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 condi- tional independence statements is studied in the presence of fixed and undetermined sets of in- complete random variables. Here, random vari- ables are termed incomplete since they admit missing data. Two different notions of implica- tion arise. In the classic notion of V -implication, a statement is implied jointly by a set of state- ments and a fixed set V of random variables. In the alternative notion of pure implication, a statement is implied by a given set of state- ments alone, leaving the set of random vari- ables undetermined. A first axiomatization for V -implication is established that distinguishes purely implied from V -implied statements. Ax- iomatic, algorithmic and logical characteriza- tions 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 in- tegrated 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 condi- tional independence statements is studied in the presence of fixed and undetermined sets of in- complete random variables. Here, random vari- ables are termed incomplete since they admit missing data. Two different notions of implica- tion arise. In the classic notion of V -implication, a statement is implied jointly by a set of state- ments and a fixed set V of random variables. In the alternative notion of pure implication, a statement is implied by a given set of state- ments alone, leaving the set of random vari- ables undetermined. A first axiomatization for V -implication is established that distinguishes purely implied from V -implied statements. Ax- iomatic, algorithmic and logical characteriza- tions 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 in- tegrated 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 condi- tional independence statements is studied in the presence of fixed and undetermined sets of in- complete random variables. Here, random vari- ables are termed incomplete since they admit missing data. Two different notions of implica- tion arise. In the classic notion of V -implication, a statement is implied jointly by a set of state- ments and a fixed set V of random variables. In the alternative notion of pure implication, a statement is implied by a given set of state- ments alone, leaving the set of random vari- ables undetermined. A first axiomatization for V -implication is established that distinguishes purely implied from V -implied statements. Ax- iomatic, algorithmic and logical characteriza- tions 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 in- tegrated 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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