ultra scalable and efficient methods for hybrid observational and experimental local causal pathway discovery discovery of causal relations from data is a fundamental objective of several scientific disciplines most causal discovery algorithms that use observational data can infer causality only up to a statistical equivalency class thus leaving many causal relations undetermined in general complete identification of causal relations requires experimentation to augment discoveries from observational data this has led to the recent development of several methods for active learning of causal networks that utilize both observational and experimental data in order to discover causal networks in this work we focus on the problem of discovering local causal pathways that contain only direct causes and direct effects of the target variable of interest and propose new discovery methods that aim to minimize the number of required experiments relax common sufficient discovery assumptions in order to increase discovery accuracy and scale to high dimensional data with thousands of variables we conduct a comprehensive evaluation of new and existing methods with data of dimensionality up to 1 000 000 variables we use both artificially simulated networks and in silico gene transcriptional networks that model the characteristics of real gene expression data

constrained likelihood for reconstructing a directed acyclic gaussian graph directed acyclic graphs are widely used to describe directional pairwise relations such relations are estimated by reconstructing a directed acyclic graph s structure which is challenging when the ordering of nodes of the graph is unknown in such a situation existing methods such as the neighbourhood and search and score methods have high estimation errors or computational complexities especially when a local or sequential approach is used to enumerate edge directions by testing or optimizing a criterion locally as a local method may break down even for moderately sized graphs we propose a novel approach to simultaneously identifying all estimable directed edges and model parameters using constrained maximum likelihood with nonconvex constraints we develop a constraint reduction method that constructs a set of active constraints from super exponentially many constraints this coupled with an alternating direction method of multipliers and a difference convex method permits efficient computation for large graph learning we show that the proposed method consistently reconstructs identifiable directions of the true graph and achieves the optimal performance in terms of parameter estimation numerically the method compares favourably with competitors a protein network is analysed to demonstrate that the proposed method can make a difference in identifying the network s structure

learning equivalence classes of bayesian network structures two bayesian network structures are said to be equivalent if the set of distributions that can be represented with one of those structures is identical to the set of distributions that can be represented with the other many scoring criteria that are used to learn bayesian network structures from data are score equivalent that is these criteria do not distinguish among networks that are equivalent in this paper we consider using a score equivalent criterion in conjunction with a heuristic search algorithm to perform model selection or model averaging we argue that it is often appropriate to search among equivalence classes of network structures as opposed to the more common approach of searching among individual bayesian network structures we describe a convenient graphical representation for an equivalence class of structures and introduce a set of operators that can be applied to that representation by a search algorithm to move among equivalence classes we show that our equivalence class operators can be scored locally and thus share the computational efficiency of traditional operators defined for individual structures we show experimentally that a greedy model selection algorithm using our representation yields slightly higher scoring structures than the traditional approach without any additional time overhead and we argue that more sophisticated search algorithms are likely to benefit much more

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

large sample learning of bayesian networks is np hard in this paper we provide new complexity results for algorithms that learn discrete variable bayesian networks from data our results apply whenever the learning algorithm uses a scoring criterion that favors the simplest structure for which the model is able to represent the generative distribution exactly our results therefore hold whenever the learning algorithm uses a consistent scoring criterion and is applied to a sufficiently large dataset we show that identifying high scoring structures is np hard even when any combination of one or more of the following hold the generative distribution is perfect with respect to some dag containing hidden variables we are given an independence oracle we are given an inference oracle we are given an information oracle we restrict potential solutions to structures in which each node has at most k parents for all k 3 our proof relies on a new technical result that we establish in the appendices in particular we provide a method for constructing the local distributions in a bayesian network such that the resulting joint distribution is provably perfect with respect to the structure of the network

learning linear cyclic causal models with latent variables identifying cause effect relationships between variables of interest is a central problem in science given a set of experiments we describe a procedure that identifies linear models that may contain cycles and latent variables we provide a detailed description of the model family full proofs of the necessary and sufficient conditions for identifiability a search algorithm that is complete and a discussion of what can be done when the identifiability conditions are not satisfied the algorithm is comprehensively tested in simulations comparing it to competing algorithms in the literature furthermore we adapt the procedure to the problem of cellular network inference applying it to the biologically realistic data of the dreamchallenges the paper provides a full theoretical foundation for the causal discovery procedure first presented by eberhardt et al 2010 and hyttinen et al 2010

characterization of closed ideals with bounded approximate identities in commutative banach algebras complemented subspaces of the group von neumann algebras and applications

similarity degree of fourier algebras we show that for a locally compact group g amongst a class which contains amenable and small invariant neighbourhood groups its fourier algebra a g satisfies a completely bounded version pisier s similarity property with similarity degree at most 2 specifically any completely bounded homomorphism a g b h admits an invertible s in b h for which s s 1 cb2 and s 1 s extends to a representation of the c algebra c0 g this significantly improves some results due to brannan and samei 2010 5 and brannan daws and samei 2013 4 we also note that a g has completely bounded similarity degree 1 if and only if it is completely isomorphic to an operator algebra if and only if g is finite

fundamental concepts of qualitative probabilistic networks abstract graphical representations for probabilistic relationships have recently received considerable attention in ai qualitative probabilistic networks abstract from the usual numeric representations by encoding only qualitative relationships which are inequality constraints on the joint probability distribution over the variables although these constraints are insufficient to determine probabilities uniquely they are designed to justify the deduction of a class of relative likelihood conclusions that imply useful decision making properties two types of qualitative relationship are defined each a probabilistic form of monotonicity constraint over a group of variables qualitative influences describe the direction of the relationship between two variables qualitative synergies describe interactions among influences the probabilistic definitions chosen justify sound and efficient inference procedures based on graphical manipulations of the network these procedures answer queries about qualitative relationships among variables separated in the network and determine structural properties of optimal assignments to decision variables

