The mother distribution is a distribution over join distributions of pairs of variables. The authors adopt the following definition of a mother distribution:

A mother distribution $P_M( P_{Π = π} ( X , Y ), G = g )$ is a distribution over pairs $( P_{Π = π} ( X , Y ), G = g )$, where $Π$ and $G$ are random variables:

• $Π = π$ indexes distribution $P_{Π = π} ( X , Y )$ over random variables $X$ and $Y$ ; $π$ may possibly take an infinite number of values (e.g. parameters of a generative process)
• $G = g$ labels causal relationship, $g ∈ \{ X → Y , X ← Y , X \leftrightarrow Y , X ⊥ Y \}$.