such graphs are called triangular systems we study consequences of triangular systems after grouping or reordering of the variables for analyses as chain graph models i e for alternative recursive factorizations of the given density using joint conditional distributions for this we introduce families of linear triangular equations which do not require assumptions of distributional form the strength of the associations that are implied by such linear families for chain graph models is derived the edge matrices of chain graphs that are implied by any triangular system are obtained by appropriately transforming the generating edge matrix it is shown how induced independences and dependences can be studied by graphs by edge matrix calculations and via the properties of densities some ways of using the results are illustrated

a unified analysis of exact methods of inverting the 2 d exponential radon transform with implications for noise control in spect exact methods of inverting the two dimensional 2 d exponential radon transform have been proposed by bellini et al 1979 and by inouye et al 1989 both of whom worked in the spatial frequency domain to estimate the 2 d fourier transform of the unattenuated sinogram by hawkins et al 1988 who worked with circularly harmonic bessel transforms and by tretiak and metz 1980 who followed filtering of appropriately modified projections by exponentially weighted backprojection with perfect sampling all four of these methods are exact in the absence of projection data noise but empirical studies have shown that they propagate noise differently and no underlying theoretical relationship among the methods has been evident here an analysis of the 2 d fourier transform of the modified sinogram reveals that all previously proposed linear methods can be interpreted as special cases of a broad class of methods and that each method in the class can be implemented in principle by any one of four distinct techniques moreover the analysis suggests a new member of the class that is predicted to have noise properties better than those of previously proposed members

minimal sufficient causation and directed acyclic graphs notions of minimal sufficient causation are incorporated within the directed acyclic graph causal framework doing so allows for the graphical representation of sufficient causes and minimal sufficient causes on causal directed acyclic graphs while maintaining all of the properties of causal directed acyclic graphs this in turn provides a clear theoretical link between two major conceptualizations of causality one counterfactual based and the other based on a more mechanistic understanding of causation the theory developed can be used to draw conclusions about the sign of the conditional covariances among variables

on the choice of a model to fit data from an exponential family let x 1 cdots x n be iid observations coming from an exponential family the problem of interest is this given a finite number of models m j smoothly curved manifolds in mathbb r k choose the best model to fit the observations with some penalty for choosing models with dimensions which are too large a result of schwarz is made more specific and is extended to the case where the models are curved manifolds if s y n j is up to a constant c n independent of the model the log of the posterior probability of the j th model where the sample mean y n 1 n sum n i 1 x i has been replaced by y schwarz suggested an asymptotic expansion of s y n j whose leading terms are gamma y n j n sup psi in m j cap theta y psi b psi frac 1 2 k j log n in the case where the models are affine subspaces of mathbb r k we establish a similar asymptotic expansion including the next term with uniform bounds for y in a compact neighborhood of nabla b theta where theta is the true value of the parameter we suggest a criterion for the choice of the best model that consists of maximizing the three leading terms in the expansion s y n j we show that the criterion gives the correct model with probabilities p n theta rightarrow 1 as n rightarrow infty

computing maximum likelihood estimates in recursive linear models with correlated errors in recursive linear models the multivariate normal joint distribution of all variables exhibits a dependence structure induced by a recursive or acyclic system of linear structural equations these linear models have a long tradition and appear in seemingly unrelated regressions structural equation modelling and approaches to causal inference they are also related to gaussian graphical models via a classical representation known as a path diagram despite the models long history a number of problems remain open in this paper we address the problem of computing maximum likelihood estimates in the subclass of bow free recursive linear models the term bow free refers to the condition that the errors for variables i and j be uncorrelated if variable i occurs in the structural equation for variable j we introduce a new algorithm termed residual iterative conditional fitting ricf that can be implemented using only least squares computations in contrast to existing algorithms ricf has clear convergence properties and yields exact maximum likelihood estimates after the first iteration whenever the mle is available in closed form

causal discovery with continuous additive noise models we consider the problem of learning causal directed acyclic graphs from an observational joint distribution one can use these graphs to predict the outcome of interventional experiments from which data are often not available we show that if the observational distribution follows a structural equation model with an additive noise structure the directed acyclic graph becomes identifiable from the distribution under mild conditions this constitutes an interesting alternative to traditional methods that assume faithfulness and identify only the markov equivalence class of the graph thus leaving some edges undirected we provide practical algorithms for finitely many samples resit regression with subsequent independence test and two methods based on an independence score we prove that resit is correct in the population setting and provide an empirical evaluation

