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Register a new userTotal positivity in Markov structures
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Added by
Piotr Zwiernik
Publication date
2015
fields
Mathematical Statistics
and research's language is
English
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We discuss properties of distributions that are multivariate totally positive of order two (MTP2) related to conditional independence. In particular, we show that any independence model generated by an MTP2 distribution is a compositional semigraphoid which is upward-stable and singleton-transitive. In addition, we prove that any MTP2 distribution satisfying an appropriate support condition is faithful to its concentration graph. Finally, we analyze factorization properties of MTP2 distributions and discuss ways of constructing MTP2 distributions; in particular we give conditions on the log-linear parameters of a discrete distribution which ensure MTP2 and characterize conditional Gaussian distributions which satisfy MTP2.
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Markov chain models are used in various fields, such behavioral sciences or econometrics. Although the goodness of fit of the model is usually assessed by large sample approximation, it is desirable to use conditional tests if the sample size is not large. We study Markov bases for performing conditional tests of the toric homogeneous Markov chain model, which is the envelope exponential family for the usual homogeneous Markov chain model. We give a complete description of a Markov basis for the following cases: i) two-state, arbitrary length, ii) arbitrary finite state space and length of three. The general case remains to be a conjecture. We also present a numerical example of conditional tests based on our Markov basis.
We derive a Markov basis consisting of moves of degree at most three for two-state toric homogeneous Markov chain model of arbitrary length without parameters for initial states. Our basis consists of moves of degree three and degree one, which alter the initial frequencies, in addition to moves of degree two and degree one for toric homogeneous Markov chain model with parameters for initial states.
We prove the conjecture by Diaconis and Eriksson (2006) that the Markov degree of the Birkhoff model is three. In fact, we prove the conjecture in a generalization of the Birkhoff model, where each voter is asked to rank a fixed number, say r, of candidates among all candidates. We also give an exhaustive characterization of Markov bases for small r.
We establish Bernstein inequalities for functions of general (general-state-space, not necessarily reversible) Markov chains. These inequalities achieve sharp variance proxies and recover the classical Bernsteins inequality under independence. The key analysis lies in upper bounding the operator norm of a perturbed Markov transition kernel by the limiting operator norm of a sequence of finite-rank and perturbed Markov transition kernels. For each finite-rank and perturbed Markov kernel, we bound its norm by the sum of two convex functions. One coincides with what delivers the classical Bernsteins inequality, and the other reflects the influence of the Markov dependence. A convex analysis on conjugates of these two functions then derives our Bernstein inequalities.
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