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

lp fourier and fourier stieltjes algebras for locally compact groups abstract let g be a locally compact group and 1 p a continuous unitary representation g b h of g is an l p representation if the matrix coefficient functions s s x x lie in l p g for sufficiently many x h the l p fourier algebra a l p g is defined to be the set of matrix coefficient functions of l p representations similarly the l p fourier stieltjes algebra b l p g is defined to be the weak closure of a l p g in the fourier stieltjes algebra b g these are always ideals in the fourier stieltjes algebra containing the fourier algebra a g in this paper we investigate how these spaces reflect properties of the underlying group and study the structural properties of these algebras as an application of this theory we characterize the fourier stieltjes ideals of sl 2 r

fourier stieltjes algebras and measure algebras on compact right topological groups abstract let g be an admissible compact hausdorff right topological group that is a group with a hausdorff topology such that for each a g the map g g a is continuous and the set of a g such that the map g a g is continuous is dense in g such groups arise in the study of distal flows in this paper we study the fourier stieltjes algebra b g the linear span of the continuous positive definite functions on g we show that b g is isomorphic with the fourier stieltjes algebra of an associated compact topological group this result is then applied to obtain some geometric properties including the weak and weak fixed point properties on b g we also study some related properties on the measure algebra m g

weak amenability of fourier algebras and local synthesis of the anti diagonal abstract we show that for a connected lie group g its fourier algebra a g is weakly amenable only if g is abelian our main new idea is to show that weak amenability of a g implies that the anti diagonal g g g 1 g g is a set of local synthesis for a g g we then show that this cannot happen if g is non abelian we conclude for a locally compact group g that a g can be weakly amenable only if it contains no closed connected non abelian lie subgroups in particular for a lie group g a g is weakly amenable if and only if its connected component of the identity g e is abelian

hyper tauberian algebras and weak amenability of figa talamanca herz algebras abstract we study certain commutative regular semisimple banach algebras which we call hyper tauberian algebras we first show that they form a subclass of weakly amenable tauberian algebras then we investigate the basic and hereditary properties of them moreover we show that if a is a hyper tauberian algebra then the linear space of bounded derivations from a into any banach a bimodule is reflexive we apply these results to the figa talamanca herz algebra a p g of a locally compact group g for p 1 we show that a p g is hyper tauberian if the principal component of g is abelian finally by considering the quantization of these results we show that for any locally compact group g a p g equipped with an appropriate operator space structure is a quantized hyper tauberian algebra this in particular implies that a p g is operator weakly amenable

bounds for the loss in probability of correct classification under model based approximation in many pattern recognition classification problem the true class conditional model and class probabilities are approximated for reasons of reducing complexity and or of statistical estimation the approximated classifier is expected to have worse performance here measured by the probability of correct classification we present an analysis valid in general and easily computable formulas for estimating the degradation in probability of correct classification when compared to the optimal classifier an example of an approximation is the naive bayes classifier we show that the performance of the naive bayes depends on the degree of functional dependence between the features and labels we provide a sufficient condition for zero loss of performance too

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

bayesian network structure learning by recursive autonomy identification we propose the recursive autonomy identification rai algorithm for constraint based cb bayesian network structure learning the rai algorithm learns the structure by sequential application of conditional independence ci tests edge direction and structure decomposition into autonomous sub structures the sequence of operations is performed recursively for each autonomous sub structure while simultaneously increasing the order of the ci test while other cb algorithms d separate structures and then direct the resulted undirected graph the rai algorithm combines the two processes from the outset and along the procedure by this means and due to structure decomposition learning a structure using rai requires a smaller number of ci tests of high orders this reduces the complexity and run time of the algorithm and increases the accuracy by diminishing the curse of dimensionality when the rai algorithm learned structures from databases representing synthetic problems known networks and natural problems it demonstrated superiority with respect to computational complexity run time structural correctness and classification accuracy over the pc three phase dependency analysis optimal reinsertion greedy search greedy equivalence search sparse candidate and max min hill climbing algorithms

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

estimating high dimensional directed acyclic graphs with the pc algorithm we consider the pc algorithm spirtes et al 2000 for estimating the skeleton and equivalence class of a very high dimensional directed acyclic graph dag with corresponding gaussian distribution the pc algorithm is computationally feasible and often very fast for sparse problems with many nodes variables and it has the attractive property to automatically achieve high computational efficiency as a function of sparseness of the true underlying dag we prove uniform consistency of the algorithm for very high dimensional sparse dags where the number of nodes is allowed to quickly grow with sample size n as fast as o na for any 0 a the sparseness assumption is rather minimal requiring only that the neighborhoods in the dag are of lower order than sample size n we also demonstrate the pc algorithm for simulated data

ideals with bounded approximate identities in fourier algebras abstract we make use of the operator space structure of the fourier algebra a g of an amenable locally compact group to prove that if h is any closed subgroup of g then the ideal i h consisting of all functions in a g vanishing on h has a bounded approximate identity this result allows us to completely characterize the ideals of a g with bounded approximate identities we also show that for several classes of locally compact groups including all nilpotent groups i h has an approximate identity with norm bounded by 2 the best possible norm bound

