A "recovery" algorithm was developed by Rebane and Pearl (1987) which rests on Wrights distinction between the three possible types of causal substructures allowed in a directed acyclic graph (DAG):

  
    
      
        X
        →
        Y
        →
        Z
      
    
    {\displaystyle X\rightarrow Y\rightarrow Z}
  

  
    
      
        X
        ←
        Y
        →
        Z
      
    
    {\displaystyle X\leftarrow Y\rightarrow Z}
  

  
    
      
        X
        →
        Y
        ←
        Z
      
    
    {\displaystyle X\rightarrow Y\leftarrow Z}
  Type 1 and type 2 represent the same statistical dependencies (i.e., 
  
    
      
        X
      
    
    {\displaystyle X}
   and 
  
    
      
        Z
      
    
    {\displaystyle Z}
   are independent given 
  
    
      
        Y
      
    
    {\displaystyle Y}
  ) and are, therefore, indistinguishable within purely cross-sectional data.
Knowing that causation is a matter of counterfactual dependence, we may reflect on the nature of counterfactual dependence to account for the nature of causation. For example, in his paper "Counterfactual Dependence and Times Arrow," Lewis sought to account for the time-directedness of counterfactual dependence in terms of the semantics of the counterfactual conditional. If correct, this theory can serve to explain a fundamental part of our experience, which is that we can only causally affect the future but not the past. Probabilistic causation
Interpreting causation as a deterministic relation means that if A causes B, then A must always be followed by B. In this sense, war does not cause deaths, nor does smoking cause cancer or emphysema. As a result, many turn to a notion of probabilistic causation. Informally, A ("The person is a smoker") probabilistically causes B ("The person has now or will have cancer at some time in the future"), if the information that A occurred increases the likelihood of Bs occurrence. Formally, P{B|A}≥ P{B} where P{B|A} is the conditional probability that B will occur given the information that A occurred, and P{B} is the probability that B will occur having no knowledge whether A did or did not occur. This intuitive condition is not adequate as a definition for probabilistic causation because of its being too general and thus not meeting our intuitive notion of cause and effect.