2,162 research outputs found
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
Basic psychological needs and the development of interest in mathematics during a learning activity
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