approximating bayesian belief networks by arc removal i propose a general framework for approximating bayesian belief networks through model simplification by arc removal given an upper bound on the absolute error allowed on the prior and posterior probability distributions of the approximated network a subset of arcs is removed thereby speeding up probabilistic inference

gene stanley the n vector model and random walks with absorbing boundaries abstract gene stanley s early contribution to the n vector model is briefly reviewed also discussed is a problem solved by mccrea and whipple in 1940 namely random walks with absorbing boundaries we have found the exact solution of the analogous problem on the triangular lattice with a true finite boundary this problem was previously considered to be intractable

a chain code for representing 3d curves abstract a chain code for representing three dimensional 3d curves is defined any 3d continuous curve can be digitalized and represented as a 3d discrete curve this 3d discrete curve is composed of constant straight line segments thus the chain elements represent the orthogonal direction changes of the constant straight line segments of the discrete curve the proposed chain code only considers relative direction changes which allows us to have a curve descriptor invariant under translation and rotation also this curve descriptor may be starting point normalized for open and closed curves and invariant under mirroring transformation in the content of this work the main characteristics of this chain code are presented this chain code is inspired by the work of guzman mcc technical report number aca 254 87 july 13 1987 for representing 3d stick bodies finally we present some results of this chain code to represent and process 3d discrete curves as linear features over the terrain by means of digital elevation model dem data also we use this chain code for representing solids composed of voxels thus each solid represents a dem which is described by only one chain

statistics of visits to sites in random walks abstract this paper analyses the statistics of visits to a site in random walks first general formulas will be derived and next they will be applied to simple symmetric random walks on finite and infinite lattices the dependence of the statistics on the lattices dimensionality will also be studied finally an application will be presented to explain the structure of the radio maps that trace the supernova remnants

improving variational methods via pairwise linear response identities inference methods are often formulated as variational approximations these approximations allow easy evaluation of statistics by marginalization or linear response but these estimates can be inconsistent we show that by introducing constraints on covariance one can ensure consistency of linear response with the variational parameters and in so doing inference of marginal probability distributions is improved for the bethe approximation and its generalizations improvements are achieved with simple choices of the constraints the approximations are presented as variational frameworks iterative procedures related to message passing are provided for finding the minima

approximate counting the lovasz local lemma and inference in graphical models in this article we introduce a new approach to approximate counting in bounded degree systems with higher order constraints our main result is an algorithm to approximately count the number of solutions to a cnf formula when the width is logarithmic in the maximum degree this closes an exponential gap between the known upper and lower bounds moreover our algorithm extends straightforwardly to approximate sampling which shows that under lovasz local lemma like conditions it is not only possible to find a satisfying assignment it is also possible to generate one approximately uniformly at random from the set of all satisfying assignments our approach is a significant departure from earlier techniques in approximate counting and is based on a framework to bootstrap an oracle for computing marginal probabilities on individual variables finally we give an application of our results to show that it is algorithmically possible to sample from the posterior distribution in an interesting class of graphical models

approximating maps for belief networks is np hard and other theorems abstract finding maximum a posteriori map assignments also called most probable explanations is an important problem on bayesian belief networks shimony has shown that finding maps is np hard in this paper we show that approximating maps with a constant ratio bound is also np hard in addition we examine the complexity of two related problems which have been mentioned in the literature we show that given the map for a belief network and evidence set or the family of maps if the optimal is not unique it remains np hard to find or approximate alternative next best explanations furthermore we show that given the map or maps for a belief network and an initial evidence set it is also np hard to find or approximate the map assignment for the same belief network with a modified evidence set that differs from the initial set by the addition or removal of even a single node assignment finally we show that our main result applies to networks with constrained in degree and out degree applies to randomized approximation and even still applies if the ratio bound instead of being constant is allowed to be a polynomial function of various aspects of the network topology

