Defining and Representing Causality

The most important feature of an influence diagram is its ability to represent all causal relationships of a phenomenon in a manner that is nonambiguous, probabilistic, and graphical. The following definitions and theorem are due to Pearl (1995):

d-separation:

Let ${\bf X}$, ${\bf Y}$, and ${\bf Z}$ be three disjoint subsets of nodes in a directed acyclic graph G, and let p be any path between a node in ${\bf X}$ and a node in ${\bf Y}$. Then ${\bf Z}$ is said to block p if there is a node w on p satisfying one of the following two conditions: (i) w has converging arrows along p, and neither w nor any of its descendants are in ${\bf Z}$, or (ii) w does not have converging arrows along p, and w is in ${\bf Z}$.Further, ${\bf Z}$ is said to d-separate ${\bf X}$ from ${\bf Y}$, in G, written (${\bf X}\coprod {\bf Y}\vert {\bf Z})_G$, if and only if ${\bf Z}$ blocks every path from a node in ${\bf X}$ to a node in ${\bf Y}$.

Causal Effect:

Given two disjoint sets of variables, ${\bf X}$ and ${\bf Y}$, the causal effect of ${\bf X}$ on ${\bf Y}$,denoted $P({\bf Y}\vert\check{{\bf x}})$,is a function from ${\bf X}$ to the space of probability distributions on ${\bf Y}$.For each realisation ${\bf x}$ of ${\bf X}$, $P({\bf Y}\vert\check{{\bf x}})$ gives the probability of ${\bf Y}={\bf y}$induced on deleting from the graph all nodes that are parents of variables in ${\bf X}$ and substituting ${\bf x}$ for ${\bf X}$ in the remainder.

Back-door Criterion:

A set of variables ${\bf Z}$ satisfies the back-door criterion relative to an ordered pair of variables $(X_i,\; Y_j)$ in a directed acyclic graph G if: (i) no node in ${\bf Z}$ is a descendant of X_i and (ii) ${\bf Z}$ blocks every path between X_i and Y_j which contains an arrow into X_i. If ${\bf X}$ and ${\bf Y}$ are two disjoint sets of nodes in G, and ${\bf Z}$ is said to satisfy the back-door criterion relative to $({\bf X},\; {\bf Y})$ if it satisfies it relative to any pair $(X_i,\; Y_j)$ such that $X_i \in {\bf X}$ and $Y_j \in {\bf Y}$.

Identifiability:

The causal effect of ${\bf X}$ on ${\bf Y}$ is said to be identifiable if the quantity $P({\bf Y}\vert\check{x})$ can be computed uniquely from any positive distribution of the observed variables that is compatible with G.

Theorem:

Computing $P({\bf Y}={\bf y}\vert \check{{\bf x}})$.

Case 1:

If a set of variables ${\bf Z}$ satisfies the back-door criterion relative to $({\bf X},\; {\bf Y})$, then the causal effect of ${\bf X}$ on ${\bf Y}$ is identifiable and is given by the formula

$$P({\bf Y}={\bf y}\vert\check{{\bf x}}) = \sum_{{\bf z}} P({\bf Y}={\bf y}\vert\check{{\bf x}},{\bf z})P({\bf Z}={\bf z}).$$

Case 2:

(Front-door) If a set of variables ${\bf Z}$ satisfies the following conditions relative to an ordered pair of variables $({\bf X}, {\bf Y})$:(i) ${\bf Z}$ intercepts all directed paths from ${\bf X}$ to ${\bf Y}$, (ii) there is no back-door path between ${\bf X}$ and ${\bf Z}$, and (iii) every back-door path between ${\bf Z}$ an ${\bf Y}$ is blocked by ${\bf X}$, then the causal effect of ${\bf X}$ on ${\bf Y}$ is identifiable and is given by the formula

$$P({\bf Y}={\bf y}\vert\check{{\bf x}}) = \sum_{{\bf z}} P({\... ...{\bf Y}={\bf y}\vert{\bf x}^{'},{\bf z})P({\bf X}={\bf x}^{'}).$$

This Theorem allows many nonexperimental samples (observational studies) to be used to test hypotheses concerning causal relationships among the influence diagram's nodes. Such samples are often encountered in ecosystem management problems. The above definition of causality will be used in the cheetah population dynamics model, below.