Causal Calculus

Sometimes experiments cannot be carried out because they are impractical or unethical, so how can one determine causality between variable if no data can be collected? One solution is to implement causal calculus. What causal calculus depends on is accepted assumptions like sprinklers on do not cause rain to fall. A graphical representation (a Bayesian Network) of this implication is shown below:

After assumptions are made, the next step is differentiate between conditional and interventional probabilities. As an example of an unethical experiment, one could study the difference between obesity rates of people who eat at McDonald's daily voluntarily [conditional probability, and expressed as P(obesity | McDonald's daily)] and people forced to eat at McDonald's daily [interventional probability, and expressed as P(obesity | do(McDonald's daily))].

The first probability can be easily carried out with a simple observational study. The second, of course, cannot be for obvious reasons. This part could only be studied by using a causal notion derived from the conditional study.

The real application of causal calculus is to characterize the confounding variables within a system, or variables that correlate to both the independent and dependent variable, which are especially problematic in risk assessments. If these confounding variables can be adequately tuned, the correct causal relationship between variables can be obtained even if they are not being directly measured.

Sources:
http://en.wikipedia.org/wiki/Causality
http://en.wikipedia.org/wiki/Bayesian_Networks
http://en.wikipedia.org/wiki/Confounding_variables
http://ftp.cs.ucla.edu/pub/stat_ser/r338-shrout.pdf