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Markov Logic Networks
 MACHINE LEARNING
, 2006
"... We propose a simple approach to combining firstorder logic and probabilistic graphical models in a single representation. A Markov logic network (MLN) is a firstorder knowledge base with a weight attached to each formula (or clause). Together with a set of constants representing objects in the ..."
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Cited by 816 (39 self)
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We propose a simple approach to combining firstorder logic and probabilistic graphical models in a single representation. A Markov logic network (MLN) is a firstorder knowledge base with a weight attached to each formula (or clause). Together with a set of constants representing objects in the domain, it specifies a ground Markov network containing one feature for each possible grounding of a firstorder formula in the KB, with the corresponding weight. Inference in MLNs is performed by MCMC over the minimal subset of the ground network required for answering the query. Weights are efficiently learned from relational databases by iteratively optimizing a pseudolikelihood measure. Optionally, additional clauses are learned using inductive logic programming techniques. Experiments with a realworld database and knowledge base in a university domain illustrate the promise of this approach.
Slice sampling
 Annals of Statistics
, 2000
"... Abstract. Markov chain sampling methods that automatically adapt to characteristics of the distribution being sampled can be constructed by exploiting the principle that one can sample from a distribution by sampling uniformly from the region under the plot of its density function. A Markov chain th ..."
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Cited by 305 (5 self)
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Abstract. Markov chain sampling methods that automatically adapt to characteristics of the distribution being sampled can be constructed by exploiting the principle that one can sample from a distribution by sampling uniformly from the region under the plot of its density function. A Markov chain that converges to this uniform distribution can be constructed by alternating uniform sampling in the vertical direction with uniform sampling from the horizontal ‘slice ’ defined by the current vertical position, or more generally, with some update that leaves the uniform distribution over this slice invariant. Variations on such ‘slice sampling ’ methods are easily implemented for univariate distributions, and can be used to sample from a multivariate distribution by updating each variable in turn. This approach is often easier to implement than Gibbs sampling, and more efficient than simple Metropolis updates, due to the ability of slice sampling to adaptively choose the magnitude of changes made. It is therefore attractive for routine and automated use. Slice sampling methods that update all variables simultaneously are also possible. These methods can adaptively choose the magnitudes of changes made to each variable, based on the local properties of the density function. More ambitiously, such methods could potentially allow the sampling to adapt to dependencies between variables by constructing local quadratic approximations. Another approach is to improve sampling efficiency by suppressing random walks. This can be done using ‘overrelaxed ’ versions of univariate slice sampling procedures, or by using ‘reflective ’ multivariate slice sampling methods, which bounce off the edges of the slice.
Regularity Properties and Pathologies of PositionSpace renormalizationGroup transformations: Scope and Limitations of Gibbsian Theory
 J. Stat.Phys
, 1993
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The multivariate Tutte polynomial (alias Potts model) for graphs and matroids, Surveys
, 2005
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The stochastic randomcluster process and the uniqueness of randomcluster measures
, 1995
"... The randomcluster model is a generalisation of percolation and ferromagnetic Potts models, due to Fortuin and Kasteleyn (see [29]). Not only is the randomcluster model a worthwhile topic for study in its own right, but also it provides much information about phase transitions in the associated phy ..."
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Cited by 95 (14 self)
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The randomcluster model is a generalisation of percolation and ferromagnetic Potts models, due to Fortuin and Kasteleyn (see [29]). Not only is the randomcluster model a worthwhile topic for study in its own right, but also it provides much information about phase transitions in the associated physical models. This paper serves two functions. First, we introduce and survey randomcluster measures from the probabilist’s point of view, giving clear statements of some of the many open problems. Secondly, we present new results for such measures, as follows. We discuss the relationship between weak limits of randomcluster measures and measures satisfying a suitable DLR condition. Using an argument based on the convexity of pressure, we prove the uniqueness of randomcluster measures for all but (at most) countably many values of the parameter p. Related results concerning phase transition in two or more dimensions are included, together with various stimulating conjectures. The uniqueness of the infinite cluster is employed in an intrinsic way, in part of these arguments. In the second part of this paper is constructed a Markov process whose levelsets are reversible Markov processes with randomcluster measures as unique equilibrium measures. This construction enables a coupling of randomcluster measures for all values of p. Furthermore it leads to a proof of the semicontinuity of the percolation probability, and provides a heuristic probabilistic justification for the widely held belief that there is a firstorder phase transition if and only if the clusterweighting factor q is sufficiently large.
Auxiliary Variable Methods for Markov Chain Monte Carlo with Applications
 Journal of the American Statistical Association
, 1997
"... Suppose one wishes to sample from the density ß(x) using Markov chain Monte Carlo (MCMC). An auxiliary variable u and its conditional distribution ß(ujx) can be defined, giving the joint distribution ß(x; u) = ß(x)ß(ujx). A MCMC scheme which samples over this joint distribution can lead to substanti ..."