markovian acyclic directed mixed graphs for discrete data acyclic directed mixed graphs admgs are graphs that contain directed rightarrow and bidirected leftrightarrow edges subject to the constraint that there are no cycles of directed edges such graphs may be used to represent the conditional independence structure induced by a dag model containing hidden variables on its observed margin the markovian model associated with an admg is simply the set of distributions obeying the global markov property given via a simple path criterion m separation we first present a factorization criterion characterizing the markovian model that generalizes the well known recursive factorization for dags for the case of finite discrete random variables we also provide a parameterization of the model in terms of simple conditional probabilities and characterize its variation dependence we show that the induced models are smooth consequently markovian admg models for discrete variables are curved exponential families of distributions

measurement bias and effect restoration in causal inference summary this paper highlights several areas where graphical techniques can be harnessed to address the problem of measurement errors in causal inference in particular it discusses the control of unmeasured confounders in parametric and nonparametric models and the computational problem of obtaining bias free effect estimates in such models we derive new conditions under which causal effects can be restored by observing proxy variables of unmeasured confounders with without external studies

ancestral graph markov models this paper introduces a class of graphical independence models that is closed under marginalization and conditioning but that contains all dag independence models this class of graphs called maximal ancestral graphs has two attractive features there is at most one edge between each pair of vertices every missing edge corresponds to an independence relation these features lead to a simple parameterization of the corresponding set of distributions in the gaussian case

standard quantum limit and heisenberg limit in function estimation unlike well established parameter estimation function estimation faces conceptual and mathematical difficulties despite its enormous potential utility we establish the fundamental error bounds on function estimation in quantum metrology for a spatially varying phase operator where various degrees of smooth functions are considered the error bounds are identified in the cases of the absence and the presence of interparticle entanglement which correspond to the standard quantum limit and the heisenberg limit respectively notably these error bounds can be reached by either position localized states or wave number localized ones in fact we show that these error bounds are theoretically optimal for any type of probe states indicating that quantum metrology on functions is also subject to the nyquist shannon sampling theorem even if classical detection is replaced by quantum measurement

learning high dimensional directed acyclic graphs with latent and selection variables we consider the problem of learning causal information between random variables in directed acyclic graphs dags when allowing arbitrarily many latent and selection variables the fci fast causal inference algorithm has been explicitly designed to infer conditional independence and causal information in such settings however fci is computationally infeasible for large graphs we therefore propose the new rfci algorithm which is much faster than fci in some situations the output of rfci is slightly less informative in particular with respect to conditional independence information however we prove that any causal information in the output of rfci is correct in the asymptotic limit we also define a class of graphs on which the outputs of fci and rfci are identical we prove consistency of fci and rfci in sparse high dimensional settings and demonstrate in simulations that the estimation performances of the algorithms are very similar all software is implemented in the r package pcalg

cumulative distribution networks and the derivative sum product algorithm models and inference for cumulative distribution functions on graphs we present a class of graphical models for directly representing the joint cumulative distribution function cdf of many random variables called cumulative distribution networks cdns unlike graphs for probability density and mass functions for cdfs the marginal probabilities for any subset of variables are obtained by computing limits of functions in the model and conditional probabilities correspond to computing mixed derivatives we will show that the conditional independence properties in a cdn are distinct from the conditional independence properties of directed undirected and factor graphs but include the conditional independence properties of bi directed graphs in order to perform inference in such models we describe the derivative sum product dsp message passing algorithm in which messages correspond to derivatives of the joint cdf we will then apply cdns to the problem of learning to rank players in multiplayer team based games and suggest several future directions for research

multimodality of the likelihood in the bivariate seemingly unrelated regressions model summary we analyse the simplest two equation seemingly unrelated regressions model and demonstrate that its likelihood may have up to five stationary points and thus there may be up to three local modes consequently the estimates obtained via iterative estimation methods may depend on starting values we further show that the probability of multi modality vanishes asymptotically monte carlo simulations suggest that multimodality rarely occurs if the seemingly unrelated regressions model is true but can become more frequent if the model is misspecified the existence of multimodality in the likelihood for seemingly unrelated regressions models contradicts several claims in the literature

