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Title: Binary hidden Markov models and varieties, AS2012 Special Volume, part 2: This issue includes a second series of papers from talks, posters and collaborations resulting from and inspired by the Algebraic Statistics in the Alleghenies Conference at Penn State, which took place in July 2012.
Creator: Critch, Andrew
Date: 2013, 2013
Description: This paper closely examines HMMs in which all the hidden random variables are...
Show moreThis paper closely examines HMMs in which all the hidden random variables are binary. Its main contributions are (1) a birational parametrization for every such HMM, with an explicit inverse for recovering the hidden parameters in terms of observables, (2) a semialgebraic model membership test for every such HMM, and (3) minimal dening equations for the 4-node fully binary model, comprising 21 quadrics and 29 cubics, which were computed using Grobner bases in the cumulant coordinates of Sturmfels and Zwiernik. The new model parameters in (1) are rationally identiable in the sense of Sullivant, Garcia-Puente, and Spielvogel, and each model's Zariski closure is therefore a rational projective variety of dimension 5. Grobner basis computations for the model and its graph are found to be considerably faster using these parameters. In the case of two hidden states, item (2) supersedes a previous algorithm of Schonhuth which is only generically dened, and the dening equations (3) yield new invariants for HMMs of all lengths 4. Such invariants have been used successfully in model selection problems in phylogenetics, and one can hope for similar applications in the case of HMMs.
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Collection: Journal of Algebraic Statistics

Title: On Polyhedral Approximations of Polytopes for Learning Bayesian Networks, AS2012 Special Volume, part 2: This issue includes a second series of papers from talks, posters and collaborations resulting from and inspired by the Algebraic Statistics in the Alleghenies Conference at Penn State, which took place in July 2012.
Creator: Studeny, Milan, Haws, David C.
Date: 2013, 2013
Description: The motivation for this paper is the geometric approach to statistical learning Bayesiannetwork (BN) structures. We review three vector...
Show moreThe motivation for this paper is the geometric approach to statistical learning Bayesiannetwork (BN) structures. We review three vector encodings of BN structures. The first one has been used by Jaakkola et al. [9] and also by Cussens [4], the other two use special integral vectors formerly introduced, called imsets [18, 20]. The topic is the comparison of outer polyhedral approximations of the corresponding polytopes. We show how to transform the inequalities suggested by Jaakkola et al. [9] into the framework of imsets. The result of our comparison is the observation that the implicit polyhedral approximation of the standard imset polytope suggested in [21] gives a tighter approximation than the (transformed) explicit polyhedral approximation from [9]. As a consequence, we confirm a conjecture from [21] that the above-mentioned implicit polyhedral approximation of the standard imset polytope is an LP relaxation of that polytope. In the end, we review recent attempts to apply the methods of integer programming to learning BN structures and discuss the task of finding suitable explicit LP relaxation in the imset-based approach.
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Collection: Journal of Algebraic Statistics

Title: Learning Coefficient in Bayesian Estimation of Restricted Boltzmann Machine, AS2012 Special Volume, part 2: This issue includes a second series of papers from talks, posters and collaborations resulting from and inspired by the Algebraic Statistics in the Alleghenies Conference at Penn State, which took place in July 2012.
Creator: Aoyagi, Miki
Date: 2013, 2013
Description: We consider the real log canonical threshold for the learning model in Bayesian estimation. This threshold corresponds to a learning...
Show moreWe consider the real log canonical threshold for the learning model in Bayesian estimation. This threshold corresponds to a learning coefficient of generalization error in Bayesian estimation, which serves to measure learning efficiency in hierarchical learning models [30, 31, 33]. In this paper, we clarify the ideal which gives the log canonical threshold of the restricted Boltzmann machine and consider the learning coefficients of this model.
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Collection: Journal of Algebraic Statistics

