Inferring species interactions from co-occurrence data is one of the most controversial tasks in community ecology. One difficulty is that a single pairwise interaction can ripple through an ecological network and produce surprising indirect consequences. For example, the negative correlation between two competing species can be reversed in the presence of a third species that is capable of outcompeting both of them. Here, I apply models from statistical physics, called Markov networks or

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... random fields, that can predict the direct and indirect consequences of any possible species interaction matrix. Interactions in these models can also be estimated from observed co-occurrence rates via maximum likelihood, controlling for indirect effects. Using simulated landscapes with known pairwise interaction strengths, I evaluated Markov networks and six existing approaches. The Markov networks consistently outperformed other methods, correctly isolating direct interactions between species pairs even when indirect interactions or abiotic factors largely overpowered them. Two computationally efficient approximations, based on controlling for indirect effects with linear or generalized linear models, also performed well. Indirect effects reliably caused a common null modeling approach to produce incorrect inferences, however.