Independence

There is no unique way to define independence relations with credal sets; in the most in-depth study of this matter, Campos and Moral have reviewed five different possible types of independence [10]. The results presented in this paper adopt Walley's definition of independence [28]. Walley's original definition was stated in terms of lower expectations; to develop a theory of convex sets of distributions, it is important to recast Walley's definition using credal sets as follows.

Consider sets of variables $\tilde{X}$, $\tilde{Y}$ and $\tilde{Z}$ and the credal sets $K(\tilde{X}, \tilde{Y}, \tilde{Z})$, $K(\tilde{X}\vert\tilde{Z})$, $K(\tilde{Y}\vert\tilde{Z})$, $K(\tilde{X}\vert\tilde{Y},\tilde{Z})$ and $K(\tilde{Y}\vert\tilde{X},\tilde{Z})$. Note that distributions in $K(\tilde{X}\vert\tilde{Z})$ and $K(\tilde{X}\vert\tilde{Y},\tilde{Z})$ are defined over the same algebra of events once $\tilde{Y}$ and $\tilde{Z}$ are fixed; likewise, distributions in $K(\tilde{Y}\vert\tilde{Z})$ and $K(\tilde{Y}\vert\tilde{X},\tilde{Z})$ are defined over the same algebra of events once $\tilde{X}$ and $\tilde{Z}$ are fixed.

Definition 1   Variables $\tilde{Y}$ are irrelevant to $\tilde{X}$ given $\tilde{Z}$ if $K(\tilde{X}\vert\tilde{Z})$ is equal to $K(\tilde{X}\vert\tilde{Y},\tilde{Z})$ regardless of the value of $\tilde{Y}$, $\tilde{Z}$. Variables $\tilde{X}$ and $\tilde{Y}$ are independent given $\tilde{Z}$ if $\tilde{X}$ is irrelevant to $\tilde{Y}$ given $\tilde{Z}$ and $\tilde{Y}$ is irrelevant to $\tilde{X}$ given $\tilde{Z}$. If $\tilde{Z}$ is empty, suppress the ``given $\tilde{Z}$'' from this definition.

This concept of independence does not imply that joint credal sets contain only joint distributions with independent marginals, nor does it imply uniqueness for the joint credal set [28, Chapter 9].