The interaction of two binary variables, assumed to be empirical observations, has three degrees of freedom when expressed as a matrix of frequencies. Usually, the size of causal influence of one variable on the other is calculated as a single value, as increase in recovery rate for a medical treatment, for example. We examine what is lost in this simplification, and propose using two interface constants to represent positive and negative implications separately. Given certain assumptions about non-causal outcomes, the set of resulting epistemologies is a continuum. We derive a variety of particular measures and contrast them with the one-dimensional index.I was moved to finish the thing, which had been languishing on my computer, because of a deadline for the AIR forum in Orlando later this month. The title of my presentation there is "Correlation, Prediction, and Causation", with the program blurb below.
Everyone knows the mantra “correlation doesn’t imply causation,” but that doesn’t make the desire to find cause-effect relationships disappear! This session will address the relationship between correlation and prediction, and take up the philosophical question of what “causation” can be thought to mean, and how we can usefully talk to decision-makers about these issues. These ideas are immediately useful in analyzing and reporting information to decision-makers, and are both practical and optimistic. The goal is to answer the question “what’s the next best thing we can try to improve our situation?” There is some math involved, but it is not necessary to understand the main ideas.In my review of literature, I turned up the tome Causality in the Sciences, which is pictured below, decorated by Greg in ILL. I'm not sure why it's upside down--some mysterious cause, no doubt.
As you can see, there is a lot to say on the subject, but there is one particular idea that seems to lie at the heart of cause-effect analysis. I didn't know about it until I read Judea Pearl's work. Here it is, in my words, not his.
If all we do is observe the world, we can never be sure what causes what because there always might be some ultimate cause hidden from us. If we watch Stanislav flip a light switch up and down and observe that a light goes on and off, this prompts the idea that the former causes the latter. But we can't be sure that Stanislav's circuit is not dead, and that in another room Nadia manipulates the live switch. Assume the two of them are timing their moves by the motion of a clock that we cannot observe. The claim that Stanislav's switch causes the light to illuminate is therefore called into doubt.
However, now imagine that we abandon our lazy recline and ask Stanislav if we can flip the switch ourselves. This we do randomly to eliminate coincidence with other variables. We have gone from observation to experiment. If the light's cycle still corresponds to the switch, then we can conclude that the switch causes the light to shine or not to shine.
In Pearl's publications, he uses a do() notation to show that a system variable is set experimentally rather than merely observed. This is a new element that cannot be accounted for in usual statistical methods.
My paper takes up the question of partial causality. Suppose the light corresponds to the switch some of the time, and not other times. What can we conclude in this case?
References can be found in the article linked above. Additionally, you may be interested in this reddit post on inferring cause in purely observational systems.