Mathematics – Statistics Theory
Scientific paper
2011-06-06
Mathematics
Statistics Theory
Scientific paper
We consider a framework for counterfactual statistical analysis with graphical models based on marked point processes. The main idea is to treat the counterfactual scenario as just another probability measure on the underlying sample space. Short-term predictions provide dynamical characterizations of the various involved factors. A hypothetical direct intervention on a factor would change its dynamics. The non-directly inter- vened factors on the other hand, should have the same dynamical characterization as in the observational scenario. We claim that any probability measure on sample space that is compatible with these characteristics yields a reasonable counterfactual model. The probability measures that govern the frequencies of the counterfac- tual observations are characterized in terms of a given martingale problem. Local independence graphs provide useful graphical models in this framework. We show that Bayesian networks appear as a special case of local independence graphs. We moreover show that Judea Pearl's do-operator and truncated factorization formula for Bayesian networks carry over to our continuous time situation and corresponding martingale measures.
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