estimating high dimensional intervention effects from observational data we assume that we have observational data generated from an unknown underlying directed acyclic graph dag model a dag is typically not identifiable from observational data but it is possible to consistently estimate the equivalence class of a dag moreover for any given dag causal effects can be estimated using intervention calculus in this paper we combine these two parts for each dag in the estimated equivalence class we use intervention calculus to estimate the causal effects of the covariates on the response this yields a collection of estimated causal effects for each covariate we show that the distinct values in this set can be consistently estimated by an algorithm that uses only local information of the graph this local approach is computationally fast and feasible in high dimensional problems we propose to use summary measures of the set of possible causal effects to determine variable importance in particular we use the minimum absolute value of this set since that is a lower bound on the size of the causal effect we demonstrate the merits of our methods in a simulation study and on a data set about riboflavin production

covariate selection for the nonparametric estimation of an average treatment effect observational studies in which the effect of a nonrandomized treatment on an outcome of interest is estimated are common in domains such as labour economics and epidemiology such studies often rel

on the causal interpretation of acyclic mixed graphs under multivariate normality abstract in multivariate statistics acyclic mixed graphs with directed and bidirected edges are widely used for compact representation of dependence structures that can arise in the presence of hidden i e latent or unobserved variables indeed under multivariate normality every mixed graph corresponds to a set of covariance matrices that contains as a full dimensional subset the covariance matrices associated with a causally interpretable acyclic digraph this digraph generally has some of its nodes corresponding to hidden variables we seek to clarify for which mixed graphs there exists an acyclic digraph whose hidden variable model coincides with the mixed graph model restricting to the tractable setting of chain graphs and multivariate normality we show that decomposability of the bidirected part of the chain graph is necessary and sufficient for equality between the mixed graph model and some hidden variable model given by an acyclic digraph

evaluation of an efficient compensation method for quantitative fan beam brain spect reconstruction fan beam collimators are designed to improve the system sensitivity and resolution for imaging small objects such as the human brain and breasts in single photon emission computed tomography spect many reconstruction algorithms have been studied and applied to this geometry to deal with every kind of degradation factor this work presents a new reconstruction approach for spect with circular orbit which demonstrated good performance in terms of both accuracy and efficiency the new approach compensates for degradation factors including noise scatter attenuation and spatially variant detector response its uniform attenuation approximation strategy avoids the additional transmission scan for the brain attenuation map hence reducing the patient radiation dose and furthermore simplifying the imaging procedure we evaluate and compare this new approach with the well established ordered subset expectation maximization iterative method using monte carlo simulations we perform quantitative analysis with regional bias variance receiver operating characteristics and hotelling trace studies for both methods the results demonstrate that our reconstruction strategy has comparable performance with a significant reduction of computing time

likelihood ratio tests and singularities many statistical hypotheses can be formulated in terms of polynomial equalities and inequalities in the unknown parameters and thus correspond to semi algebraic subsets of the parameter space we consider large sample asymptotics for the likelihood ratio test of such hypotheses in models that satisfy standard probabilistic regularity conditions we show that the assumptions of chernoff s theorem hold for semi algebraic sets such that the asymptotics are determined by the tangent cone at the true parameter point at boundary points or singularities the tangent cone need not be a linear space and limiting distributions other than chi square distributions may arise while boundary points often lead to mixtures of chi square distributions singularities give rise to nonstandard limits we demonstrate that minima of chi square random variables are important for locally identifiable models and in a study of the factor analysis model with one factor we reveal connections to eigenvalues of wishart matrices

marginalization and collapsibility in graphical interaction models the behaviour of a graphical interaction model under marginalization is discussed a graphical interaction model is called collapsible onto a set of variables if the class of marginal distributions is the same as that implied by the related subgraph the necessary and sufficient condition for collapsibility is found and it is shown that collapsibility is equivalent to a range of other important statistical properties of the model

model selection for gaussian concentration graphs gga multivariate gaussian graphical markov model for an undirected graph g also called a covariance selection model or concentration graph model is defined in terms of the markov properties i e conditional independences associated with g which in turn are equivalent to specified zeros among the set of pairwise partial correlation coefficients by means of fisher s z transformation and sidak s correlation inequality conservative simultaneous confidence intervals for the entire set of partial correlations can be obtained leading to a simple method for model selection that controls the overall error rate for incorrect edge inclusion the simultaneous p values corresponding to the partial correlations are partitioned into three disjoint sets a significant set s an indeterminate set i and a nonsignificant set n our model selection method selects two graphs a graph g si whose edges correspond to the set s i and a more conservative graph g s whose edges correspond to s only similar considerations apply to covariance graph models which are defined in terms of marginal independence rather than conditional independence the method is applied to some well known examples and to simulated data