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Causality, or causation, is the relationship between causes and effects. In common parlance, an event or state of affairs A is a cause of an event B if A is a reason that brings about the effect B. For instance, one might say "my pushing the accelerator caused the car to go faster". But this definition is somewhat circular; what does it then really mean to say that A is a reason that B occurs? An important question in philosophy and other fields is to clarify the relationship between causes and effects, as well as how (and even if!) causes can bring about effects.
David Hume held that causes and effects are not real (or at least not knowable), but imagined by our mind to make sense of the observation that A often occurs together with or slightly before B. All we can observe are correlations, not causations.
The establishing of cause and effect, even with this relaxed reading, is notoriously difficult, expressed by the widely accepted statement "correlation does not imply causation". For instance, the observation that smokers have a dramatically increased lung cancer rate does not establish that smoking must be a cause of that increased cancer rate: maybe there exists a certain genetic defect which both causes cancer and a yearning for nicotine.
That said, under certain assumptions, parts of the causal structure among several variables can be learned from full covariance or case data by the techniques of Path analysis and more generally, Bayesian networks. Generally these inference algorithms search through the many possible causal structures among the variables, and remove ones which are strongly incompatible with the observed correlations. In general this leaves a set of possible causal relations, which should then be tested by designing appropriate experiments. If experimental data is already available, the algorithms can take advantage of that as well.
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