Title: Phylogenetic invariants for group-based models, AS2012 Special Volume, part 1: This issue includes a second series of papers from talks, posters and collaborations resulting from and inspired by the Algebraic Statistics in the Alleghenies Conference at Penn State, which took place in July 2012.
Creator: Donten-Bury, Maria, Michalek, Mateusz
Date: 2012, 2012
Description: In this paper we investigate properties of algebraic varieties representing group-based phylogenetic models. We propose a method of generating...
Show moreIn this paper we investigate properties of algebraic varieties representing group-based phylogenetic models. We propose a method of generating many phylogenetic invariants. We prove that we obtain all invariants for any tree for the two-state Jukes-Cantor model. We conjecture that for a large class of models our method can give all phylogenetic invariants for any tree. We show that for 3-Kimura our conjecture is equivalent to the conjecture of Sturmfels and Sullivant [22, Conjecture 2]. This, combined with the results in [22], would make it possible to determine all phylogenetic invariants for any tree for 3-Kimura model, and also other phylogenetic models. Next we give the (first) examples of non-normal varieties associated to general group-based model for an abelian group. Following Kubjas [17] we prove that for many group-based models varieties associated to trees with the same number of leaves do not have to be deformation equivalent.
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Collection: Journal of Algebraic Statistics

Title: Properties of semi-elementary imsets as sums of elementary imsets
Creator: Kashimura, Takuya, Sei, Tomonari, Takemura, Akimichi, Tanaka, Kentaro
Date: 2011, 2011
Description: We study properties of semi-elementary imsets and elementary imsets introduced by Studeny [10]. The rules of the semi-graphoid axiom ...
Show moreWe study properties of semi-elementary imsets and elementary imsets introduced by Studeny [10]. The rules of the semi-graphoid axiom (decomposition, weak union and contraction) for conditional independence statements can be translated into a simple identity among three semi-elementary imsets. By recursively applying the identity, any semi-elementary imset can be written as a sum of elementary imsets, which we call a representation of the semi-elementary imset. A semi-elementary imset has many representations. We study properties of the set of possible representations of a semi-elementary imset and prove that all representations are connected by relations among four elementary imsets.
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Collection: Journal of Algebraic Statistics

Title: Higher Connectivity of Fiber Graphs of Gröbner Bases, AS2012 Special Volume, part 2: This issue includes a second series of papers from talks, posters and collaborations resulting from and inspired by the Algebraic Statistics in the Alleghenies Conference at Penn State, which took place in July 2012.
Creator: Potka, Samu
Date: 2013, 2013
Description: Fiber graphs of Gröbner bases from contingency tables are important in statistical hypothesis testing, where one studies random walks on these...
Show moreFiber graphs of Gröbner bases from contingency tables are important in statistical hypothesis testing, where one studies random walks on these graphs using the Metropolis-Hastings algorithm. The connectivity of the graphs has implications on how fast the algorithm converges. In this paper, we study a class of ber graphs with elementary combinatorial techniques and provide results that support a recent conjecture of Engström: the connectivity is given by the minimum vertex degree.
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Collection: Journal of Algebraic Statistics

Title: An Iterative Method Converging to a Positive Solution of Certain Systems of Polynomial Equations
Creator: Cartwright, Dustin
Date: 2011, 2011
Description: We present a numerical algorithm for finding real non-negative solutions to a certain class of polynomial equations. Our methods are based on...
Show moreWe present a numerical algorithm for finding real non-negative solutions to a certain class of polynomial equations. Our methods are based on the expectation maximization and iterative proportional fitting algorithms, which are used in statistics to find maximum likelihood parameters for certain classes of statistical models. Since our algorithm works by iteratively improving an approximate solution, we find approximate solutions in the cases when there are no exact solutions, such as overconstrained systems.
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Collection: Journal of Algebraic Statistics

