10* - Causal Graphs and Conditional Probabilistic Dependencies

Summary

Independence Implies Screening-Off

If causes are INUS conditions, the following two theorems (which are sketched by Papineau (1985b, p. 63 and 1985a, p. 279)) establish that effects are probabilistically dependent on their causes and that common causes screen off their effects.

Theorem 10.1: If is probabilistically independent of S and Z, and

Theorem 10.2: If (3) C is probabilistically independent of S, Z, Y, and W, (4) S, Z, Y, and W are probabilistically independent of one another, and (5) all probabilities are intermediate, then C and ∈C screen off A and B.

Proof: Given premises 1–3 and 5, Given premise which by 1, 2, 3, and 5 equals. Since Z and W are independent, and.

In Papineau's view, screening-off is explained by the independence among different deterministic causal facts.

Causal Graphs and Probability Distributions – Some Formal Results

In this section I shall sketch the proof of the striking theorem discussed in §10.3.

Theorem 10.3 CM and F imply

1. (Direct causal connection) For all x andy in V, x and y are adjacent if and only if they are probabilistically dependent conditional on every subset of <I>V that does not include them, and

2. (Direct causation) For all x, y and z in V, if JC and y and y and z are adjacent, and JC and z are not adjacent, then V causally depends on x and z if and only if x and z are probabilistically dependent conditional on every subset of Fthat contains y but not y or z.