Sequential importance sampling for multiway tables

Chen, Yuguo; Dinwoodie, Ian H.; Sullivant, Seth

doi:10.1214/009053605000000822

Cited by 71 publications

(81 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…. , u 4 Proof. We proceed by induction on k. For the base case k = 1, note that ∂∆ 1 is two isolated points.…”

Section: β-Avoiding Simplicial Complexesmentioning

confidence: 98%

“…In previous work of Rauh and the second author [11], normality was identified as a key property of hierarchical models to be able to apply the toric fiber product construction to calculate a Markov basis. If A C,d is unimodular, it is also easy to solve the integer programs that arise in sequential importance sampling [4]. It is a major open problem to classify the normal hierarchical models.…”

Section: Definition 12mentioning

confidence: 99%

See 1 more Smart Citation

Unimodular binary hierarchical models

Bernstein

Sullivant

2017

Journal of Combinatorial Theory, Series B

Self Cite

View full text Add to dashboard Cite

Abstract. Associated to each simplicial complex is a binary hierarchical model. We classify the simplicial complexes that yield unimodular binary hierarchical models. Our main theorem provides both a construction of all unimodular binary hierarchical models, together with a characterization in terms of excluded minors, where our definition of a minor allows the taking of links and induced complexes. A key tool in the proof is the lemma that the class of unimodular binary hierarchical models is closed under the Alexander duality operation on simplicial complexes.

show abstract

“…. , u 4 Proof. We proceed by induction on k. For the base case k = 1, note that ∂∆ 1 is two isolated points.…”

Section: β-Avoiding Simplicial Complexesmentioning

confidence: 98%

Section: Definition 12mentioning

confidence: 99%

Unimodular binary hierarchical models

Bernstein

Sullivant

2017

Journal of Combinatorial Theory, Series B

Self Cite

View full text Add to dashboard Cite

show abstract

“…In particular, a set of minimal Markov bases allows us to build a connected Markov chain and perform a random walk over all the points in the fiber that have the same fixed marginals and/or conditionals. Thus we can either enumerate or sample from the space of tables via Sequential Importance Sampling (SIS) or Markov Chain Monte Carlo (MCMC) sampling, e.g., see Chen et al [5]. Some disadvantages of the algebraic approach are that (1) calculation of Markov bases are computationally infeasible even for tables of small dimension, and (2) for conditionals, Markov bases are extremely sensitive to rounding of cell probabilities.…”

Section: Link Between Maximum Likelihood Estimates and Cellmentioning

confidence: 99%

“…In theory we could enumerate the number of possible tables utilizing algebraic techniques and software such as LattE [8] or sampling techniques such as MCMC and SIS [5]. Due to the large dimension of the solution polytope for this example, however, LattE is currently unable the execute the computation because the space of possible tables is extremely large.…”

Section: Examplementioning

confidence: 99%

Algebraic Statistics and Contingency Table Problems: Log-Linear Models, Likelihood Estimation, and Disclosure Limitation

Dobra

Fienberg

Rinaldo

et al. 2008

Emerging Applications of Algebraic Geometry

View full text Add to dashboard Cite

Abstract. Contingency tables have provided a fertile ground for the growth of algebraic statistics. In this paper we briefly outline some features of this work and point to open research problems. We focus on the problem of maximum likelihood estimation for log-linear models and a related problem of disclosure limitation to protect the confidentiality of individual responses. Risk of disclosure has often been measured either formally or informally in terms of information contained in marginal tables linked to a log-linear model analysis and has focused on disclosure potential of small cell counts, especially those equal to 1 or 2. One way to assess risk is to compute bounds for cell entries given a set of released marginals. Both of these methodologies become complicated for large sparse tables. This paper revisits the problem of computing bounds for cell entries and picks up on a theme first suggested in Fienberg [21] that there is an intimate link between the ideas on bounds and the existence of maximum likelihood estimates, and shows how these ideas can be made rigorous through the underlying mathematics of the same geometric/algebraic framework. We illustrate the linkages through a series of examples. We also discuss the more complex problem of releasing marginal and conditional information. We illustrate the statistical features of the methodology on two examples and then conclude with a series of open problems.

show abstract

“…Related algorithms using elegant random scan Gibbs samplers were given by ; McDonald, Smith, and Forster (1999); . Furthermore, relevant recent developments in sequential importance sampling (Chen, Diaconis, Holmes, and Liu 2005;Chen, Dinwoodie, and Sullivant 2006) are applicable to this setting. We refer the reader to Caffo and Booth (2003) for an overview of Monte Carlo algorithms in this area.…”

Section: The Softwarementioning

confidence: 99%

Exact Hypothesis Tests for Log-linear Models withexactLoglinTest

Caffo¹

2006

J. Stat. Soft.

View full text Add to dashboard Cite

This manuscript overviews exact testing of goodness of fit for log-linear models using the R package exactLoglinTest. This package evaluates model fit for Poisson log-linear models by conditioning on minimal sufficient statistics to remove nuisance parameters. A Monte Carlo algorithm is proposed to estimate P values from the resulting conditional distribution. In particular, this package implements a sequentially rounded normal approximation and importance sampling to approximate probabilities from the conditional distribution. Usually, this results in a high percentage of valid samples. However, in instances where this is not the case, a Metropolis Hastings algorithm can be implemented that makes more localized jumps within the reference set. The manuscript details how some conditional tests for binomial logit models can also be viewed as conditional Poisson log-linear models and hence can be performed via exactLoglinTest. A diverse battery of examples is considered to highlight use, features and extensions of the software. Notably, potential extensions to evaluating disclosure risk are also considered.

show abstract

Sequential importance sampling for multiway tables

Cited by 71 publications

References 28 publications

Unimodular binary hierarchical models

Unimodular binary hierarchical models

Algebraic Statistics and Contingency Table Problems: Log-Linear Models, Likelihood Estimation, and Disclosure Limitation

Exact Hypothesis Tests for Log-linear Models withexactLoglinTest

Contact Info

Product

Resources

About