Title: Geometry of Higher-Order Markov Chains, AS2012 Special Volume, part 1: This issue includes a second series of papers from talks, posters and collaborations resulting from and inspired by the Algebraic Statistics in the Alleghenies Conference at Penn State, which took place in July 2012.
Creator: Sturmfels, Bernd
Date: 2012, 2012
Description: We determine an explicit Gr ?obner basis, consisting of linear forms and determinantal quadrics, for the prime ideal of Raftery’s mixture...
Show moreWe determine an explicit Gr ?obner basis, consisting of linear forms and determinantal quadrics, for the prime ideal of Raftery’s mixture transition distribution model for Markov chains. When the states are binary, the corresponding projective variety is a linear space, the model itself consists of two simplices in a cross-polytope, and the likelihood function typically has two local maxima. In the general non-binary case, the model corresponds to a cone over a Segre variety.
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Collection: Journal of Algebraic Statistics

Title: Betti Numbers of Cut Ideals of Trees, AS2012 Special Volume, part 2: This issue includes a second series of papers from talks, posters and collaborations resulting from and inspired by the Algebraic Statistics in the Alleghenies Conference at Penn State, which took place in July 2012.
Creator: Potka, Samu, Sarmiento, Camilo
Date: 2013, 2013
Description: Cut ideals, introduced by Sturmfels and Sullivant, are used in phylogenetics and algebraic statistics. We study the minimal free resolutions...
Show moreCut ideals, introduced by Sturmfels and Sullivant, are used in phylogenetics and algebraic statistics. We study the minimal free resolutions of cut ideals of tree graphs. By employing basic methods from topological combinatorics, we obtain upper bounds for the Betti numbers of this type of ideals. These take the form of simple formulas on the number of vertices, which arise from the enumeration of induced subgraphs of certain incomparability graphs associated to the edge sets of trees.
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Collection: Journal of Algebraic Statistics

Title: The geometry of Sloppiness
Creator: Dufresne, Emilie, Harrington , Heather A, Raman, Dhruva V
Date: 2018, 2018-09-24
Description: The use of mathematical models in the sciences often requires the estimation of unknown parameter values from data. Sloppiness provides...
Show moreThe use of mathematical models in the sciences often requires the estimation of unknown parameter values from data. Sloppiness provides information about the uncertainty of this task. In this paper, we develop a precise mathematical foundation for sloppiness and define rigorously its key concepts, such as `model manifold', in relation to concepts of structural identifiability. We redefine sloppiness conceptually as a comparison between the premetric on parameter space induced by measurement noise and a reference metric. This opens up the possibility of alternative quantification of sloppiness, beyond the standard use of the Fisher Information Matrix, which assumes that parameter space is equipped with the usual Euclidean and the measurement error is infinitesimal. Applications include parametric statistical models, explicit time dependent models, and ordinary differential equation models.
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Collection: Journal of Algebraic Statistics

Title: Mixture models for rating data: the method of moments via Groebner bases
Creator: Iannario, Maria, Simone, Rosaria
Date: 2017, 2017-12-26
Description: A recent thread of research in ordinal data analysis involves a class of mixture models that designs the responses as the combination of the...
Show moreA recent thread of research in ordinal data analysis involves a class of mixture models that designs the responses as the combination of the two main aspects driving the decision pro- cess: a feeling and an uncertainty components. This novel paradigm has been proven flexible to account also for overdispersion. In this context, Groebner bases are exploited to estimate model parameters by implementing the method of moments. In order to strengthen the validity of the moment procedure so derived, alternatives parameter estimates are tested by means of a simulation experiment. Results show that the moment estimators are satisfactory per se, and that they significantly reduce the bias and perform more efficiently than others when they are set as starting values for the Expectation-Maximization algorithm.
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Collection: Journal of Algebraic Statistics

