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RichardKennaway comments on Causal Diagrams and Causal Models - Less Wrong
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Well, in some sense this is why causal inference is hard. Most of the time if you see independence that really does mean there is nothing there. The reasonable default is the null hypothesis: there is no causal effect. However, if you are poking around because you suspect there is something there, then not seeing any correlations does not mean you should give up. What it does mean is you should think about causal structure and specifically about confounders.
What people do about confounders is:
(a) Try to measure them somehow (epidemiology, medicine). If you can measure confounders you can adjust for them, and then the effect cancellation will go away.
(b) Try to find an instrumental variable (econometrics). If you can find a good instrument, you can get a causal effect with some parametric assumptions, even if there are unmeasured confounders.
(c) Try to randomize (statistics). This explicitly cuts out all confounding.
(d) You can sometimes get around unmeasured confounders by using strong mediating variables by means of "front-door" type methods. These methods aren't really well known, and aren't commonly used.
There is no royal road: getting rid of confounders is the entire point of causal inference. People have been thinking of clever ways to do it for close to a hundred years now. If you have infinite samples, and know where unobserved confounding is, there is an algorithm for getting the causal effect from observational data by being sneaky. This algorithm only succeeds sometimes, and if it doesn't, there is no other way in general to do it (e.g. it's "complete"). More in my thesis, if you are curious.
Your thesis deals only with acyclic causal graphs. What is the current state of the art for cyclic causal graphs? You'll know already that I've been looking at that, and I have various papers of other people that attempt to take steps in that direction, but my impression is that none of them actually get very far and there is nothing like a set of substantial results that one can point to. Even my own, were they in print yet, are primarily negative.
The recent stuff I have seen is negative results:
(a) Can't assign Pearlian semantics to cyclic graphs.
(b) If you assign equilibrium semantics, you might as well use a dynamic causal Bayesian network, a cyclic graph does not buy you anything.
(c) A graph representing the Markov property of the equilibrium distribution of a Markov chain represented by a causal DBN is an interesting open question. (This graph wouldn't have a causal interpretation of course).