residence times and other functionals of reflected brownian motion we study the residence and local times for a brownian particle confined by reflecting boundaries and propose a general solution to the problem of finding the related probability distribution its fourier transform characteristic function and laplace transform survival probability are obtained in a compact matrix form involving the laplace operator eigenbasis explicit combinatorial relations are derived for the moments and the probability distribution is shown to be nearly gaussian when the exploration time is long enough when the eigenbasis or a part of it is known the numerical computation of the residence time distributions is straightforward and accurate the present approach can also be applied to investigate other functionals of reflected brownian motion describing in particular restricted diffusion in an external field or potential e g nuclei diffusing in an inhomogeneous magnetic field theoretical results for the local times are confronted with monte carlo simulations on the unit interval disk and sphere

the complexity of approximating maps for belief networks with bounded probabilities abstract probabilistic inference and maximum a posteriori map explanation are two important and related problems on bayesian belief networks both problems are known to be np hard for both approximation and exact solution in 1997 dagum and luby showed that efficiently approximating probabilistic inference is possible for belief networks in which all probabilities are bounded away from 0 in this paper we show that the corresponding result for map explanation does not hold finding or approximating maps for belief networks remains np hard for belief networks with probabilities bounded within the range l u for any 0 l our results cover both deterministic and randomized approximation

first exit times and residence times for discrete random walks on finite lattices in this paper we derive explicit formulas for the surface averaged first exit time of a discrete random walk on a finite lattice we consider a wide class of random walks and lattices including random walks in a nontrivial potential landscape we also compute quantities of interest for modeling surface reactions and other dynamic processes such as the residence time in a subvolume the joint residence time of several particles and the number of hits on a reflecting surface

lossless compression of map contours by context tree modeling of chain codes we consider lossless compression of digital contours in map images the problem is attacked by the use of context based statistical modeling and entropy coding of the chain codes we propose to generate an optimal n ary incomplete context tree by first constructing a complete tree up to a predefined depth and creating the optimal tree by pruning out nodes that do not provide improvement in compression we apply this method for both vector and raster maps experiments show that the proposed method gives lower bit rates than the existing methods of chain codes compression for the set of test data

approximating probabilistic inference in bayesian belief networks is np hard abstract it is known that exact computation of conditional probabilities in belief networks is np hard many investigators in the ai community have tacitly assumed that algorithms for performing approximate inference with belief networks are of polynomial complexity indeed special cases of approximate inference can be performed in time polynomial in the input size however we have discovered that the general problem of approximating conditional probabilities with belief networks like exact inference resides in the np hard complexity class we develop a complexity analysis to elucidate the difficulty of approximate probabilistic inference more specifically we show that the existence of a polynomial time relative approximation algorithm for major classes of problem instances implies that np p we present our proof and explore the implications of the result

a new chain code abstract a new chain code for shapes composed of regular cells is defined this boundary chain code is based on the numbers of cell vertices which are in touch with the bounding contour of the shape this boundary chain code is termed vertex chain code vcc the vcc is invariant under translation and rotation also it may be starting point normalized and invariant under mirroring transformation using this concept of chain code it is possible to relate the chain length to the contact perimeter which corresponds to the sum of the boundaries of neighboring cells of the shape bribiesca e comp math appl 33 11 1997 1 9 also to relate the chain nodes to the contact vertices which correspond to the vertices of neighboring cells so in this way these relations among the chain and the characteristics of interior of the shape allow us to obtain interesting properties this work is motivated by the idea of obtaining various shape features computed directly from the vcc without going to cartesian coordinate representation finally in order to illustrate the capabilities of the vcc we present some results using real shapes

the computational complexity of probabilistic inference using bayesian belief networks research note abstract bayesian belief networks provide a natural efficient method for representing probabilistic dependencies among a set of variables for these reasons numerous researchers are exploring the use of belief networks as a knowledge representation in artificial intelligence algorithms have been developed previously for efficient probabilistic inference using special classes of belief networks more general classes of belief networks however have eluded efforts to develop efficient inference algorithms we show that probabilistic inference using belief networks is np hard therefore it seems unlikely that an exact algorithm can be developed to perform probabilistic inference efficiently over all classes of belief networks this result suggests that research should be directed away from the search for a general efficient probabilistic inference algorithm and toward the design of efficient special case average case and approximation algorithms