learning causal bayesian network structures from experimental data we propose a method for the computational inference of directed acyclic graphical structures given data from experimental interventions order space markov chain monte carlo equi energy sampling importance weighting and stream based computation are combined to create a fast algorithm for learning causal bayesian network structures

learning factor graphs in polynomial time and sample complexity we study the computational and sample complexity of parameter and structure learning in graphical models our main result shows that the class of factor graphs with bounded degree can be learned in polynomial time and from a polynomial number of training examples assuming that the data is generated by a network in this class this result covers both parameter estimation for a known network structure and structure learning it implies as a corollary that we can learn factor graphs for both bayesian networks and markov networks of bounded degree in polynomial time and sample complexity importantly unlike standard maximum likelihood estimation algorithms our method does not require inference in the underlying network and so applies to networks where inference is intractable we also show that the error of our learned model degrades gracefully when the generating distribution is not a member of the target class of networks in addition to our main result we show that the sample complexity of parameter learning in graphical models has an o 1 dependence on the number of variables in the model when using the kl divergence normalized by the number of variables as the performance criterion

a note on p completely bounded homomorphisms of the figa talamanca herz algebras abstract for a locally compact group g and 1 p let a p g be the figa talamanca herz algebras which include in particular the fourier algebra of g a g p 2 it is shown that for any amenable group h a proper affine map y h g induces a p completely contractive algebra homomorphism a p g a p h by setting u u on y and u 0 off of y moreover we show that if both g and h are amenable then any p completely contractive algebra homomorphism a p g a p h is of this form these results are the analogs in the context of the figa talamanca herz algebras of the ones in the fourier algebra setting p 2 initiated by the author and continued with n spronk which in turn generalize results of p j cohen and b host from abelian group algebra setting

polyhedral aspects of score equivalence in bayesian network structure learning this paper deals with faces and facets of the family variable polytope and the characteristic imset polytope which are special polytopes used in integer linear programming approaches to statistically learn bayesian network structure a common form of linear objectives to be maximized in this area leads to the concept of score equivalence se both for linear objectives and for faces of the family variable polytope we characterize the linear space of se objectives and establish a one to one correspondence between se faces of the family variable polytope the faces of the characteristic imset polytope and standardized supermodular functions the characterization of se facets in terms of extremality of the corresponding supermodular function gives an elegant method to verify whether an inequality is se facet defining for the family variable polytope we also show that when maximizing an se objective one can eliminate linear constraints of the family variable polytope that correspond to non se facets however we show that solely considering se facets is not enough as a counter example shows one has to consider the linear inequality constraints that correspond to facets of the characteristic imset polytope despite the fact that they may not define facets in the family variable mode

the similarity problem for fourier algebras and corepresentations of group von neumann algebras let g be a locally compact group and let a g and vn g be its fourier algebra and group von neumann algebra respectively in this paper we consider the similarity problem for a g is every bounded representation of a g on a hilbert space h similar to a representation we show that the similarity problem for a g has a negative answer if and only if there is a bounded representation of a g which is not completely bounded for groups with small invariant neighborhoods i e sin groups we show that a representation a g b h is similar to a representation if and only if it is completely bounded this in particular implies that corepresentations of vn g associated to non degenerate completely bounded representations of a g are similar to unitary corepresentations we also show that if g is a sin maximally almost periodic or totally disconnected group then a representation of a g is a representation if and only if it is a complete contraction these results partially answer questions posed in effros and ruan 2003 7 and spronk 2002 25

weak and cyclic amenability for fourier algebras of connected lie groups abstract using techniques of non abelian harmonic analysis we construct an explicit non zero cyclic derivation on the fourier algebra of the real a x b group in particular this provides the first proof that this algebra is not weakly amenable using the structure theory of lie groups we deduce that the fourier algebras of connected semisimple lie groups also support non zero cyclic derivations and are likewise not weakly amenable our results complement earlier work of johnson 1994 15 plymen 2001 18 and forrest samei and spronk 2009 9 as an additional illustration of our techniques we construct an explicit non zero cyclic derivation on the fourier algebra of the reduced heisenberg group providing the first example of a connected nilpotent group whose fourier algebra is not weakly amenable

finite size lyapunov exponent for levy processes the finite size lyapunov exponent fsle is the exponential rate at which two particles separate from a distance of r to a x r a 1 and provides a measure of dispersive mixing in chaotic systems it is shown analytically that for particle trajectories governed by symmetric alpha stable levy motion the fsle is proportional to the diffusion coefficient and inversely proportional to r alpha this power law provides an easy method to determine the parameters for levy processes and hence has applications to superdiffusion in the atmospheric oceanic and terrestrial sciences

a probabilistic graphical model based stochastic input model construction abstract model reduction techniques have been widely used in modeling of high dimensional stochastic input in uncertainty quantification tasks however the probabilistic modeling of random variables projected into reduced order spaces presents a number of computational challenges due to the curse of dimensionality the underlying dependence relationships between these random variables are difficult to capture in this work a probabilistic graphical model based approach is employed to learn the dependence by running a number of conditional independence tests using observation data thus a probabilistic model of the joint pdf is obtained and the pdf is factorized into a set of conditional distributions based on the dependence structure of the variables the estimation of the joint pdf from data is then transformed to estimating conditional distributions under reduced dimensions to improve the computational efficiency a polynomial chaos expansion is further applied to represent the random field in terms of a set of standard random variables this technique is combined with both linear and nonlinear model reduction methods numerical examples are presented to demonstrate the accuracy and efficiency of the probabilistic graphical model based stochastic input models

