What are the main differences between Granger's and Pearl's causality frameworks? Recently, I ran across several papers and online resources that mention Granger causality. Brief browsing through the corresponding Wikipedia article left me with the impression that this term refers to causality in the context of time series (or, more generally, stochastic processes). Moreover, reading this nice blog post created additional confusion about how to view this approach.
I'm by no means a person knowledgeable about causality, as my fuzzy understanding of the concept consists of partly common sense, common knowledge, some exposure to latent variable modelling and structural equation modelling (SEM) and reading a bit from Judea Pearl's work on causality - not THE book of his, but more along the lines of an interesting overview paper by Pearl (2009), which for some reason, surprisingly, doesn't mention Granger causality at all.
In this context, I'm wondering whether Granger causality is something more general than a time series (stochastic) framework and, if such, what is its relation (commonalities and differences) to Pearl's causality framework, based on the structural causal model (SCM), which, as far as I understand, is, in turn, based on direct acyclic graphs (DAGs) and counterfactuals. It seems that Granger causality can be classified as a general approach to causal inference for dynamic systems, considering the existence of dynamic causal modelling (DCM) approach (Chicharro & Panzeri, 2014). However, my concern is about whether (and, if so, how) it is possible to compare the two approaches, one of which is based on stochastic process analysis and the other is not.
More generally, what do you think would be a sensible high-level approach - if one is possible - for considering all currently existing causality theories within a single comprehensive causality framework (as different perspectives)? This question is largely triggered by my attempt to read an excellent and comprehensive paper by Chicharro and Panzeri (2014) as well as reviewing an interesting causal inference course at the University of California, Berkeley (Petersen & Balzer, 2014).
References
Chicharro, D., & Panzeri, S. (2014). Algorithms of causal inference for the analysis of effective connectivity among brain regions. Frontiers in Neuroinformatics, 8(64). doi: 10.3389/fninf.2014.00064 Retrieved from http://journal.frontiersin.org/article/10.3389/fninf.2014.00064/pdf
Pearl, J. (2009). Causal inference in statistics: An overview. Statistics Surveys, 3, 96–146. doi:10.1214/09-SS057 Retrieved from http://projecteuclid.org/download/pdfview_1/euclid.ssu/1255440554
Petersen, M., & Balzer, L. (2014). Introduction to causal inference. University of California, Berkeley. [Website] Retrieved from http://www.ucbbiostat.com
 A: Granger causality is essentially usefulness for forecasting: X is said to Granger-cause Y if Y can be better predicted using the histories of both X and Y than it can by using the history of Y alone. GC has very little to do with causality in Pearl's counterfactual sense, which involves comparisons of different states of the world that could have occurred. So Peeps Granger-cause Easter, but they do not cause it. Of course, the two will overlap in a world where there are no potential causes other than X, but that is not a very likely setting and a fundamentally untestable one. Another less restrictive way they can coincide is, if, conditional on the realised history of Y and X, the next realisation of X is independent of the potential outcomes. This point is made in Lechner, M. (2010), "The Relation of Different Concepts of Causality Used in Time Series and
Microeconometrics," Econometric Reviews, 30, 109-127 (WP link), which is written in the potential outcomes framework, rather than Pearl's DAGs.
Addendum:
Let me make an implicit assumption more explicit. The crucial ingredient for my claim is that Easter does not have a fixed date. Suppose you knew nothing about Easter and wanted to forecast its date next year. From historical data (history of Y), you can see that Easter takes place in the spring. But can we do better than that? Using Peeps sales or marketing data (X) from near the holiday, we can see that peeps do Grange-cause it since that data is useful for forecasting Easter more precisely.
The corollary is that Christmas trees sales do not Granger-cause Christmas since if you know that Christmas took place on December 25th for centuries (adjusting for various calendar reforms and church schisms), tree sales do not help.
A: Pearl provides a calculus for reasoning about causality, Granger provides a method for discovering potential causal relations.  I will elaborate:
Pearl's work is based on what he has termed "Structural Causal Models", which is a triple M = (U, V, F).  In this model U is the collection of the exogenous (background, or driving) unobserved variables, V is the collection of endogenous (determined in some way by variables from U and V) variables, and F is a collection of functions f1, f2, ..., for each Vi in V.  The variable Vi is fully determined as Vi = fi(U, V \ Vi), that is the arguments to fi are some of the variables in U, and some of the variables in V, but not Vi itself.  In order to turn this into a probabilistic model, U is augmented with a probability distribution.  An example is given where U1 is a court order for a man's execution, V are the actions of a captain (V1) and two riflemen (V2,V3) in a firing squad as well as the living/dead state of the person to whom the court order pertains (V3).  If the judge orders the man shot (U1 = 'execute'), then this causes the captain to issue the order to fire, which causes the riflemen to shoot the prisoner, and hence causing his death.  If the court order is not given, the captain remains silent, the riflemen don't shoot, and the prisoner is left alive.
Pearl argues how his model can be used to reason about causation, design experiments, predict the effects of intervention, and answer counter-factual questions.  Intervention is distinct from anything in probability theory.  In doing intervention we interact with the model and hold a variable constant (which is more than merely observing that the variable is in a particular state, as with probabilistic conditioning), and Pearl describes how to "perform surgery" on the model in order to predict the outcome of this intervention.  Counter-factuals are even more difficult to answer, as we want to know what would have been the outcome of an experiment had something not been the case, even though it was.  This is what Pearl's models are about.
Granger Causality on the other hand is a statistical method, and makes no attempt to "prove" causation.  If we have a whole bunch of processes, we can use Granger causality to obtain a graph of "plausible causal relations", which may be interpreted as potentially genuine causes, or to provide measures of their interconnectedness, or detect the flow of energy or information amongst the processes.  In the case of literal causation, you can imagine a situation in which experiments (necessary for the methods of Pearl) are very costly.  In which case, you may be able to still observe the system and apply Granger-Causality to narrow things down to potential causes.  After doing this, you can have some sense of where to appropriate additional resources.
One question that immediately comes to mind when reading about Pearl's causal models is "how does one build the model in the first place?".  This would be accomplished through a combination of domain expertise and hypothesizing, but Granger-Causality could potentially provide some more information about how to construct the Pearl causal model as well.
Since I don't have enough reputation to comment, I will add here a criticism of Dimitriy V. Masterov's answer:  Peeps do not Granger-Cause Easter.  Easter occurs periodically, even though the occurrence of Peeps is closely correlated with that of Easter, the history of the occurrences of Easter is enough to predict it's future occurrence.  Information about Peeps does not add any additional information about Easter.  I think this is a key point: Granger-Causality is much more than mere correlation.  Processes that are correlated may not have any Granger-Causal relation, and processes with a Granger-Causal relation may not be correlated.