an optimal approximation algorithm for bayesian inference abstract approximating the inference probability pr x x e e in any sense even for a single evidence node e is np hard this result holds for belief networks that are allowed to contain extreme conditional probabilities that is conditional probabilities arbitrarily close to 0 nevertheless all previous approximation algorithms have failed to approximate efficiently many inferences even for belief networks without extreme conditional probabilities we prove that we can approximate efficiently probabilistic inference in belief networks without extreme conditional probabilities we construct a randomized approximation algorithm the bounded variance algorithm that is a variant of the known likelihood weighting algorithm the bounded variance algorithm is the first algorithm with provably fast inference approximation on all belief networks without extreme conditional probabilities from the bounded variance algorithm we construct a deterministic approximation algorithm using current advances in the theory of pseudorandom generators in contrast to the exponential worst case behavior of all previous deterministic approximations the deterministic bounded variance algorithm approximates inference probabilities in worst case time that is subexponential 2 log n d for some integer d that is a linear function of the depth of the belief network

and or branch and bound search for combinatorial optimization in graphical models this is the first of two papers presenting and evaluating the power of a new framework for combinatorial optimization in graphical models based on and or search spaces we introduce a new generation of depth first branch and bound algorithms that explore the and or search tree using static and dynamic variable orderings the virtue of the and or representation of the search space is that its size may be far smaller than that of a traditional or representation which can translate into significant time savings for search algorithms the focus of this paper is on linear space search which explores the and or search tree in the second paper we explore memory intensive and or search algorithms in conjunction with the and or search space we investigate the power of the mini bucket heuristics in both static and dynamic setups we focus on two most common optimization problems in graphical models finding the most probable explanation in bayesian networks and solving weighted csps in extensive empirical evaluations we demonstrate that the new and or branch and bound approach improves considerably over the traditional or search strategy and show how various variable ordering schemes impact the performance of the and or search scheme

exact solution for random walks on the triangular lattice with absorbing boundaries the problem of a random walk on a finite triangular lattice with a single interior source point and zig zag absorbing boundaries is solved exactly this problem has been previously considered intractable

on the joint residence time of n independent two dimensional brownian motions we study the behaviour of several joint residence times of n independent brownian particles in a disc of radius r in two dimensions we consider i the time tn t spent by all n particles simultaneously in the disc within the time interval 0 t ii the time t m n t which at least m out of n particles spend together in the disc within the time interval 0 t and iii the time m n t which exactly m out of n particles spend together in the disc within the time interval 0 t we obtain very simple exact expressions for the expectations of these three residence times in the limit t

on the hardness of approximate reasoning abstract many ai problems when formalized reduce to evaluating the probability that a propositional expression is true in this paper we show that this problem is computationally intractable even in surprisingly restricted cases and even if we settle for an approximation to this probability we consider various methods used in approximate reasoning such as computing degree of belief and bayesian belief networks as well as reasoning techniques such as constraint satisfaction and knowledge compilation that use approximation to avoid computational difficulties and reduce them to model counting problems over a propositional domain we prove that counting satisfying assignments of propositional languages is intractable even for horn and monotone formulae and even when the size of clauses and number of occurrences of the variables are extremely limited this should be contrasted with the case of deductive reasoning where horn theories and theories with binary clauses are distinguished by the existence of linear time satisfiability algorithms what is even more surprising is that as we show even approximating the number of satisfying assignments i e approximating approximate reasoning is intractable for most of these restricted theories we also identify some restricted classes of propositional formulae for which efficient algorithms for counting satisfying assignments can be given

diagnosis under compound effects and multiple causes by means of the conditional causal possibility approach abstract the paper addresses uncertain reasoning on a causal model given by two layered networks where nodes in one layer express possible causes and those in the other are possible effects uncertainty of causalities is expressed in a novel manner i e by conditional causal possibilities the expression has two advantages over the conventional way with conditional possibilities it expresses the exact degrees of possibility of causalities and the number of necessary conditional causal possibilities is far smaller than that of conditional possibilities however it has a weakness that it cannot handle causalities with compound effects such as synergistic and canceling effects by multiple causes the paper discusses the weakness and proposes a solution first it discusses how to deal with the compound effects and proposes a new causal model with conditional causal possibilities by multiple causes then it defines a causality consistency problem that calculates possibility of a hypothesis given some observed events and shows a way to solve the problem