fractional brownian motion versus the continuous time random walk a simple test for subdiffusive dynamics fractional brownian motion with hurst index less then 1 2 and continuous time random walk with heavy tailed waiting times and the corresponding fractional fokker planck equation are two different processes that lead to a subdiffusive behavior widespread in complex systems we propose a simple test based on the analysis of the so called p variations which allows distinguishing between the two models on the basis of one realization of the unknown process we apply the test to the data of golding and cox phys rev lett 96 098102 2006 describing the motion of individual fluorescently labeled mrna molecules inside live e coli cells it is found that the data does not follow heavy tailed continuous time random walk the test shows that it is likely that fractional brownian motion is the underlying process

identifiability of gaussian structural equation models with equal error variances summary we consider structural equation models in which variables can be written as a function of their parents and noise terms which are assumed to be jointly independent corresponding to each structural equation model is a directed acyclic graph describing the relationships between the variables in gaussian structural equation models with linear functions the graph can be identified from the joint distribution only up to markov equivalence classes assuming faithfulness in this work we prove full identifiability in the case where all noise variables have the same variance the directed acyclic graph can be recovered from the joint gaussian distribution our result has direct implications for causal inference if the data follow a gaussian structural equation model with equal error variances then assuming that all variables are observed the causal structure can be inferred from observational data only we propose a statistical method and an algorithm based on our theoretical findings

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

methods for causal inference from gene perturbation experiments and validation inferring causal effects from observational and interventional data is a highly desirable but ambitious goal many of the computational and statistical methods are plagued by fundamental identifiability issues instability and unreliable performance especially for large scale systems with many measured variables we present software and provide some validation of a recently developed methodology based on an invariance principle called invariant causal prediction icp the icp method quantifies confidence probabilities for inferring causal structures and thus leads to more reliable and confirmatory statements for causal relations and predictions of external intervention effects we validate the icp method and some other procedures using large scale genome wide gene perturbation experiments in saccharomyces cerevisiae the results suggest that prediction and prioritization of future experimental interventions such as gene deletions can be improved by using our statistical inference techniques

figa talamanca herz algebras for restricted inverse semigroups and clifford semigroups abstract we develop the figa talamanca herz algebras and the space of p pseudomeasures to inverse semigroups with restricted semigroup algebras let 1 p q be such that 1 p 1 q 1 we define the banach algebra of p pseudomeasures pm p s and the figa talamanca herz algebras a q s then we show that a q s pm p s we characterize pm p s and a q s for a clifford semigroup in the sense of p pseudomeasures and figa talamanca herz algebras of maximal subgroups of s respectively we also show that the character space of a q s is equal to s for a clifford semigroup s as an example of these banach algebras and restricted semigroup algebras we discuss uniformly locally finite inverse semigroups

weak amenability for fourier algebras of 1 connected nilpotent lie groups a special case of a conjecture raised by forrest and runde 2005 10 asserts that the fourier algebra of every non abelian connected lie group fails to be weakly amenable this was already known to hold in the non abelian compact cases by earlier work of johnson 1994 13 and plymen unpublished note in recent work choi and ghandehari 2014 4 the authors verified this conjecture for the real ax b group and hence by structure theory for any semisimple lie group in this paper we verify the conjecture for all 1 connected non abelian nilpotent lie groups by reducing the problem to the case of the heisenberg group as in our previous paper an explicit non zero derivation is constructed on a dense subalgebra and then shown to be bounded using harmonic analysis en route we use the known fusion rules for schrodinger representations to give a concrete realization of the dual convolution for this group as a kind of twisted operator valued convolution we also give some partial results for solvable groups which give further evidence to support the general conjecture

completely bounded homomorphisms of the fourier algebras abstract for locally compact groups g and h let a g denote the fourier algebra of g and b h the fourier stieltjes algebra of h any continuous piecewise affine map y h g where y is an element of the open coset ring induces a completely bounded homomorphism a g b h by setting u u on y and u 0 off of y we show that if g is amenable then any completely bounded homomorphism a g b h is of this form and this theorem fails if g contains a discrete nonabelian free group our result generalises results of cohen amer j math 82 1960 213 226 host bull soc math france 1986 114 and of the first author j funct anal 2004 213 we also obtain a description of all the idempotents in the fourier stieltjes algebras which are contractive or positive definite

random renormalization group operators applied to stochastic dynamics let x t be a fixed point the renormalization group operator rgo r p r x t x rt r p scaling laws for the probability density mean first passage times finite size lyapunov exponents of such fixed points are reviewed in anticipation of more general results a generalized rgo mathcal r p n where p is a random variable is introduced scaling laws associated with these random rgos rrgos are demonstrated numerically and applied to subdiffusion in bacterial cytoplasm and a process modeling the transition from subdiffusion to classical diffusion the scaling laws for the rrgo are not simple power laws but are a weighted average of power laws the weighting used in the scaling laws can be determined adaptively via bayes theorem

mean first passage time for anomalous diffusion when the random force acting on a particle diffusing in an interval 0 l and subjected to a constant external force is a gaussian white noise the brownian mean squared displacement is described by the seminal relation x 2 2 mathrm dt ensuremath gamma with ensuremath gamma 1 however for more complicated random forces the diffusion may be slower ensuremath gamma l1 subdiffusion or faster ensuremath gamma g1 superdiffusion than the normal diffusion for both these cases we calculated the mean free passage time mfpt the time needed to reach one of the traps at boundaries the simple formulas for the different diffusive regimes are compared quantitatively for the simplest case of the absence of an external field and for an initial position in the middle of the interval it turns out that the mfpt s for anomalous diffusion can be both larger or smaller than that for normal diffusion depending on the values of the length of the interval and the diffusion coefficient moreover the mfpt can show nonmonotonic changes with the degree of departure from normal diffusion

