A general definition of influence between stochastic processes |
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Authors: | Anne Gégout-Petit Daniel Commenges |
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Institution: | 1.IMB, UMR 5251, INRIA CQFD,Talence,France;2.Université Victor Segalen Bordeaux 2,Bordeaux,France;3.INSERM, U 897,Bordeaux,France |
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Abstract: | We extend the study of weak local conditional independence (WCLI) based on a measurability condition made by (Commenges and
Gégout-Petit J R Stat Soc B 71:1–18) to a larger class of processes that we call D¢{\bf {\mathcal{D}'}}. We also give a definition related to the same concept based on certain likelihood processes, using the Girsanov theorem.
Under certain conditions, the two definitions coincide on D¢{\bf {\mathcal{D}'}}. These results may be used in causal models in that we define what may be the largest class of processes in which influences
of one component of a stochastic process on another can be described without ambiguity. From WCLI we can construct a concept
of strong local conditional independence (SCLI). When WCLI does not hold, there is a direct influence while when SCLI does
not hold there is direct or indirect influence. We investigate whether WCLI and SCLI can be defined via conventional independence
conditions and find that this is the case for the latter but not for the former. Finally we recall that causal interpretation
does not follow from mere mathematical definitions, but requires working with a good system and with the true probability. |
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