number of times a site is visited in two dimensional random walks in this paper formulas are derived to compute the mean number of times a site has been visited in a random walk on a two dimensional lattice asymmetric random walks are considered with or without drift for different boundary conditions it is shown that in case of absorbing boundaries the mean number of visits reaches stationary values over the lattice comparisons with a monte carlo simulation are also presented

dynamic uncertain causality graph for knowledge representation and probabilistic reasoning statistics base matrix and application graphical models for probabilistic reasoning are now in widespread use many approaches have been developed such as bayesian network a newly developed approach named as dynamic uncertain causality graph ducg is initially presented in a previous paper in which only the inference algorithm in terms of individual events and probabilities is addressed in this paper we first explain the statistic basis of ducg then we extend the algorithm to the form of matrices of events and probabilities it is revealed that the representation of ducg can be incomplete and the exact probabilistic inference may still be made a real application of ducg for fault diagnoses of a generator system of a nuclear power plant is demonstrated which involves variables most inferences take with a laptop computer the causal logic between inference result and observations is graphically displayed to users so that they know not only the result but also why the result obtained

finding maps for belief networks is np hard given a probabilistic world model an important problem is to find the maximum a posteriori probability map instantiation of all the random variables given the evidence numerous researchers using such models employ some graph representation for the distributions such as a bayesian belief network this representation simplifies the complexity of specifying the distributions from exponential in n the number of variables in the model to linear in n in many interesting cases we show however that finding the map is np hard in the general case when these representations are used even if the size of the representation happens to be linear in n furthermore minor modifications to the proof show that the problem remains np hard for various restrictions of the topology of the graphs the same technique can be applied to the results of a related paper by cooper to further restrict belief network topology in the proof that probabilistic inference is np hard

a new look at survey propagation and its generalizations this article provides a new conceptual perspective on survey propagation which is an iterative algorithm recently introduced by the statistical physics community that is very effective in solving random k sat problems even with densities close to the satisfiability threshold we first describe how any sat formula can be associated with a novel family of markov random fields mrfs parameterized by a real number 0 1 we then show that applying belief propagation a well known message passing technique for estimating marginal probabilities to this family of mrfs recovers a known family of algorithms ranging from pure survey propagation at one extreme e 1 to standard belief propagation on the uniform distribution over sat assignments at the other extreme e 0 configurations in these mrfs have a natural interpretation as partial satisfiability assignments on which a partial order can be defined we isolate cores as minimal elements in this partial ordering which are also fixed points of survey propagation and the only assignments with positive probability in the mrf for e 1 our experimental results for k e 3 suggest that solutions of random formulas typically do not possess non trivial cores this makes it necessary to study the structure of the space of partial assignments for 0 can be viewed as a smoothed version of the uniform distribution over satisfying assignments e 0 finally we isolate properties of gibbs sampling and message passing algorithms that are typical for an ensemble of k sat problems

a one pass algorithm for local symmetry of contours from chain codes this paper presents a one pass algorithm for computing the local symmetry of contours from their chain code representation this algorithm embeds the local symmetry deficiency lsd computation into our chain codes extraction process the existing algorithms need to extract chain codes first and use them as input to compute lsd the new algorithm uses a binary image as input and can complete the code extraction and the lsd computation in one pass it can also extract multiple contours in an image in one pass the algorithm is suitable for parallel implementation

a random polynomial time algorithm for approximating the volume of convex bodies a randomized polynomial time algorithm for approximating the volume of a convex body k in n dimensional euclidean space is presented the proof of correctness of the algorithm relies on recent theory of rapidly mixing markov chains and isoperimetric inequalities to show that a certain random walk can be used to sample nearly uniformly from within k