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

controlling the false discovery rate of the association causality structure learned with the pc algorithm in real world applications graphical statistical models are not only a tool for operations such as classification or prediction but usually the network structures of the models themselves are also of great interest e g in modeling brain connectivity the false discovery rate fdr the expected ratio of falsely claimed connections to all those claimed is often a reasonable error rate criterion in these applications however current learning algorithms for graphical models have not been adequately adapted to the concerns of the fdr the traditional practice of controlling the type i error rate and the type ii error rate under a conventional level does not necessarily keep the fdr low especially in the case of sparse networks in this paper we propose embedding an fdr control procedure into the pc algorithm to curb the fdr of the skeleton of the learned graph we prove that the proposed method can control the fdr under a user specified level at the limit of large sample sizes in the cases of moderate sample size about several hundred empirical experiments show that the method is still able to control the fdr under the user specified level and a heuristic modification of the method is able to control the fdr more accurately around the user specified level the proposed method is applicable to any models for which statistical tests of conditional independence are available such as discrete models and gaussian models

optimal structure identification with greedy search in this paper we prove the so called meek conjecture in particular we show that if a dag h is an independence map of another dag g then there exists a finite sequence of edge additions and covered edge reversals in g such that 1 after each edge modification h remains an independence map of g and 2 after all modifications g h as shown by meek 1997 this result has an important consequence for bayesian approaches to learning bayesian networks from data in the limit of large sample size there exists a two phase greedy search algorithm that when applied to a particular sparsely connected search space provably identifies a perfect map of the generative distribution if that perfect map is a dag we provide a new implementation of the search space using equivalence classes as states for which all operators used in the greedy search can be scored efficiently using local functions of the nodes in the domain finally using both synthetic and real world datasets we demonstrate that the two phase greedy approach leads to good solutions when learning with finite sample sizes

introduction to causal inference the goal of many sciences is to understand the mechanisms by which variables came to take on the values they have that is to find a generative model and to predict what the values of those variables would be if the naturally occurring mechanisms were subject to outside manipulations the past 30 years has seen a number of conceptual developments that are partial solutions to the problem of causal inference from observational sample data or a mixture of observational sample and experimental data particularly in the area of graphical causal modeling however in many domains problems such as the large numbers of variables small samples sizes and possible presence of unmeasured causes remain serious impediments to practical applications of these developments the articles in the special topic on causality address these and other problems in applying graphical causal modeling algorithms this introduction to the special topic on causality provides a brief introduction to graphical causal modeling places the articles in a broader context and describes the differences between causal inference and ordinary machine learning classification and prediction problems

six problems for causal inference from fmri neuroimaging e g fmri data are increasingly used to attempt to identify not only brain regions of interest rois that are especially active during perception cognition and action but also the qualitative causal relations among activity in these regions known as effective connectivity friston 1994 previous investigations and anatomical and physiological knowledge may somewhat constrain the possible hypotheses but there often remains a vast space of possible causal structures to find actual effective connectivity relations search methods must accommodate indirect measurements of nonlinear time series dependencies feedback multiple subjects possibly varying in identified regions of interest and unknown possible location dependent variations in bold response delays we describe combinations of procedures that under these conditions find feed forward sub structure characteristic of a group of subjects the method is illustrated with an empirical data set and confirmed with simulations of time series of non linear randomly generated effective connectivities with feedback subject to random differences of bold delays with regions of interest missing at random for some subjects measured with noise approximating the signal to noise ratio of the empirical data

mind change optimal learning of bayes net structure from dependency and independency data this paper analyzes the problem of learning the structure of a bayes net in the theoretical framework of gold s learning paradigm bayes nets are one of the most prominent formalisms for knowledge representation and probabilistic and causal reasoning we follow constraint based approaches to learning bayes net structure where learning is based on observed conditional dependencies and independencies between variables of interest e g the data are of the form x is dependent on y given any assignment to variables s or of the form x is independent of y given any assignment to variables s applying learning criteria in this model leads to the following results 1 the mind change complexity of identifying a bayes net graph over variables v from either dependency data or from independency data are v 2 the maximum number of edges 2 there is a unique fastest mind change optimal bayes net learner for either data type convergence speed is evaluated using gold s dominance notion of uniformly faster convergence for dependency data the optimal learner conjectures a graph if it is the unique bayes net pattern that satisfies the observed dependencies with a minimum number of edges and outputs no guess otherwise for independency data the optimal learner conjectures a graph if it is the unique bayes net pattern that satisfies the observed dependencies with a maximum number of edges and outputs no guess otherwise we investigate the complexity of computing the output of the fastest mind change optimal learner for either data type and show that each of these two problems is np hard assuming p rp to our knowledge these are the first np hardness results concerning the existence of a uniquely optimal bayes net structure

