On the rank of 2×2×2 probability tables

Iván Pérez, Jiřı́ Vomlel
Proceedings of The 11th International Conference on Probabilistic Graphical Models, PMLR 186:361-372, 2022.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v186-perez22a, title = {On the rank of 2×2×2 probability tables}, author = {P{\'e}rez, Iv{\'a}n and Vomlel, Ji\v{r}\'{\i}}, booktitle = {Proceedings of The 11th International Conference on Probabilistic Graphical Models}, pages = {361--372}, year = {2022}, editor = {Salmerón, Antonio and Rumı́, Rafael}, volume = {186}, series = {Proceedings of Machine Learning Research}, month = {05--07 Oct}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v186/perez22a/perez22a.pdf}, url = {https://proceedings.mlr.press/v186/perez22a.html}, abstract = {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.} }
Endnote
%0 Conference Paper %T On the rank of 2×2×2 probability tables %A Iván Pérez %A Jiřı́ Vomlel %B Proceedings of The 11th International Conference on Probabilistic Graphical Models %C Proceedings of Machine Learning Research %D 2022 %E Antonio Salmerón %E Rafael Rumı́ %F pmlr-v186-perez22a %I PMLR %P 361--372 %U https://proceedings.mlr.press/v186/perez22a.html %V 186 %X 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.
APA
Pérez, I. & Vomlel, J.. (2022). On the rank of 2×2×2 probability tables. Proceedings of The 11th International Conference on Probabilistic Graphical Models, in Proceedings of Machine Learning Research 186:361-372 Available from https://proceedings.mlr.press/v186/perez22a.html.

Related Material