Abstract
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The problem
of maximum-likelihood learning of the structure of an unknown discrete
distribution from samples is considered when the distribution is Markov on a
tree. Large deviation analysis of the error in estimation of the set of edges
of the tree is performed. Necessary and sufficient conditions are provided to
ensure that this error probability decays exponentially. These conditions are
based on the mutual information between each pair of variables being distinct
from that of other pairs. The rate of error decay, or error exponent, is
derived using the large-deviation principle. The error exponent is
approximated using Euclidean information theory and is given by a ratio, to
be interpreted as the signal-to-noise ratio (SNR) for learning .Numerical
experiments show the SNR approximation is accurate. |
Index Terms—Large-deviations, Tree
structure learning, Error exponents, Euclidean Information Theory.