information:extraction, stylistic:style_transfer, information:filtering
Fixing versus conditioning

The basic premise of the causal approach is that it isnot always appropriate to "control" for the mediator M when we seek to estimate the direct effect of X on Y (see the Figure above).The classical rationale for "controlling" for M " is that, if we succeed in preventing M from changing, then whatever changes we measure in Y are attributable solely to variations in X and we are justified then in proclaiming the effect observed as "direct effect of X on Y." Unfortunately,"controlling for M " does not physically prevent M from changing;it merely narrows the analyst's attention to casesof equal M values. Moreover, the language of probability theory does not possess the notation to express the idea of "preventing M from changing" or "physically holding M constant".The only operator probability provides is "Conditioning"which is what we do when we "control" for M, or add M as a regressor in the equation for Y. The result is that, instead of physically holding M constant (say at M = m) and comparing Y for units under X = 1' to those under X = 0, we allow M to vary but ignore all units except those in which M achieves the value M = m. These two operations are fundamentally different, and yield different results, except in the case of no omitted variables. Improperly conditioning mediated effects can be a type of bad control.