Title: Ideal-Theoretic Strategies for Asymptotic Approximation of Marginal Likelihood Integrals
Creator: Lin, Shaowei
Date: 2017, 2017-02-08
Description: The accurate asymptotic evaluation of marginal likelihood integrals is a fundamental problem in Bayesian statistics. Following the approach...
Show moreThe accurate asymptotic evaluation of marginal likelihood integrals is a fundamental problem in Bayesian statistics. Following the approach introduced by Watanabe, we translate this into a problem of computational algebraic geometry, namely, to determine the real log canonical threshold of a polynomial ideal, and we present effective methods for solving this problem. Our results are based on resolution of singularities. They apply to parametric models where the Kullback-Leibler distance is upper and lower bounded by scalar multiples of some sum of squared real analytic functions. Such models include finite state discrete models.
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Collection: Journal of Algebraic Statistics

Title: Unimodular hierarchical models and their Graver bases
Creator: Bernstein, Daniel Irving, O'Neill, Christopher
Date: 2017, 2017-12-26
Description: Given a simplicial complex whose vertices are labeled with positive integers, one can associate a vector configuration whose corresponding...
Show moreGiven a simplicial complex whose vertices are labeled with positive integers, one can associate a vector configuration whose corresponding toric variety is the Zariski closure of a hierarchical model. We classify all the vertex-weighted simplicial complexes that give rise to unimodular vector configurations. We also provide a combinatorial characterization of their Graver bases.
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Collection: Journal of Algebraic Statistics

Title: Mixtures and products in two graphical models
Creator: Seigal,Anna, Montufar, Guido
Date: 2018, 2018-09-24
Description: We compare two statistical models of three binary random variables. One is a mixture model and the other is a product of mixtures model called...
Show moreWe compare two statistical models of three binary random variables. One is a mixture model and the other is a product of mixtures model called a restricted Boltzmann machine. Although the two models we study look different from their parametrizations, we show that they represent the same set of distributions on the interior of the probability simplex, and are equal up to closure. We give a semi-algebraic description of the model in terms of six binomial inequalities and obtain closed form expressions for the maximum likelihood estimates. We briefly discuss extensions to larger models.
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Collection: Journal of Algebraic Statistics

Title: Markov bases for two-way change-point models of ladder determinantal tables
Creator: Aoki, Satoshi, Hibi, Takayuki
Date: 2017, 2017-02-08
Description: To evaluate the goodness-of-fit of a statistical model to given data, calculating a conditional p value by a Markov chain Monte Carlo method...
Show moreTo evaluate the goodness-of-fit of a statistical model to given data, calculating a conditional p value by a Markov chain Monte Carlo method is one of the effective approaches. For this purpose, a Markov basis plays an important role because it guarantees the connectivity of the chain, which is needed for unbiasedness of the estimation, and therefore is investigated in various settings such as incomplete tables or subtable sum constraints. In this paper, we consider the two-way change-point model for the ladder determinantal table, which is an extension of these two previous works, i.e., works on incomplete tables by Aoki and Takemura (2005, J. Stat. Comput. Simulat.) and subtable some constraints by Hara, Takemura and Yoshida (2010, J. Pure Appl. Algebra). Our main result is based on the theory of Gr ?obner basis for the distributive lattice. We give a numerical example for actual data.
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Collection: Journal of Algebraic Statistics

Title: Cubature Rules and Expected Value of Some Complex Functions, Special Volume in honor of memory of S.E.Fienberg
Creator: Fassino, Claudia, Riccomagno, Eva, Rogantin, Maria Piera
Date: 2019, 2019-04-12
Description: The expected value of some complex valued random vectors is computed by means of the indicator function of a designed experiment as known in...
Show moreThe expected value of some complex valued random vectors is computed by means of the indicator function of a designed experiment as known in algebraic statistics. The general theory is set-up and results are obtained for finite discrete random vectors and the Gaussian random vector. The precision space of some cubature rules/designed experiments is determined.
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Collection: Journal of Algebraic Statistics