completely bounded disjointness preserving operators between fourier algebras in this paper we characterize surjective completely bounded disjointness preserving linear operators on fourier algebras of locally compact amenable groups we show that such operators are given by weighted homomorphisms induced by piecewise affine proper maps

efficient structure learning of bayesian networks using constraints this paper addresses the problem of learning bayesian network structures from data based on score functions that are decomposable it describes properties that strongly reduce the time and memory costs of many known methods without losing global optimality guarantees these properties are derived for different score criteria such as minimum description length or bayesian information criterion akaike information criterion and bayesian dirichlet criterion then a branch and bound algorithm is presented that integrates structural constraints with data in a way to guarantee global optimality as an example structural constraints are used to map the problem of structure learning in dynamic bayesian networks into a corresponding augmented bayesian network finally we show empirically the benefits of using the properties with state of the art methods and with the new algorithm which is able to handle larger data sets than before

exact bayesian structure discovery in bayesian networks learning a bayesian network structure from data is a well motivated but computationally hard task we present an algorithm that computes the exact posterior probability of a subnetwork e g a directed edge a modified version of the algorithm finds one of the most probable network structures this algorithm runs in time o n 2n nk 1c m where n is the number of network variables k is a constant maximum in degree and c m is the cost of computing a single local marginal conditional likelihood for m data instances this is the first algorithm with less than super exponential complexity with respect to n exact computation allows us to tackle complex cases where existing monte carlo methods and local search procedures potentially fail we show that also in domains with a large number of variables exact computation is feasible given suitable a priori restrictions on the structures combining exact and inexact methods is also possible we demonstrate the applicability of the presented algorithm on four synthetic data sets with 17 22 37 and 100 variables

stable non gaussian random processes stochastic models with infinite variance stable random variables on the real line multivariate stable distributions stable stochastic integrals dependence structures of multivariate stable distributions non linear regression complex stable stochastic integrals and harmonizable processes self similar processes chentsov random fields introduction to sample path properties boundedness continuity and oscillations measurability integrability and absolute continuity boundedness and continuity via metric entropy integral representation historical notes and extensions

bais a bayesian artificial immune system for the effective handling of building blocks significant progress has been made in theory and design of artificial immune systems aiss for solving hard problems accurately however an aspect not yet widely addressed by the research reported in the literature is the lack of ability of the aiss to deal effectively with building blocks partial high quality solutions coded in the antibody the available aiss present mechanisms for evolving the population that do not take into account the relationship among the variables of the problem potentially causing the disruption of high quality partial solutions this paper proposes a novel ais with abilities to identify and properly manipulate building blocks in optimization problems instead of using cloning and mutation to generate new individuals our algorithm builds a probabilistic model representing the joint probability distribution of the promising solutions and subsequently uses this model for sampling new solutions the probabilistic model used is a bayesian network due to its capability of properly capturing the most relevant interactions among the variables therefore our algorithm called bayesian artificial immune system bais represents a significant attempt to improve the performance of immune inspired algorithms when dealing with building blocks and hence to solve efficiently hard optimization problems with complex interactions among the variables the performance of bais compares favorably with that produced by contenders such as state of the art estimation of distribution algorithms

sub local constraint based learning of bayesian networks using a joint dependence criterion constraint based learning of bayesian networks bn from limited data can lead to multiple testing problems when recovering dense areas of the skeleton and to conflicting results in the orientation of edges in this paper we present a new constraint based algorithm light mutual min lmm for improved accuracy of bn learning from small sample data lmm improves the assessment of candidate edges by using a ranking criterion that considers conditional independence on neighboring variables at both sides of an edge simultaneously the algorithm also employs an adaptive relaxation of constraints that selectively allows some nodes not to condition on some neighbors this relaxation aims at reducing the incorrect rejection of true edges connecting high degree nodes due to multiple testing lmm additionally incorporates a new criterion for ranking v structures that is used to recover the completed partially directed acyclic graph cpdag and to resolve conflicting v structures a common problem in small sample constraint based learning using simulated data each of these components of lmm is shown to significantly improve network inference compared to commonly applied methods when learning from limited data including more accurate recovery of skeletons and cpdags compared to the pc maxmin and maxmin hill climbing algorithms a proof of asymptotic correctness is also provided for lmm for recovering the correct skeleton and cpdag

on inclusion driven learning of bayesian networks two or more bayesian network structures are markov equivalent when the corresponding acyclic digraphs encode the same set of conditional independencies therefore the search space of bayesian network structures may be organized in equivalence classes where each of them represents a different set of conditional independencies the collection of sets of conditional independencies obeys a partial order the so called inclusion order this paper discusses in depth the role that the inclusion order plays in learning the structure of bayesian networks in particular this role involves the way a learning algorithm traverses the search space we introduce a condition for traversal operators the inclusion boundary condition which when it is satisfied guarantees that the search strategy can avoid local maxima this is proved under the assumptions that the data is sampled from a probability distribution which is faithful to an acyclic digraph and the length of the sample is unbounded the previous discussion leads to the design of a new traversal operator and two new learning algorithms in the context of heuristic search and the markov chain monte carlo method we carry out a set of experiments with synthetic and real world data that show empirically the benefit of striving for the inclusion order when learning bayesian networks from data

bnc pso structure learning is a very important problem in the field of bayesian networks bns it is also an active research area for more than 2 decades therefore many approaches have been proposed in order to find an optimal structure based on training samples in this paper a particle swarm optimization pso based algorithm is proposed to solve the bn structure learning problem named bnc pso bayesian network construction algorithm using pso edge inserting deleting is employed in the algorithm to make the particles have the ability to achieve the optimal solution while a cycle removing procedure is used to prevent the generation of invalid solutions then the theorem of markov chain is used to prove the global convergence of our proposed algorithm finally some experiments are designed to evaluate the performance of the proposed pso based algorithm experimental results indicate that bnc pso is worthy of being studied in the field of bns construction meanwhile it can significantly increase nearly 15 in the scoring metric values comparing with other optimization based algorithms