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Cited by 84 (1 self)
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Suppose one wishes to sample from the density ß(x) using Markov chain Monte Carlo (MCMC). An auxiliary variable u and its conditional distribution ß(ujx) can be defined, giving the joint distribution ß(x; u) = ß(x)ß(ujx). A MCMC scheme which samples over this joint distribution can lead to substantial gains in efficiency compared to standard approaches. The revolutionary algorithm of Swendsen and Wang (1987) is one such example. In addition to reviewing the SwendsenWang algorithm and its generalizations, this paper introduces a new auxiliary variable method called partial decoupling. Two applications in Bayesian image analysis are considered. The first is a binary classification problem in which partial decoupling out performs SW and single site Metropolis. The second is a PET reconstruction which uses the gray level prior of Geman and McClure (1987). A generalized SwendsenWang algorithm is developed for this problem, which reduces the computing time to the point that MCMC is a viabl...
Generalizing swendsenwang to sampling arbitrary posterior probabilities
, 2007
"... Abstract—Many vision tasks can be formulated as graph partition problems that minimize energy functions. For such problems, the Gibbs sampler [9] provides a general solution but is very slow, while other methods, such as Ncut [24] and graph cuts [4], [22], are computationally effective but only work ..."
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Cited by 77 (17 self)
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Abstract—Many vision tasks can be formulated as graph partition problems that minimize energy functions. For such problems, the Gibbs sampler [9] provides a general solution but is very slow, while other methods, such as Ncut [24] and graph cuts [4], [22], are computationally effective but only work for specific energy forms [17] and are not generally applicable. In this paper, we present a new inference algorithm that generalizes the SwendsenWang method [25] to arbitrary probabilities defined on graph partitions. We begin by computing graph edge weights, based on local image features. Then, the algorithm iterates two steps. 1) Graph clustering: It forms connected components by cutting the edges probabilistically based on their weights. 2) Graph relabeling: It selects one connected component and flips probabilistically, the coloring of all vertices in the component simultaneously. Thus, it realizes the split, merge, and regrouping of a “chunk ” of the graph, in contrast to Gibbs sampler that flips a single vertex.We prove that this algorithm simulates ergodic and reversibleMarkov chain jumps in the space of graph partitions and is applicable to arbitrary posterior probabilities or energy functions defined on graphs. We demonstrate the algorithm on two typical problems in computer vision—image segmentation and stereo vision. Experimentally, we show that it is 100400 times faster in CPU time than the classical Gibbs sampler and 2040 times faster then the DDMCMC segmentation algorithm [27]. For stereo, we compare performance with graph cuts and belief propagation. We also show that our algorithm can automatically infer generativemodels and obtain satisfactory results (better than the graphic cuts or belief propagation) in the same amount of time.
Data clustering using a model granular magnet
 Neural Computation
, 1997
"... We present a new approach to clustering, based on the physical properties of an inhomogeneous ferromagnet. No assumption is made regarding the underlying distribution of the data. We assign a Potts spin to each data point and introduce an interaction between neighboring points, whose strength is a d ..."
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Cited by 72 (4 self)
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We present a new approach to clustering, based on the physical properties of an inhomogeneous ferromagnet. No assumption is made regarding the underlying distribution of the data. We assign a Potts spin to each data point and introduce an interaction between neighboring points, whose strength is a decreasing function of the distance between the neighbors. This magnetic system exhibits three phases. At very low temperatures, it is completely ordered; all spins are aligned. At very high temperatures, the system does not exhibit any ordering, and in an intermediate regime, clusters of relatively strongly coupled spins become ordered, whereas different clusters remain uncorrelated. This intermediate phase is identified by a jump in the order parameters. The spinspin correlation function is used to partition the spins and the corresponding data points into clusters. We demonstrate on three synthetic and three real data sets how the method works. Detailed comparison to the performance of other techniques clearly indicates the relative success of our method. 1
The RandomCluster Model
, 2008
"... The class of randomcluster models is a unification of a variety of stochastic processes of significance for probability and statistical physics, including percolation, Ising, and Potts models; in addition, their study has impact on the theory of certain random combinatorial structures, and of elec ..."
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Cited by 69 (21 self)
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The class of randomcluster models is a unification of a variety of stochastic processes of significance for probability and statistical physics, including percolation, Ising, and Potts models; in addition, their study has impact on the theory of certain random combinatorial structures, and of electrical networks. Much (but not all) of the physical theory of Ising/Potts models is best implemented in the context of the randomcluster representation. This systematic summary of randomcluster models includes accounts of the fundamental methods and inequalities, the uniqueness and specification of infinitevolume measures, the existence and nature of the phase transition, and the structure of the subcritical and supercritical phases. The theory for twodimensional lattices is better developed than for three and more dimensions. There is a rich collection of open problems, including some of substantial significance for the general area of disordered systems, and these are highlighted when encountered. Amongst the major open questions, there is the problem of ascertaining the exact nature of the phase transition for general values of the clusterweighting factor q, and the problem of proving that the critical randomcluster model in two
Convergence of slice sampler Markov chains
, 1998
"... In this paper, we analyse theoretical properties of the slice sampler. We find that the algorithm has extremely robust geometric ergodicity properties. For the case of just one auxiliary variable, we demonstrate that the algorithm is stochastic monotone, and deduce analytic bounds on the total varia ..."
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Cited by 63 (9 self)
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In this paper, we analyse theoretical properties of the slice sampler. We find that the algorithm has extremely robust geometric ergodicity properties. For the case of just one auxiliary variable, we demonstrate that the algorithm is stochastic monotone, and deduce analytic bounds on the total variation distance from stationarity of the method using FosterLyapunov drift condition methodology.