On the rank of 2×2×2 probability tables

Result code in IS VaVaI

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Result on the web

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DOI - Digital Object Identifier

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Result language

angličtina

Original language name

On the rank of 2×2×2 probability tables

Original language description

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.

Czech name

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Czech description

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Type

D - Article in proceedings

CEP classification

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OECD FORD branch

10103 - Statistics and probability

Project

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Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2022

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Article name in the collection

Proceedings of Machine Learning Research, Volume 186 : Proceedings of The 11th International Conference on Probabilistic Graphical Models

ISBN

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ISSN

2640-3498

e-ISSN

2640-3498

Number of pages

12

Pages from-to

361-372

Publisher name

PMLR

Place of publication

Almerı́a

Event location

Almería

Event date

Oct 5, 2022

Type of event by nationality

WRD - Celosvětová akce

UT code for WoS article

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