bibtype	C - Conference Paper (international conference)
ARLID	0561326
utime	20240111141109.3
mtime	20220920235959.9
title (primary) (eng)	On the rank of 2×2×2 probability tables
specification	page_count 12 s. media_type E
serial	ARLID cav_un_epca*0561323 ISSN Proceedings of Machine Learning Research, Volume 186 : Proceedings of The 11th International Conference on Probabilistic Graphical Models title Proceedings of Machine Learning Research, Volume 186 : Proceedings of The 11th International Conference on Probabilistic Graphical Models page_num 361-372 publisher place Almerı́a name PMLR year 2022 editor name1 Salmerón name2 Antonio editor name1 Rumí name2 Rafael
keyword	Tensor rank
keyword	Conditional probability tables
keyword	Monotonicity
keyword	Educational testing
author (primary)	ARLID cav_un_auth*0436638 name1 Peréz name2 I. country MX
author	ARLID cav_un_auth*0101228 name1 Vomlel name2 Jiří institution UTIA-B full_dept (cz) Matematická teorie rozhodování full_dept Department of Decision Making Theory department (cz) MTR department MTR full_dept Department of Decision Making Theory share 50 fullinstit Ústav teorie informace a automatizace AV ČR, v. v. i.
source	source_type online url http://library.utia.cas.cz/separaty/2022/MTR/vomlel-0561326.pdf
cas_special	abstract (eng) Bayesian networks for real-world problems typically satisfy the property of positive monotonicity (in the context of educational testing, it is commonly assumed that answering correctly a question A increases the probability of answering correctly another question B). In this paper, we focus on the study of relations between positive monotonic influences on three-variable patterns and a family of 2×2×2 tensors. In this study, we use the Kruskal polynomial, well-known in the psychometrics community, which is equivalent to Cayley’s hyperdeterminant (homogeneous polynomial of degree 4 in the 8 entries of a 2×2×2 tensor). It is known that when the Kruskal polynomial is positive, the rank of the tensor is two. We show that when a probability table associated with three random variables obeys the positive monotonicity property, its corresponding 2×2×2 tensor has rank two. Moreover, it can be decomposed using only nonnegative tensors, which can each be given a statistical interpretation. We study two concepts of monotonicity in sets of three random variables, strong monotonicity (any two variables have a positive influence on the third one), and weak monotonicity (just one pair of variables that have a positive influence on the third one), and we give an example to show they do not coincide. Furthermore, we proved that the strong monotonicity property implies that the tensor rank is at most two. We also performed experiments with real data to test the monotonicity properties. The real datasets were formed by information from the Czech high school final exam from the years 2016 to 2022. These datasets are representative since the sample size (number of students) for each year is very large (N > 10000) and information comes from students of all regions of the Czech Republic. In this datasets, we observed that almost all 2×2×2 tensors are monotone and all their corresponding 2×2×2 tensors have nonnegative decomposition. action ARLID cav_un_auth*0436551 name International Conference on Probabilistic Graphical Models dates 20221005 mrcbC20-s 20221007 place Almería country ES RIV BA FORD0 10000 FORD1 10100 FORD2 10103 reportyear 2023 num_of_auth 2 presentation_type PR inst_support RVO:67985556 permalink https://hdl.handle.net/11104/0334057 cooperation ARLID cav_un_auth*0372408 name Karlova Universita v Praze country CZ confidential S arlyear 2022 mrcbU14 SCOPUS mrcbU24 PUBMED mrcbU34 WOS mrcbU56 online mrcbU63 cav_un_epca*0561323 Proceedings of Machine Learning Research, Volume 186 : Proceedings of The 11th International Conference on Probabilistic Graphical Models PMLR 2022 Almerı́a 361 372 2640-3498 mrcbU67 Salmerón Antonio 340 mrcbU67 Rumí Rafael 340