Results 1  10
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34
Exact Sampling with Coupled Markov Chains and Applications to Statistical Mechanics
, 1996
"... For many applications it is useful to sample from a finite set of objects in accordance with some particular distribution. One approach is to run an ergodic (i.e., irreducible aperiodic) Markov chain whose stationary distribution is the desired distribution on this set; after the Markov chain has ..."
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Cited by 406 (13 self)
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For many applications it is useful to sample from a finite set of objects in accordance with some particular distribution. One approach is to run an ergodic (i.e., irreducible aperiodic) Markov chain whose stationary distribution is the desired distribution on this set; after the Markov chain has run for M steps, with M sufficiently large, the distribution governing the state of the chain approximates the desired distribution. Unfortunately it can be difficult to determine how large M needs to be. We describe a simple variant of this method that determines on its own when to stop, and that outputs samples in exact accordance with the desired distribution. The method uses couplings, which have also played a role in other sampling schemes; however, rather than running the coupled chains from the present into the future, one runs from a distant point in the past up until the present, where the distance into the past that one needs to go is determined during the running of the al...
Algebraic Algorithms for Sampling from Conditional Distributions
 Annals of Statistics
, 1995
"... We construct Markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic. Examples include generating tables with fixed row and column sums and higher dimensional analogs. The algorithms involve finding bases for associated polynomial ideals and so a ..."
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Cited by 192 (16 self)
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We construct Markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic. Examples include generating tables with fixed row and column sums and higher dimensional analogs. The algorithms involve finding bases for associated polynomial ideals and so an excursion into computational algebraic geometry.
An Interruptible Algorithm for Perfect Sampling via Markov Chains
 Annals of Applied Probability
, 1998
"... For a large class of examples arising in statistical physics known as attractive spin systems (e.g., the Ising model), one seeks to sample from a probability distribution # on an enormously large state space, but elementary sampling is ruled out by the infeasibility of calculating an appropriate nor ..."
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Cited by 84 (7 self)
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For a large class of examples arising in statistical physics known as attractive spin systems (e.g., the Ising model), one seeks to sample from a probability distribution # on an enormously large state space, but elementary sampling is ruled out by the infeasibility of calculating an appropriate normalizing constant. The same difficulty arises in computer science problems where one seeks to sample randomly from a large finite distributive lattice whose precise size cannot be ascertained in any reasonable amount of time. The Markov chain Monte Carlo (MCMC) approximate sampling approach to such a problem is to construct and run "for a long time" a Markov chain with longrun distribution #. But determining how long is long enough to get a good approximation can be both analytically and empirically difficult. Recently, Jim Propp and David Wilson have devised an ingenious and efficient algorithm to use the same Markov chains to produce perfect (i.e., exact) samples from #. However, the running t...
Sequential Monte Carlo methods for statistical analysis of tables
 J. Amer. Statist. Assoc
"... We describe a sequential importance sampling (SIS) procedure for analyzing twoway zero–one or contingency tables with fixed marginal sums. An essential feature of the new method is that it samples the columns of the table progressively according to certain special distributions. Our method produces ..."
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Cited by 51 (10 self)
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We describe a sequential importance sampling (SIS) procedure for analyzing twoway zero–one or contingency tables with fixed marginal sums. An essential feature of the new method is that it samples the columns of the table progressively according to certain special distributions. Our method produces Monte Carlo samples that are remarkably close to the uniform distribution, enabling one to approximate closely the null distributions of various test statistics about these tables. Our method compares favorably with other existing Monte Carlobased algorithms, and sometimes is a few orders of magnitude more efficient. In particular, compared with Markov chain Monte Carlo (MCMC)based approaches, our importance sampling method not only is more efficient in terms of absolute running time and frees one from pondering over the mixing issue, but also provides an easy and accurate estimate of the total number of tables with fixed marginal sums, which is far more difficult for an MCMC method to achieve.
Minimal basis for connected Markov chain over 3 × 3 × K contingency tables with fixed twodimensional marginals
, 2002
"... We consider connected Markov chain for sampling 3 × 3 × K contingency tables having fixed twodimensional marginal totals. Such sampling arises inperforming various tests of... ..."
