Markov random fields are used to model high dimensional distributions in a number of applied areas. Much recent interest has been devoted to the reconstruction of the dependency structure from independent samples from the Markov random fields. We analyze a simple algorithm for reconstructing the underlying graph defining a Markov random field on @math nodes and maximum degree @math given observations. We show that under mild non-degeneracy conditions it reconstructs the generating graph with high probability using @math samples where @math depend on the local interactions. For most local interaction @math are of order @math . Our results are optimal as a function of @math up to a multiplicative constant depending on @math and the strength of the local interactions. Our results seem to be the first results for general models that guarantee that the generating model is reconstructed. Furthermore, we provide explicit @math running time bound. In cases where the measure on the graph has correlation decay, the running time is @math for all fixed @math . We also discuss the effect of observing noisy samples and show that as long as the noise level is low, our algorithm is effective. On the other hand, we construct an example where large noise implies non-identifiability even for generic noise and interactions. Finally, we briefly show that in some simple cases, models with hidden nodes can also be recovered.
@cite_5: We study distorted metrics on binary trees in the context of phylogenetic reconstruction. Given a binary tree T on n leaves with a path metric d, consider the pairwise distances d(u,v) between leaves. It is well known that these determine the tree and the d length of all edges. Here, we consider distortions @math of d such that, for all leaves u and v, it holds that @math if either d(u,v)< M + f 2 or @math , where d satisfies f ≤ d(e) ≤ g for all edges e. Given such distortions, we show how to reconstruct in polynomial time a forest T1, ... ,Tα such that the true tree T may be obtained from that forest by adding α-1 edges and α-1 ≤ 2-Ω(M g) n. Our distorted metric result implies a reconstruction algorithm of phylogenetic forests with a small number of trees from sequences of length logarithmic in the number of species. The reconstruction algorithm is applicable for the general Markov model. Both the distorted metric result and its applications to phylogeny are almost tight.
@cite_4: A method is presented to approximate optimally an n -dimensional discrete probability distribution by a product of second-order distributions, or the distribution of the first-order tree dependence. The problem is to find an optimum set of n - 1 first order dependence relationship among the n variables. It is shown that the procedure derived in this paper yields an approximation of a minimum difference in information. It is further shown that when this procedure is applied to empirical observations from an unknown distribution of tree dependence, the procedure is the maximum-likelihood estimate of the distribution.
@cite_6: 
@cite_2: One of the major tasks of evolutionary biology is the reconstruction of phylogenetic trees from molecular data. The evolutionary model is given by a Markov chain on the true evolutionary tree. Given samples from this Markov chain at the leaves of the tree, the goal is to reconstruct the evolutionary tree.It is well known that in order to reconstruct a tree on n leaves, sequences of length Ω(log n) are needed. It was conjectured by M. Steel that for the CFN evolutionary model, if the mutation probability on all edges of the tree is less than p* = (√2-1) 23 2, then the tree can be recovered from sequences of length O(log n). This was proven by the second author in the special case where the tree is "balanced". The second author also proved that if all edges have mutation probability larger than p* then the length needed is nΩ(1). This "phase-transition" in the number of samples needed is closely related to the phase transition for the reconstruction problem (or extremality of free measure) studied extensively in statistical physics, probability and computer science.Here we complete the proof of Steel's conjecture and give a reconstruction algorithm using optimal (up to a multiplicative constant) sequence length. Our results further extend to obtain an optimal reconstruction algorithm for the Jukes-Cantor model with short edges. All reconstruction algorithms run in polynomial time.
Chow and Liu @cite_4 considered the problem of estimating Markov random fields whose underlying graphs are trees, and provided an efficient (polynomial-time) algorithm based on the fact that in the tree case maximum-likelihood estimation amounts to the computation of a maximum-weight spanning tree with edge weights equal to pairwise empirical mutual information. Unfortunately, their approach does not generalize to the estimation of Markov random fields whose graphs have cycles. Much work in mathematical biology is devoted to reconstructing tree Markov fields when there are hidden models. For trees, given data that is generated from the model, the tree can be reconstructed efficiently from samples at a subset of the nodes given mild non-degeneracy conditions. See @cite_6 @cite_5 @cite_2 for some of the most recent and tightest results in this setup.