Maximum likelihood pedigree reconstruction using integer programming

Cussens, James

doi:10.29007/sghd

Cited by 13 publications

(42 citation statements)

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“…As shown in [Cussens, 2011;, the sub-IP in Figure 3 either detects a cluster constraint to add as a cut if this is possible, or is infeasible if no such cut exists.…”

Section: Complexity Of the Gobnilp Sub-ipsmentioning

confidence: 99%

“…Inequalities (4) are just one way that cycles can be prevented [Cussens, 2010;Peharz and Pernkopf, 2012;Cussens et al, 2013]. These particular constraints are known as cluster constraints [Jaakkola et al, 2010] as each corresponds to the constraint that any cluster (set) of nodes must have at least one node with no parents in that cluster.…”

Section: Bn Structure Learningmentioning

confidence: 99%

“…3 The GOBNILP System GOBNILP [Cussens, 2011] is a program that has been developed to find optimal Bayesian networks by solving an IP formulation of the BNSL problem using the SCIP IP system [Achterberg, 2007]. The search is implemented as a branch-and-cut approach, the essentials of which are outlined next.…”

Section: Bn Structure Learningmentioning

confidence: 99%

“…As shown in a recent study [Malone et al, 2014], perhaps the most successful exact approach to BNSL is provided by the GOBNILP system [Cussens, 2011]. GOBNILP implements a branch-and-cut approach to BNSL, using state-of-the-art IP solving techniques together with specialised BNSL cutting planes.…”

Section: Introductionmentioning

confidence: 99%

“…Due to NP-hardness, much work on BNSL has focused on developing approximate, local search style algorithms [Tsamardinos et al, 2006] that in general cannot guarantee that optimal structures in terms of the objective function are found. Recently, despite its complexity, several advances in exact approaches to BNSL have surfaced [Koivisto and Sood, 2004;Silander and Myllymäki, 2006;Cussens, 2011;de Campos and Ji, 2011;Yuan and Malone, 2013;van Beek and Hoffmann, 2015], ranging from problem-specific dynamic programming branch-and-bound algorithms to approaches based on A * -style state-space search, constraint programming, and integer linear programming (IP), which can, with certain restrictions, learn provably-optimal BN structures with tens to hundreds of nodes.…”

Section: Introductionmentioning

confidence: 99%

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Bayesian Network Structure Learning with Integer Programming: Polytopes, Facets and Complexity (Extended Abstract)

Cussens

Järvisalo

Korhonen

et al. 2017

Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence

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Developing accurate algorithms for learning structures of probabilistic graphical models is an important problem within modern AI research. Here we focus on score-based structure learning for Bayesian networks as arguably the most central class of graphical models. A successful generic approach to optimal Bayesian network structure learning (BNSL), based on integer programming (IP), is implemented in the GOBNILP system. Despite the recent algorithmic advances, current understanding of foundational aspects underlying the IP based approach to BNSL is still somewhat lacking. In this paper, we provide theoretical contributions towards understanding fundamental aspects of cutting planes and the related separation problem in this context, ranging from NP-hardness results to analysis of polytopes and the related facets in connection to BNSL.

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“…As shown in [Cussens, 2011;, the sub-IP in Figure 3 either detects a cluster constraint to add as a cut if this is possible, or is infeasible if no such cut exists.…”

Section: Complexity Of the Gobnilp Sub-ipsmentioning

confidence: 99%

Section: Bn Structure Learningmentioning

confidence: 99%

Section: Bn Structure Learningmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Bayesian Network Structure Learning with Integer Programming: Polytopes, Facets and Complexity (Extended Abstract)

Cussens

Järvisalo

Korhonen

et al. 2017

Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence

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Maximum Likelihood Pedigree Reconstruction Using Integer Linear Programming

Cussens

Bartlett

Jones

et al. 2012

Genetic Epidemiology

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Large population biobanks of unrelated individuals have been highly successful in detecting common genetic variants affecting diseases of public health concern. However, they lack the statistical power to detect more modest gene-gene and gene-environment interaction effects or the effects of rare variants for which related individuals are ideally required. In reality, most large population studies will undoubtedly contain sets of undeclared relatives, or pedigrees. Although a crude measure of relatedness might sometimes suffice, having a good estimate of the true pedigree would be much more informative if this could be obtained efficiently. Relatives are more likely to share longer haplotypes around disease susceptibility loci and are hence biologically more informative for rare variants than unrelated cases and controls. Distant relatives are arguably more useful for detecting variants with small effects because they are less likely to share masking environmental effects. Moreover, the identification of relatives enables appropriate adjustments of statistical analyses that typically assume unrelatedness. We propose to exploit an integer linear programming optimisation approach to pedigree learning, which is adapted to find valid pedigrees by imposing appropriate constraints. Our method is not restricted to small pedigrees and is guaranteed to return a maximum likelihood pedigree. With additional constraints, we can also search for multiple high-probability pedigrees and thus account for the inherent uncertainty in any particular pedigree reconstruction. The true pedigree is found very quickly by comparison with other methods when all individuals are observed. Extensions to more complex problems seem feasible.

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Polyhedral aspects of score equivalence in Bayesian network structure learning

Cussens¹,

Haws²,

Studený³

2016

Math. Program.

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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.

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Maximum likelihood pedigree reconstruction using integer programming

Cited by 13 publications

References 10 publications

Bayesian Network Structure Learning with Integer Programming: Polytopes, Facets and Complexity (Extended Abstract)

Bayesian Network Structure Learning with Integer Programming: Polytopes, Facets and Complexity (Extended Abstract)

Maximum Likelihood Pedigree Reconstruction Using Integer Linear Programming

Polyhedral aspects of score equivalence in Bayesian network structure learning

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