Some Examples and Counterexamples on the Relationship Between Conditional and Unconditional Independence

A. García Nogales, P. Pérez Fernández

Some examples and counterexamples are given to illustrate a result included in Nogales and Pérez (2019, to appear in Statistica Neerlandica) on the relationship between conditional and unconditional independence. The mentioned result reads as follows: Let X, Y, Z be three random variables. If X and Y are conditionally independent given Z, then X and Y are independent if and only if are independent the conditional distributions of X and Y given Z. This last condition is equivalent to the uncorrelatedness of the conditional expectations given Z of every pair of bounded real random functions of X and Y. A general discrete framework is built to describe the cited examples and counterexamples.

Palabras clave: conditional independence Markov kernel

A. García Galindo, O. González Velasco, J. M. Sánchez Santos, J. De Las Rivas Sanz, E. Sánchez Luis

P. Pérez Fernández, A. García Nogales

C. E. Carleos Artime, N. Corral Blanco, S. Álvarez Morán, A. Shatla

P. Román Román, S. Román- Román, J. J. Serrano Pérez, F. Torres Ruiz

I. Morillo, L. Bermúdez, D. Karlis

R. Caballero Águila, A. Hermoso Carazo, J. Linares Pérez

R. Caballero Águila, D. A. Hermoso Carazo, D. J. Linares Pérez

M. J. García-Ligero Ramírez, M. A. Hermoso Carazo, J. Linares Pérez

R. Caballero Águila, M. A. Hermoso Carazo, J. Linares Pérez

J. A. Cristóbal Cristóbal, P. Olave

A. Vaamonde Liste, M. Meijide Vecino, P. Muñoz Dueñas

A. Pérez-González, T. R. Cotos Yáñez, W. González–Manteiga, R. M. Crujeiras

A. V. García Luengo, I. Oña Casado

R. Mata Crespo

L. Davila Pena, B. Casas Méndez

T. R. Cotos Yáñez, M. A. Mosquera Rodríguez, A. Pérez González, B. Reguengo Lareo

J. García Rodríguez, B. Sinova Fernández, C. E. Carleos Artime

C. González Martín, A. Sedeño Noda

S. A. Mulema, A. Carrión García