Reference no: EM131319030

In the Bayesian graphical model community, the task of inferring which way the arrows point - that is, which nodes are parents, and which children - is one on which much has been written. Inferring causation is tricky because of ‘likelihood equivalence'. Two graphical models are likelihood-equivalent if for any setting of the parameters of either, there exists a setting of the parameters of the other such that the two joint probability distributions of all observables are identical. An example of a pair of likelihood-equivalent models are A → B and B → A. The model A → B asserts that A is the parent of B, or, in very sloppy terminology, ‘A causes B'. An example of a situation where ‘B → A' is true is the case where B is the variable ‘burglar in house' and A is the variable ‘alarm is ringing'. Here it is literally true that B causes A. But this choice of words is confusing if applied to another example, R → D, where R denotes ‘it rained this morning' and D denotes ‘the pavement is dry'. ‘R causes D' is confusing. I'll therefore use the words ‘B is a parent of A' to denote causation. Some statistical methods that use the likelihood alone are unable to use data to distinguish between likelihood-equivalent models. In a Bayesian approach, on the other hand, two likelihood-equivalent models may nevertheless be somewhat distinguished, in the light of data, since likelihood-equivalence does not force a Bayesian to use priors that assign equivalent densities over the two parameter spaces of the models.