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Minimal Markov bases of configurations of integer vectors correspond to minimal binomial generating sets of the assocciated lattice ideal. We give necessary and sufficient conditions for the elements of a minimal Markov basis to be (a) inside the universal Gr{ o}bner basis and (b) inside the Graver basis. We study properties of Markov bases of generalized Lawrence liftings for arbitrary matrices $Ainmathcal{M}_{mtimes n}(Bbb{Z})$ and $Binmathcal{M}_{ptimes n}(Bbb{Z})$ and show that in cases of interest the {em complexity} of any two Markov bases is the same.
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We study how to lift Markov bases and Grobner bases along linear maps of lattices. We give a lifting algorithm that allows to compute such bases iteratively provided a certain associated semigroup is normal. Our main application is the toric fiber product of toric ideals, where lifting gives Markov bases of the factor ideals that satisfy the compatible projection property. We illustrate the technique by computing Markov bases of various infinite families of hierarchical models. The methodology also implies new finiteness results for iterated toric fiber products.
Let n be a positive integer, and let A be a strongly commutative differential graded (DG) algebra over a commutative ring R. Assume that (a) B=A[X_1,...,X_n] is a polynomial extension of A, where X_1,...,X_n are variables of positive degrees; or (b) A is a divided power DG R-algebra and B=A<X_1,...,X_n> is a free extension of A obtained by adjunction of variables X_1,...,X_n of positive degrees. In this paper, we study naive liftability of DG modules along the natural injection A-->B using the notions of diagonal ideals and homotopy limits. We prove that if N is a bounded below semifree DG B-module such that Ext_B^i(N, N)=0 for all i>0, then N is naively liftable to A. This implies that N is a direct summand of a DG B-module that is liftable to A. Also, the relation between naive liftability of DG modules and the Auslander-Reiten Conjecture has been described.
We consider the connections among `clumped residual allocation models (RAMs), a general class of stick-breaking processes including Dirichlet processes, and the occupation laws of certain discrete space time-inhomogeneous Markov chains related to simulated annealing and other applications. An intermediate structure is introduced in a given RAM, where proportions between successive indices in a list are added or clumped together to form another RAM. In particular, when the initial RAM is a Griffiths-Engen-McCloskey (GEM) sequence and the indices are given by the random times that an auxiliary Markov chain jumps away from its current state, the joint law of the intermediate RAM and the locations visited in the sojourns is given in terms of a `disordered GEM sequence, and an induced Markov chain. Through this joint law, we identify a large class of `stick breaking processes as the limits of empirical occupation measures for associated time-inhomogeneous Markov chains.
Assessing the magnitude of cause-and-effect relations is one of the central challenges found throughout the empirical sciences. The problem of identification of causal effects is concerned with determining whether a causal effect can be computed from a combination of observational data and substantive knowledge about the domain under investigation, which is formally expressed in the form of a causal graph. In many practical settings, however, the knowledge available for the researcher is not strong enough so as to specify a unique causal graph. Another line of investigation attempts to use observational data to learn a qualitative description of the domain called a Markov equivalence class, which is the collection of causal graphs that share the same set of observed features. In this paper, we marry both approaches and study the problem of causal identification from an equivalence class, represented by a partial ancestral graph (PAG). We start by deriving a set of graphical properties of PAGs that are carried over to its induced subgraphs. We then develop an algorithm to compute the effect of an arbitrary set of variables on an arbitrary outcome set. We show that the algorithm is strictly more powerful than the current state of the art found in the literature.
Our purpose is to prove central limit theorem for countable nonhomogeneous Markov chain under the condition of uniform convergence of transition probability matrices for countable nonhomogeneous Markov chain in Ces`aro sense. Furthermore, we obtain a corresponding moderate deviation theorem for countable nonhomogeneous Markov chain by Gartner-Ellis theorem and exponential equivalent method.
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