probabilistic conflicts in a search algorithm for estimating posterior probabilities in bayesian networks abstract this paper presents a search algorithm for estimating posterior probabilities in discrete bayesian networks it shows how conflicts as used in consistency based diagnosis can be adapted to speed up the search this algorithm is especially suited to the case where there are skewed distributions although nothing about the algorithm or the definitions depends on skewness of distributions the general idea is to forward simulate the network based on the normal values for each variable the value with high probability given its parents when a predicted value is at odds with the observations we analyse which variables were responsible for the expectation failure these form a conflict and continue forward simulation considering different values for these variables this results in a set of possible worlds from which posterior probabilities together with error bounds can be derived empirical results with bayesian networks having tens of thousands of nodes are presented

mean first passage times in confined media from markovian to non markovian processes we review recent theoretical works that enable the accurate evaluation of the mean first passage time mfpt of a random walker to a target in confinement for markovian memory less and non markovian walkers for the markovian problem we present a general theory which allows one to accurately evaluate the mfpt and its extensions to related first passage observables such as splitting probabilities and occupation times we show that this analytical approach provides a universal scaling dependence of the mfpt on both the volume of the confining domain and the source target distance in the case of general scale invariant processes this analysis is applicable to a broad range of stochastic processes characterized by length scale invariant properties and reveals the key role that can be played by the starting position of the random walker we then present an extension to non markovian walks by taking the specific example of a tagged monomer of a polymer chain looking for a target in confinement we show that the mfpt can be calculated accurately by computing the distribution of the positions of all the monomers in the chain at the instant of reaction such a theory can be used to derive asymptotic relations that generalize the scaling dependence with the volume and the initial distance to the target derived for markovian walks finally we present an application of this theory to the problem of the first contact time between the two ends of a polymer chain and review the various theoretical approaches of this nonmarkovian problem

on the complexity of inference about probabilistic relational models abstract we investigate the complexity of probabilistic inference from knowledge bases that encode probability distributions on finite domain relational structures our interest here lies in the complexity in terms of the domain under consideration in a specific application instance we obtain the result that assuming netime etime this problem is not polynomial for reasonably expressive representation systems the main consequence of this result is that it is unlikely to find inference techniques with a better worst case behavior than the commonly employed strategy of constructing standard bayesian networks over ground atoms knowledge based model construction

approximating probabilistic inference in bayesian belief networks a belief network comprises a graphical representation of dependencies between variables of a domain and a set of conditional probabilities associated with each dependency unless rho np an efficient exact algorithm does not exist to compute probabilistic inference in belief networks stochastic simulation methods which often improve run times provide an alternative to exact inference algorithms such a stochastic simulation algorithm d bnras which is a randomized approximation scheme is presented to analyze the run time belief networks are parameterized by the dependence value d sub xi which is a measure of the cumulative strengths of the belief network dependencies given background evidence xi this parameterization defines the class of f dependence networks the run time of d bnras is polynomial when f is a polynomial function thus the results prove the existence of a class of belief networks for which inference approximation is polynomial and hence provably faster than any exact algorithm

possibilistic causality consistency problem based on asymmetrically valued causal model the paper addresses uncertain reasoning based on causal knowledge given by two layered networks where nodes in one layer express possible causes and those in the other are possible effects uncertainties of the causalities are given by conditional causal possibilities which were proposed to express the exact degrees of possibility of causalities the expression of the uncertainty also has an advantage over the conventional conditional possibilities in the number of possibilistic values that should be given as a priori knowledge the number of conditional causal possibilities given as knowledge is far smaller than that of conventional conditional possibilities the paper starts with the definition of a causal model called asymmetrically valued causal model the conditional causal possibilities are defined on the causal model and their mathematical properties are discussed then the paper defines the possibilistic causality consistency problem based on the proposed model and shows how to solve the problem the discussed problem is one to calculate the conditional possibility of a hypothesis about presence and absence of unknown events when the states of some other events are known

approximate belief updating in max 2 connected bayes networks is np hard a max 2 connected bayes network is one where there are at most 2 distinct directed paths between any two nodes we show that even for this restricted topology null evidence belief updating is hard to approximate