Title: Strongly Robust Toric Ideals in Codimension 2, Special Volume in honor of memory of S.E.Fienberg
Creator: Sullivant ,Seth
Date: 2019, 2019-04-12
Description: A homogeneous ideal is robust if its universal Gr ?obner basis is also a minimal generating set. For toric ideals, one has the stronger...
Show moreA homogeneous ideal is robust if its universal Gr ?obner basis is also a minimal generating set. For toric ideals, one has the stronger definition: A toric ideal is strongly robust if its Graver basis equals the set of indispensable binomials. We characterize the codimension 2 strongly robust toric ideals by their Gale diagrams. This gives a positive answer to a question of Petrovi?, Thoma, and Vladoiu in the case of codimension 2 toric ideals.
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Collection: Journal of Algebraic Statistics

Title: Maximum Likelihood for Matrices with Rank Constraints
Creator: Hauenstein, Jonathan, Rodriguez, Jose Israel, Sturmfels, Bernd
Date: 2014, 2014-04-30
Description: Maximum likelihood estimation is a fundamental optimization problem in statistics. We study this problem on manifolds of matrices with bounded...
Show moreMaximum likelihood estimation is a fundamental optimization problem in statistics. We study this problem on manifolds of matrices with bounded rank. These represent mixtures of distributions of two independent discrete random variables. We determine the maximum likelihood degree for a range of determinantal varieties, and we apply numerical algebraic geometry to compute all critical points of their likelihood functions. This led to the discovery of maximum likelihood duality between matrices of complementary ranks, a result proved subsequently by Draisma and Rodriguez.
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Collection: Journal of Algebraic Statistics

Title: Generic Identification of Binary-Valued Hidden Markov Processes
Creator: Schönhuth, Alexander
Date: 2014, 2014-04-30
Description: The generic identification problem is to decide whether a stochastic process (X_t) is a hidden Markov process and if yes to infer its...
Show moreThe generic identification problem is to decide whether a stochastic process (X_t) is a hidden Markov process and if yes to infer its parameters for all but a subset of parametrizations that form a lower-dimensional subvariety in parameter space. Partial answers so far available depend on extra assumptions on the processes, which are usually centered around stationarity. Here we present a general solution for binary-valued hidden Markov processes. Our approach is rooted in algebraic statistics hence it is geometric in nature. We find that the algebraic varieties associated with the probability distributions of binary-valued hidden Markov processes are zero sets of determinantal equations which draws a connection to well-studied objects from algebra. As a consequence, our solution allows for algorithmic implementation based on elementary (linear) algebraic routines.
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Collection: Journal of Algebraic Statistics

Title: Estimation for Dyadic-Dependent Exponential Random Graph Models
Creator: Yang, Xiaolin, Rinaldo, Alessandro, Fienberg, Stephen E.
Date: 2014, 2014-04-30
Description: Graphs are the primary mathematical representation for networks, with nodes or vertices corresponding to units (e.g., individuals) and edges...
Show moreGraphs are the primary mathematical representation for networks, with nodes or vertices corresponding to units (e.g., individuals) and edges corresponding to relationships. Exponential Random Graph Models (ERGMs) are widely used for describing network data because of their simple structure as an exponential function of a sum of parameters multiplied by their corresponding sufficient statistics. As with other exponential family settings the key computational difficulty is determining the normalizing constant for the likelihood function, a quantity that depends only on the data. In ERGMs for network data, the normalizing constant in the model often makes the parameter estimation intractable for large graphs, when the model involves dependence among dyads in the graph. One way to deal with this problem is to approximate the likelihood function by something tractable, e.g., by using the method of pseudo-likelihood estimation suggested in the early literature. In this paper, we describe the family of ERGMs and explain the increasing complexity that arises from imposing different edge dependence and homogeneous parameter assumptions. We then compare maximum likelihood (ML) and maximum pseudo-likelihood (MPL) estimation schemes with respect to existence and related degeneracy properties for ERGMs involving dependencies among dyads.
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Collection: Journal of Algebraic Statistics

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