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Cited by 44 (16 self)
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We consider connected Markov chain for sampling 3 × 3 × K contingency tables having fixed twodimensional marginal totals. Such sampling arises inperforming various tests of...
Inference in Curved Exponential Family Models for Networks
 Journal of Computational and Graphical Statistics
, 2006
"... Network data arise in a wide variety of applications. Although descriptive statistics for networks abound in the literature, the science of fitting statistical models to complex network data is still in its infancy. The models considered in this article are based on exponential families; therefore, ..."
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Cited by 42 (9 self)
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Network data arise in a wide variety of applications. Although descriptive statistics for networks abound in the literature, the science of fitting statistical models to complex network data is still in its infancy. The models considered in this article are based on exponential families; therefore, we refer to them as exponential random graph models (ERGMs). Although ERGMs are easy to postulate, maximum likelihood estimation of parameters in these models is very difficult. In this article, we first review the method of maximum likelihood estimation using Markov chain Monte Carlo in the context of fitting linear ERGMs. We then extend this methodology to the situation where the model comes from a curved exponential family. The curved exponential family methodology is applied to new specifications of ERGMs, proposed by Snijders et al. (2004), having nonlinear parameters to represent structural properties of networks such as transitivity and heterogeneity of degrees. We review the difficult topic of implementing likelihood ratio tests for these models, then apply all these modelfitting and testing techniques to the estimation of linear and nonlinear parameters for a collaboration network between partners in a New England law firm.
Markov chain Monte Carlo exact tests for incomplete twoway contingency tables
, 2002
"... We consider testing the quasiindependence hypothesis for twoway contingency tables which contain some structural zero cells. For sparse contingency tables where the large sample... ..."
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Cited by 23 (14 self)
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We consider testing the quasiindependence hypothesis for twoway contingency tables which contain some structural zero cells. For sparse contingency tables where the large sample...
Sequential importance sampling for multiway tables
 Annals of Statistics
, 2005
"... We describe an algorithm for the sequential sampling of entries in multiway contingency tables with given constraints. The algorithm can be used for computations in exact conditional inference. To justify the algorithm, a theory relates sampling values at each step to properties of the associated to ..."
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Cited by 21 (3 self)
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We describe an algorithm for the sequential sampling of entries in multiway contingency tables with given constraints. The algorithm can be used for computations in exact conditional inference. To justify the algorithm, a theory relates sampling values at each step to properties of the associated toric ideal using computational commutative algebra. In particular, the property of interval cell counts at each step is related to exponents on lead indeterminates of a lexicographic Gröbner basis. Also, the approximation of integer programming by linear programming for sampling is related to initial terms of a toric ideal. We apply the algorithm to examples of contingency tables which appear in the social and medical sciences. The numerical results demonstrate that the theory is applicable and that the algorithm performs well. 1. Introduction. Sampling
Markov Chain Monte Carlo for Statistical Inference
 University of Washington, Center for
, 2000
"... These notes provide an introduction to Markov chain Monte Carlo methods that are useful in both Bayesian and frequent... ..."
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Cited by 18 (0 self)
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These notes provide an introduction to Markov chain Monte Carlo methods that are useful in both Bayesian and frequent...
Invariant minimal Markov basis for sampling contingency tables with fixed marginals
, 2003
"... In this paper we define an invariant Markov basis for a connected Markov chain over the set of contingency tables with fixed marginals and derive some characterizations of minimality of the invariant basis. We also give a necessary and sufficient condition for uniqueness of invariant minimal Markov ..."
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Cited by 11 (7 self)
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In this paper we define an invariant Markov basis for a connected Markov chain over the set of contingency tables with fixed marginals and derive some characterizations of minimality of the invariant basis. We also give a necessary and sufficient condition for uniqueness of invariant minimal Markov basis. The invariance here refers to permutation of indices of each axis of the contingency tables. If the categories of each axis do not have any order relations among them, it is natural to consider the action of the symmetric group on each axis of the contingency table. A general algebraic algorithm for obtaining a Markov basis was given by Diaconis and Sturmfels (1998). Their algorithm is based on computing Gröbner basis of a wellspecified polynomial ideal. However the reduced Gröbner basis depends on the particular term order and is not symmetric. Therefore it is of interest to consider properties of invariant Markov basis. We study minimality of invariant Markov basis using techniques of Takemura and Aoki (2003).