Probabilistic reasoning in intelligent systems: networks of plausible inference
Probabilistic reasoning in intelligent systems: networks of plausible inference
Learning equivalence classes of bayesian-network structures
The Journal of Machine Learning Research
Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing)
Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing)
Graphical Models, Exponential Families, and Variational Inference
Foundations and Trends® in Machine Learning
A large-deviation analysis for the maximum likelihood learning of tree structures
ISIT'09 Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 2
Spectral efficiency in the wideband regime
IEEE Transactions on Information Theory
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The problem of learning tree-structured Gaussian graphical models from i.i.d. samples is considered. The influence of the tree structure and the parameters of the Gaussian distribution on the learning rate as the number of samples increases is discussed. Specifically, the error exponent corresponding to the event that the estimated tree structure differs from the actual unknown tree structure of the distribution is analyzed. Finding the error exponent reduces to a least-squares problem in the very noisy learning regime. In this regime, it is shown that universally, the extremal tree structures which maximize and minimize the error exponent are the star and the Markov chain for any fixed set of correlation coefficients on the edges of the tree. In other words, the star and the chain graphs represent the hardest and the easiest structures to learn in the class of tree-structured Gaussian graphical models. This result can also be intuitively explained by correlation decay: pairs of nodes which are far apart, in terms of graph distance, are unlikely to be mistaken as edges by the maximum-likelihood estimator in the asymptotic regime.