On the closure of relational models

Relational models for contingency tables are generalizations of log-linear models, allowing effects associated with arbitrary subsets of cells in a possibly incomplete table, and not necessarily containing the overall effect. In this generality, the MLEs under Poisson and multinomial sampling are not always identical. This paper deals with the theory of maximum likelihood estimation in the case when there are observed zeros in the data. A unique MLE to such data is shown to always exist in the set of pointwise limits of sequences of distributions in the original model. This set is equal to the closure of the original model with respect to the Bregman information divergence. The same variant of iterative scaling may be used to compute the MLE in the original model and in its closure

Faithfulness and learning hypergraphs from discrete distributions

The concepts of faithfulness and strong-faithfulness are important for statistical learning of graphical models. Graphs are not sufficient for describing the association structure of a discrete distribution. Hypergraphs representing hierarchical log-linear models are considered instead, and the concept of parametric (strong-) faithfulness with respect to a hypergraph is introduced. Strong-faithfulness ensures the existence of uniformly consistent parameter estimators and enables building uniformly consistent procedures for a hypergraph search. The strength of association in a discrete distribution can be quantified with various measures, leading to different concepts of strong-faithfulness. Lower and upper bounds for the proportions of distributions that do not satisfy strong-faithfulness are computed for different parameterizations and measures of association.Comment: 23 pages, 6 figure

Relational models for contingency tables

The paper considers general multiplicative models for complete and incomplete contingency tables that generalize log-linear and several other models and are entirely coordinate free. Sufficient conditions of the existence of maximum likelihood estimates under these models are given, and it is shown that the usual equivalence between multinomial and Poisson likelihoods holds if and only if an overall effect is present in the model. If such an effect is not assumed, the model becomes a curved exponential family and a related mixed parameterization is given that relies on non-homogeneous odds ratios. Several examples are presented to illustrate the properties and use of such models

Entropy and Hausdorff Dimension in Random Growing Trees

We investigate the limiting behavior of random tree growth in preferential attachment models. The tree stems from a root, and we add vertices to the system one-by-one at random, according to a rule which depends on the degree distribution of the already existing tree. The so-called weight function, in terms of which the rule of attachment is formulated, is such that each vertex in the tree can have at most K children. We define the concept of a certain random measure mu on the leaves of the limiting tree, which captures a global property of the tree growth in a natural way. We prove that the Hausdorff and the packing dimension of this limiting measure is equal and constant with probability one. Moreover, the local dimension of mu equals the Hausdorff dimension at mu-almost every point. We give an explicit formula for the dimension, given the rule of attachment

Being a Progressive in Divinitia

In Liberalism’s Religion, Cécile Laborde defends a theory of liberal secularism that is compatible with a minimal separation of religion and politics. According to her view, liberal state—she calls it Divinitia—that symbolically establishes the historic majority’s religious doctrine and inspires some of its legislation on a conservative interpretation of such religious tradition can be legitimate. In this article I analyse how is it like to belong to the minority of liberal progressive citizens in a country like Divinitia. I argue that their political activism will be defeated by Divinitia’s status quo on at least four different grounds. First, in virtue of being a minority, liberal progressive citizens would rarely obtain democratic victories; second, the conservative majority could rightly argue that they do not have reasons to compromise their views in order to accommodate progressives’; third, the conservative majority can rightly complain that counter-majoritarian initiatives advanced by progressives are unfair; and four, Divinitia’s public reason reproduces an asymmetry, for religiously inspired reasons can be accessible and therefore justificatory in politics, while the reasons progressives would desire to present in public deliberation would not be accessible to their conservative fellow citizens

Being a Progressive in Divinitia

In Liberalism’s Religion, Cécile Laborde defends a theory of liberal secularism that is compatible with a minimal separation of religion and politics. According to her view, liberal state—she calls it Divinitia—that symbolically establishes the historic majority’s religious doctrine and inspires some of its legislation on a conservative interpretation of such religious tradition can be legitimate. In this article I analyse how is it like to belong to the minority of liberal progressive citizens in a country like Divinitia. I argue that their political activism will be defeated by Divinitia’s status quo on at least four different grounds. First, in virtue of being a minority, liberal progressive citizens would rarely obtain democratic victories; second, the conservative majority could rightly argue that they do not have reasons to compromise their views in order to accommodate progressives’; third, the conservative majority can rightly complain that counter-majoritarian initiatives advanced by progressives are unfair; and four, Divinitia’s public reason reproduces an asymmetry, for religiously inspired reasons can be accessible and therefore justificatory in politics, while the reasons progressives would desire to present in public deliberation would not be accessible to their conservative fellow citizens

Ecological peculiarities of natural populations of hyperhalobe microalga Dunaliella salina Teod. in solar salt work ponds of the South of Ukraine and Russia

The paper presents the results of expedition research of some solar salt works of the South of Ukraine (Kherson region, AR Crimea) and lake Baskunchak (Astrachan region, Russia), as well as stationary observations on populations of the microalga Dunaliella salina Teod. in the ponds of Heroyske salt works (Gola Prystan’ district, Kherson region) that we carried out in 2005–2008. We discuss the approaches to modeling natural environment in D. salina laboratory culture to develop and optimize the process of industrial culturing in the open culture