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

column and row operator spaces over qslp spaces and their use in abstract harmonic analysis the notions of column and row operator space were extended by a lambert from hilbert spaces to general banach spaces in this paper we use column and row spaces over quotients of subspaces of general lp spaces to equip several banach algebras occurring naturally in abstract harmonic analysis with canonical yet not obvious operator space structures that turn them into completely bounded banach algebras we use these operator space structures to gain new insights on those algebras

search for additive nonlinear time series causal models pointwise consistent feasible procedures for estimating contemporaneous linear causal structure from time series data have been developed using multiple conditional independence tests but no such procedures are available for non linear systems we describe a feasible procedure for learning a class of non linear time series structures which we call additive non linear time series we show that for data generated from stationary models of this type two classes of conditional independence relations among time series variables and their lags can be tested efficiently and consistently using tests based on additive model regression combining results of statistical tests for these two classes of conditional independence relations and the temporal structure of time series data a new consistent model specification procedure is able to extract relatively detailed causal information we investigate the finite sample behavior of the procedure through simulation and illustrate the application of this method through analysis of the possible causal connections among four ocean indices several variants of the procedure are also discussed

conditional likelihood maximisation a unifying framework for information theoretic feature selection we present a unifying framework for information theoretic feature selection bringing almost two decades of research on heuristic filter criteria under a single theoretical interpretation this is in response to the question what are the implicit statistical assumptions of feature selection criteria based on mutual information to answer this we adopt a different strategy than is usual in the feature selection literature instead of trying to define a criterion we derive one directly from a clearly specified objective function the conditional likelihood of the training labels while many hand designed heuristic criteria try to optimize a definition of feature relevancy and redundancy our approach leads to a probabilistic framework which naturally incorporates these concepts as a result we can unify the numerous criteria published over the last two decades and show them to be low order approximations to the exact but intractable optimisation problem the primary contribution is to show that common heuristics for information based feature selection including markov blanket algorithms as a special case are approximate iterative maximisers of the conditional likelihood a large empirical study provides strong evidence to favour certain classes of criteria in particular those that balance the relative size of the relevancy redundancy terms overall we conclude that the jmi criterion yang and moody 1999 meyer et al 2008 provides the best tradeoff in terms of accuracy stability and flexibility with small data samples

orthogonality and disjointness preserving linear maps between fourier and fourier stieltjes algebras of locally compact groups this paper is devoted to the study of orthogonality and disjointness preserving linear maps between fourier and fourier stieltjes algebras of locally compact groups we show that a linear bijection a g1 a g2 resp b g1 b g2 between two fourier algebras resp fourier stieltjes algebras of locally compact groups will induce a topological group isomorphism between g1 and g2 provided that preserves both disjointness and some kind of orthogonality this improves earlier results of j j font and m s monfared where amenability of the groups or continuity of the linear maps are assumed we also study the structure of bounded and unbounded disjointness preserving linear functionals of fourier algebras in the development general results about disjointness and orthogonality preserving linear maps between c algebras w algebras and their preduals are obtained

finding optimal bayesian networks using precedence constraints we consider the problem of finding a directed acyclic graph dag that optimizes a decomposable bayesian network score while in a favorable case an optimal dag can be found in polynomial time in the worst case the fastest known algorithms rely on dynamic programming across the node subsets taking time and space 2n to within a factor polynomial in the number of nodes n in practice these algorithms are feasible to networks of at most around 30 nodes mainly due to the large space requirement here we generalize the dynamic programming approach to enhance its feasibility in three dimensions first the user may trade space against time second the proposed algorithms easily and efficiently parallelize onto thousands of processors third the algorithms can exploit any prior knowledge about the precedence relation on the nodes underlying all these results is the key observation that given a partial order p on the nodes an optimal dag compatible with p can be found in time and space roughly proportional to the number of ideals of p which can be significantly less than 2n considering sufficiently many carefully chosen partial orders guarantees that a globally optimal dag will be found aside from the generic scheme we present and analyze concrete tradeoff schemes based on parallel bucket orders

