We can first think more generally about what a causal diagram really is. Then let's discuss how one might practically use them as an informative prior, and jointly with observational data, to confidently predict causal effects.
A causal diagram is a directed acyclic graph (DAG) representation of the functional relationships between the variables (i.e. nodes) within the distribution. And the structure of the graph serves to encode the conditional dependence or independence among the variables. The diagram essentially asserts our assumptions about the world in a easy-to-understand visual format. Provided with a joint distribution p(a,b,c), the same distribution can be written as either:
So which causal diagram is the correct one for the joint distribution? The example demonstrates that the mapping of causal diagrams to our observational data is many to one. Multiple correct hypothesis are plausible, and it is usually impossible to definitely choose between them just by looking at the observational data only.
How can we then use observational data to infer the correct diagram? For a said causal diagram, we mimic the effects of a intervention by conditioning on a variable (i.e. we force it to take a particular value). The action is encapsulated by the do-operator in p(Y|do(X)) and more formally by do-calculas, a tool for causal inference that allows us to disambiguate what needs to be estimated from the observational data. The front- and back-door approaches are but just two doors through which we can eliminate all the do's in our quest to climb Mount Intervention.
Suffice to say, by removing all incoming edges to the node of interest, an intervention modifies the original joint distribution to become the post-interventional distribution. A causal query becomes identifiable if we can remove all do-operators and therefore we can use the observational data to estimate causal effect. Else the causal query is considered non-identifiable and a real-world interventional experiment would be required for determining the causal effect.
While a researcher may never be completely persuaded in the soundness and integrity of the causal diagram they've constructed, they do have mechanisms in place to empirically test a partial collection of relationships between the sets of variables. If the dependencies and independencies are not present in the observational data, this might be a signal that the diagram is inaccurate. The researcher can then iteratively test and update the causal diagram to be more inline with the information contained within the observational data (and domain knowledge if applicable).