mini buckets a general scheme for bounded inference this article presents a class of approximation algorithms that extend the idea of bounded complexity inference inspired by successful constraint propagation algorithms to probabilistic inference and combinatorial optimization the idea is to bound the dimensionality of dependencies created by inference algorithms this yields a parameterized scheme called mini buckets that offers adjustable trade off between accuracy and efficiency the mini bucket approach to optimization problems such as finding the most probable explanation mpe in bayesian networks generates both an approximate solution and bounds on the solution quality we present empirical results demonstrating successful performance of the proposed approximation scheme for the mpe task both on randomly generated problems and on realistic domains such as medical diagnosis and probabilistic decoding

and or search spaces for graphical models the paper introduces an and or search space perspective for graphical models that include probabilistic networks directed or undirected and constraint networks in contrast to the traditional or search space view the and or search tree displays some of the independencies present in the graphical model explicitly and may sometimes reduce the search space exponentially indeed most algorithmic advances in search based constraint processing and probabilistic inference can be viewed as searching an and or search tree or graph familiar parameters such as the depth of a spanning tree treewidth and pathwidth are shown to play a key role in characterizing the effect of and or search graphs vs the traditional or search graphs we compare memory intensive and or graph search with inference methods and place various existing algorithms within the and or search space

random walks on finite lattice tubes exact results are obtained for random walks on finite lattice tubes with a single source and absorbing lattice sites at the ends explicit formulas are derived for the absorption probabilities at the ends and for the expectations that a random walk will visit a particular lattice site before being absorbed results are obtained for lattice tubes of arbitrary size and each of the regular lattice types square triangular and honeycomb the results include an adjustable parameter to model the effects of strain such as surface curvature on the surface diffusion results for the triangular lattice tubes and the honeycomb lattice tubes model diffusion of adatoms on single walled zig zag carbon nanotubes with open ends

symmetry detection in 3d chain coded discrete curves and trees symmetry detection in 2d and 3d shapes has been a classical problem in computer vision computational geometry and pattern recognition one of the reasons of this interest is because symmetry is related to the balance and harmony and in most cases it is considered like a beautiful proportion in this paper we present different methods to detect mirror and rotational symmetries in 2d and 3d open and closed curves and in trees represented by means of the orthogonal direction change chain code these methods detect both global and local symmetries they can also detect the number and position of the axes of symmetry they are simple easy to implement and fast to execute highlightssymmetry detection for 3d curves and tree objects via chain coding is presented symmetry is related to the balance and harmony the symmetry of plane and space filling curves is shown the symmetry detection on plane and space filling trees is computed

subdiffusion in a bounded domain with a partially absorbing reflecting boundary the exit time of a subdiffusive process from a bounded domain with a partially absorbing reflecting boundary is considered the short time and long time behaviors of the exit time probability density are investigated by using a spectral decomposition on the basis of the laplace operator eigenfunctions rotation invariant domains are analyzed in depth in order to illustrate the use of theoretical formulas and to compare them to numerical simulations the asymptotic results obtained are relevant for describing subdiffusion inside a living cell with a semipermeable membrane in a chemical reactor filled with catalytic grains of finite reactivity or in mineral or biological samples which are probed by nuclear magnetic resonance measurements subject to surface relaxation

iterative state space reduction for flexible computation abstract flexible computation is a general framework for decision making under limited computational resources it enables an agent to allocate limited computational resources to maximize its overall performance or utility in this paper we present a strategy for flexible computation which we call iterative state space reduction the main ideas are to reduce a problem space that is difficult to search to one that is relatively easy to explore to use the optimal solution from the reduced space as an approximate solution to the original problem and to iteratively apply multiple reductions to progressively find better solutions the basic operation for state space reduction is heuristic node pruning which excludes the nodes that do not seem to lead to high quality solutions from further examination iterative state space reduction can be combined with a state space search such as best first search or depth first branch and bound dfbnb the resulting algorithm can run as an anytime algorithm which provides a tradeoff between solution quality and computation based on an analytical modal we analyze the probability that one iteration of the iterative process finds a solution furthermore by combining it within dfbnb we apply iterative state space reduction to three combinatorial problems the maximum boolean satisfiability the symmetric tsp and asymmetric tsp our experimental results show that iterative state space reduction is effective and efficient making dfbnb find better solutions with less computation