Saturated Conditional Independence with Fixed and Undetermined Sets of Incomplete Random Variables

Henning Koehler Massey University, Sebastian Link
Proceedings of the 30th Conference on Uncertainty in Artificial Intelligence, PMLR R12:361-370, 2014.

Abstract

The implication problem for saturated conditional independence statements is studied in the presence of fixed and undetermined sets of incomplete random variables. Here, random variables are termed incomplete since they admit missing data. Two different notions of implication arise. In the classic notion of V -implication, a statement is implied jointly by a set of statements and a fixed set V of random variables. In the alternative notion of pure implication, a statement is implied by a given set of statements alone, leaving the set of random variables undetermined. A first axiomatization for V -implication is established that distinguishes purely implied from V -implied statements. Axiomatic, algorithmic and logical characterizations of pure implication are established. Pure implication appeals to applications in which the existence of random variables is uncertain, for example, when independence statements are integrated from different sources, when random variables are unknown or shall remain hidden.

Cite this Paper

BibTeX
@InProceedings{pmlr-vR12-university14l, title = {Saturated Conditional Independence with Fixed and Undetermined Sets of Incomplete Random Variables}, author = {University, Henning Koehler Massey and Link, Sebastian}, booktitle = {Proceedings of the 30th Conference on Uncertainty in Artificial Intelligence}, pages = {361--370}, year = {2014}, editor = {Zhang, Nevin L. and Tian, Jin}, volume = {R12}, series = {Proceedings of Machine Learning Research}, month = {23--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/r12/main/assets/university14l/university14l.pdf}, url = {https://proceedings.mlr.press/r12/university14l.html}, abstract = {The implication problem for saturated conditional independence statements is studied in the presence of fixed and undetermined sets of incomplete random variables. Here, random variables are termed incomplete since they admit missing data. Two different notions of implication arise. In the classic notion of V -implication, a statement is implied jointly by a set of statements and a fixed set V of random variables. In the alternative notion of pure implication, a statement is implied by a given set of statements alone, leaving the set of random variables undetermined. A first axiomatization for V -implication is established that distinguishes purely implied from V -implied statements. Axiomatic, algorithmic and logical characterizations of pure implication are established. Pure implication appeals to applications in which the existence of random variables is uncertain, for example, when independence statements are integrated from different sources, when random variables are unknown or shall remain hidden.}, note = {Reissued by PMLR on 04 October 2026.} }
Endnote
%0 Conference Paper %T Saturated Conditional Independence with Fixed and Undetermined Sets of Incomplete Random Variables %A Henning Koehler Massey University %A Sebastian Link %B Proceedings of the 30th Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2014 %E Nevin L. Zhang %E Jin Tian %F pmlr-vR12-university14l %I PMLR %P 361--370 %U https://proceedings.mlr.press/r12/university14l.html %V R12 %X The implication problem for saturated conditional independence statements is studied in the presence of fixed and undetermined sets of incomplete random variables. Here, random variables are termed incomplete since they admit missing data. Two different notions of implication arise. In the classic notion of V -implication, a statement is implied jointly by a set of statements and a fixed set V of random variables. In the alternative notion of pure implication, a statement is implied by a given set of statements alone, leaving the set of random variables undetermined. A first axiomatization for V -implication is established that distinguishes purely implied from V -implied statements. Axiomatic, algorithmic and logical characterizations of pure implication are established. Pure implication appeals to applications in which the existence of random variables is uncertain, for example, when independence statements are integrated from different sources, when random variables are unknown or shall remain hidden. %Z Reissued by PMLR on 04 October 2026.
APA
University, H.K.M. & Link, S.. (2014). Saturated Conditional Independence with Fixed and Undetermined Sets of Incomplete Random Variables. Proceedings of the 30th Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research R12:361-370 Available from https://proceedings.mlr.press/r12/university14l.html. Reissued by PMLR on 04 October 2026.

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