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

beurling fourier algebras operator amenability and arens regularity abstract we introduce the class of beurling fourier algebras on locally compact groups and show that they are non commutative analogs of classical beurling algebras we obtain various results with regard to the operator amenability operator weak amenability and arens regularity of beurling fourier algebras on compact groups and show that they behave very similarly to the classical beurling algebras of discrete groups we then apply our results to study explicitly the beurling fourier algebras on su 2 the 2 2 special unitary group we demonstrate that how beurling fourier algebras are closely connected to the amenability of the fourier algebra of su 2 another major consequence of our results is that our investigation allows us to construct families of unital infinite dimensional closed arens regular subalgebras of the fourier algebra of certain products of su 2

projectivity of modules over fourier algebras in this paper we will study the homological properties of various natural modules associated to the fourier algebra of a locally compact group in particular we will focus on the question of identifying when such modules will be projective in the category of operator spaces we will show that pro jectivity often implies that the underlying group is discrete and give evidence to show that amenability also plays an important role over the past thirty years there has been a rich body of work dedicated to understanding the cohomological properties of the various banach algebras arising in the study of locally compact groups the seminal paper in this respect is certainly barry johnson s memoir 22 in which he introduces the concept of amenability for a banach algebra and shows that for a locally compact group g the group algebra l 1 g is amenable if and only if g is amenable in the classical sense while applications of cohomology to harmonic anlysis may be more celebrated there have also been a number of significant studies related to the homological properties of the algebras arising from groups recently h g dales and m e polyakov 12 gave a detailed study of the homological properties of modules over the group algebra of a locally compact group in their work they focused primarily on the question of whether or not certain natural left l 1 g modules are respectively projective injective or flat they were able to show for example that when viewed as the dual left module of l 1 g l g is projective precisely when the group g is finite in stark contrast they prove that l g is always injective they also showed that the measure algebra m g is projective precisely when g is discrete while in this case injectivity is equivalent to the group g being amenable when g is abelian the classical fourier transform identifies l 1 g with a com

cohen host type idempotent theorems for representations on banach spaces and applications to figa talamanca herz algebras abstract let g be a locally compact group and let r g denote the ring of subsets of g generated by the left cosets of open subsets of g the cohen host idempotent theorem asserts that a set lies in r g if and only if its indicator function is a coefficient function of a unitary representation of g on some hilbert space we prove related results for representations of g on certain banach spaces we apply our cohen host type theorems to the study of the figa talamanca herz algebras a p g with p 1 for arbitrary g we characterize those closed ideals of a p g that have an approximate identity bounded by 1 in terms of their hulls furthermore we characterize those g such that a p g is 1 amenable for some and equivalently for all p 1 these are precisely the abelian groups

a renormalization group classification of nonstationary and or infinite second moment diffusive processes anomalous diffusion processes are often classified by their mean square displacement if the mean square displacement grows linearly in time the process is considered classical if it grows like t with 1 the process is considered subdiffusive or superdiffusive respectively processes with infinite mean square displacement are considered superdiffusive we begin by examining the ways in which power law mean square displacements can arise namely via non zero drift nonstationary increments and correlated increments subsequently we describe examples which illustrate that the above classification scheme does not work well when nonstationary increments are present finally we introduce an alternative classification scheme based on renormalization groups this scheme classifies processes with stationary increments such as brownian motion and fractional brownian motion in the same groups as the mean square displacement scheme but does a better job of classifying processes with nonstationary increments and or processes with infinite second moments such as stable levy motion a numerical approach to analyzing data based on the renormalization group classification is also presented

characterization and greedy learning of interventional markov equivalence classes of directed acyclic graphs the investigation of directed acyclic graphs dags encoding the same markov property that is the same conditional independence relations of multivariate observational distributions has a long tradition many algorithms exist for model selection and structure learning in markov equivalence classes in this paper we extend the notion of markov equivalence of dags to the case of interventional distributions arising from multiple intervention experiments we show that under reasonable assumptions on the intervention experiments interventionalmarkov equivalence defines a finer partitioning of dags than observational markov equivalence and hence improves the identifiability of causal models we give a graph theoretic criterion for two dags being markov equivalent under interventions and show that each interventional markov equivalence class can analogously to the observational case be uniquely represented by a chain graph called interventional essential graph also known as cpdag in the observational case these are key insights for deriving a generalization of the greedy equivalence search algorithm aimed at structure learning from interventional data this new algorithm is evaluated in a simulation study

cam causal additive models high dimensional order search and penalized regression we develop estimation for potentially high dimensional additive structural equation models a key component of our approach is to decouple order search among the variables from feature or edge selection in a directed acyclic graph encoding the causal structure we show that the former can be done with nonregularized restricted maximum likelihood estimation while the latter can be efficiently addressed using sparse regression techniques thus we substantially simplify the problem of structure search and estimation for an important class of causal models we establish consistency of the restricted maximum likelihood estimator for low and high dimensional scenarios and we also allow for misspecification of the error distribution furthermore we develop an efficient computational algorithm which can deal with many variables and the new method s accuracy and performance is illustrated on simulated and real data

consistent feature selection for pattern recognition in polynomial time we analyze two different feature selection problems finding a minimal feature set optimal for classification minimal optimal vs finding all features relevant to the target variable all relevant the latter problem is motivated by recent applications within bioinformatics particularly gene expression analysis for both problems we identify classes of data distributions for which there exist consistent polynomial time algorithms we also prove that all relevant is much harder than minimal optimal and propose two consistent polynomial time algorithms we argue that the distribution classes considered are reasonable in many practical cases so that our results simplify feature selection in a wide range of machine learning tasks

amenability and weak amenability of the fourier algebra let g be a locally compact group we show that its fourier algebra a g is amenable if and only if g has an abelian subgroup of finite index and that its fourier stieltjes algebra b g is amenable if and only if g has a compact abelian subgroup of finite index we then show that a g is weakly amenable if the component of the identity of g is abelian and we prove some partial results towards the converse