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The fact that we only consider models that can be described by at least one Bayesian network leaves many interesting dependency models out. The exact characterization of models that are left out (or that are in) is somewhat technical, but certain interesting subsets of discarded models can be characterized intuitively.
How do we characterize those models describable by Bayesian networks? One characterization can be based on the notion of causality. If the variables of our model are in causal relationships with each other, and if in our domain there are no latent variables (i.e., variables that for some reason are not included in our data) that have causal influence on the variables of our model, then the dependencies caused by these causal relationships can be described by a Bayesian network.
Sad to say latent variables often induce sets of dependency statements, that cannot be described accurately by any Bayesian network. That severely restricts our ability to automatically infer something about causalities just based on statistical dependencies.
» References to limitations of Bayesian nets
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