Reconstruction of Markov Random Fields from Samples: Some Observations and Algorithms

• Published in: SIAM journal on computing Vol. 42; no. 2; pp. 563 - 578
• Main Authors: Bresler, Guy, Mossel, Elchanan, Sly, Allan
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2013
• Subjects: Algorithms; Electrical engineering; Graphs; Independent sample; Markov processes; Mathematical analysis; Noise; Noise levels; Probability distribution; Random variables; Reconstruction; Running; Theorems
• ISSN: 0097-5397; 1095-7111
• DOI: 10.1137/100796029

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 $n$ nodes and maximum degree $d$ given observations. We show that under mild nondegeneracy conditions it reconstructs the generating graph with high probability using $\Theta(d \epsilon^{-2}\delta^{-4} \log n)$ samples, where $\epsilon,\delta$ depend on the local interactions. For most local interactions $\epsilon,\delta$ are of order $\exp(-O(d))$. Our results are optimal as a function of $n$ up to a multiplicative constant depending on $d$ 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 $O(n^{d+2} \epsilon^{-2}\delta^{-4} \log n)$ running-time bound. In cases where the measure on the graph has correlation decay, the running time is $O(n^2 \log n)$ for all fixed $d$. 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 nonidentifiability even for generic noise and interactions. Finally, we briefly show that in some simple cases, models with hidden nodes can also be recovered. [PUBLICATION ABSTRACT]