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This problem is called causal discovery.

Learning causality from observational data is hard, but there are statistical methods to help with that. Of course, you want to verify statistical findings with experiments, but first you want to find most likely causal relations from large number of correlations.

Causal direction between the two variables can sometimes be identified by observation because there is more information in the data than correlation coefficient and it can be asymmetrical.

Here is nice intro:

Causal discovery and inference: concepts and recent methodological advances Peter Spirtes and Kun Zhang http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4841209/

Nice paper and method:

Distinguishing Causes from Effects using Nonlinear Acyclic Causal Models https://www.cs.helsinki.fi/u/ahyvarin/papers/Zhang09NIPSwork...

Good paper. Aapo is at the ML group at UCL now, starting this semester.

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