How can you tell whether one event causes another? For instance, assume that you observe that when one event happens, another event is more likely to happen. For instance, when it rains the ground gets wet. Generally, rain will cause the ground to get wet. Use of umbrellas is also highly correlated with the ground getting wet. Even though umbrella use and a wet ground are correlated, umbrellas of course do not cause the ground to get wet. In this example, one can use intuition and experience to determine which event causes another. How does one define causality mathematically?
Reichenbach’s Common Cause Principle states that the correlation between events A and B indicates either that A causes B, or that B causes A, or that A and B have a common cause. Mathematically, consider the case where two events are likely to occur at once. Mathematically, this means that Pr(A,B) > Pr(A) × Pr(B). If there is a common cause, C, then this common cause has the following properties:
- P(A|C) > P(A|~C)
- P(B|C) > P(B|~C)
- P(A,B|C) = P(A|C)× P(B|C)
- P(A,B|~C) = P(A|~C)× P(B|~C)
The first condition states that event A is more likely to occur when the common cause, C, is present. The second condition states that event B is also more likely to occur when C is present. The third and fourth conditions state that once we know that the common cause has occured, A and B are independent. In other words, C causes both A and B; and correlation between A and B disappears once we know that C.
Consider our example above, where the use of umbrellas is A and the ground getting wet is B. These events are highly correlated. Let’s check if the event “rain” meets the common cause criteria. The use of umbrellas is more likely when rain occurs (condition 1) and the ground is more likely to be wet when it rains (condition 2). Further, once we know that it is raining, the use of umbrellas does not effect the probability the ground gets wet or conversely the ground being wet does not effect the probability that umbrellas are used. In other words, umbrella use and the ground being wet are independent once we know whether (condition 3) or not (condition 4) it is raining.
Extensions of the Common Cause Principle
In many cases, however, there may not be a single common cause C but a set of common causes {C1, C2,…, Cn)of an event. One can extend this principle more generally to cases with multiple causes using the causal Markov condition. The causal Markov condition “…holds of a set of quantities {Q1,…,Qn} if and only if the values of any quantity Qi in that set, conditional upon the values of all the quantities in the set that are direct causes of Qi, are probabilistically independent of the values of all quantities in the set other than Qi‘s effects.[3] The causal Markov condition implies the following version of the common cause principle: If Qi and Qj are correlated and Qi is not a cause of Qj, and Qj is not a cause of Qi, then there are common causes of Qi and Qj in the set {Q1,…,Qn} such that Qi and Qj are independent conditional upon these common causes.”
One can also apply temporal restrictions on causality. For instance, assume that the police catch 100% of people who are speeding. In this case, does the car’s speeding cause the ticket or does the ticket cause the speeding. Since they are perfectly correlated and there may be no common cause, this is unclear. However, since the speeding ticket is always issued after the speeding, intuition says that driving fast causes the speeding ticket; speeding tickets cannot cause people to drive faster (the economic intuition is that they would incentive people to drive slower). Thus, one could restrict the eligible common causes to those which occur prior to the resulting events.
A note on Markov Causation and Markov Processes
Markov processes are unique in that they are memoryless. In other words, only current state you are in has any effect on the probability of which state you will be in the next period. One example of a Markov process is a random walk (picture). Although the naming convention is confusing, Markov processes generically do not satisfy the causal Markov condition.
More information on the common cause principle can be found here.