## Conditional Independence Relations Have No Finite Complete Characterization (1990)

Citations: | 43 - 6 self |

### BibTeX

@MISC{Studeny90conditionalindependence,

author = {Milan Studeny},

title = {Conditional Independence Relations Have No Finite Complete Characterization},

year = {1990}

}

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### Abstract

The hypothesis of existence of a finite characterization of conditional--independence relations (CIRs) is refused. This result is shown to be equivalent with the non--existence of a simple deductive system describing relationships among CI--statements (it is certain type of syntactic description). However, under the assumption that CIRs are grasped the existence of a countable characterization of CIRs is shown. Finally, the problem of characterization of CIRs is shown to be diverse from an analogical problem of axiomatization EMVDs arising in the theory of relational databases. INTRODUCTION Let [¸ i ] i2N be a random vector (2 card N ! 1) and let us suppose for simplicity that its components are finite--valued random variables. Then we can define a ternary disjoint relation I on expN (disjoint means that its domain is the set of triplets of pairwise disjoint subsets of N ): I(A; BjC) holds iff [¸ i ] i2A is conditionally independent of [¸ i ] i2B given [¸ i ] i2C . We shall ca...