Thanks for reading.

You might enjoy Sander Greenland's essay here: http://bayes.cs.ucla.edu/TRIBUTE/festschrift-complete.pdf Sander can be pretty bleak!

I tried to read that, but I think I didn't understand too much of it or its connection to this topic. I'll save that whole festschrift for later, there were some interesting titles in the table of contents.

I am not sure exactly what you mean, but I can think of a formalization where this is not hard to show.

I agree I did sort of conflate causal networks and Bayesian networks in general... I didn't realize there was no clean way of having both at the same time.

It might help if I describe a concrete way to test my claim using just causal networks: generate a randomly connected causal network with x nodes and y arrows, where each arrow has some random noise in it; count how many pairs of nodes are in a causal relationship; now, 1000 times initialize the root nodes to random values and generate a possible state of the network & storing the values for each node; count how many pairwise correlations there are between all the nodes using the 1000 samples (using an appropriate significance test & alpha if one wants); divide # of causal relationships by # of correlations, store; return to the beginning and resume with x+1 nodes and y+1 arrows... As one graphs each x against its respective estimated fraction, does the fraction head toward 0 as x increases? My thesis is it does.

So e.g. Simpson's paradox is surprising because we intuitively think of a conditional distribution (where conditioning can change anything!) as a kind of "interventional distribution" (no Simpson's type reversal under interventions: http://ftp.cs.ucla.edu/pub/stat_ser/r414.pdf).

Interesting, and it reminds me of what happens in physics classes: people learn how to memorize teachers' passwords, but go on thinking in folk-Aristotelian physics fashion, as revealed by simple multiple-choice tests designed to hone in on the appealing folk-physics misconceptions vs 'unnatural' Newtonian mechanics. That's a plausible explanation, but I wonder if anyone has established more directly that people really do reason causally even when they know they're not supposed to? Offhand, it doesn't really sound like any bias I can think of. It shouldn't be too hard to develop such a test for teachers of causality material, just take common student misconceptions or deadends and refine them into a multiple-choice test. I'd bet stats 101 courses have as much problems as intro physics courses.

I am not sure exactly what you mean, but I can think of a formalization where this is not hard to show. We say A "structurally causes" B in a DAG G if and only if there is a directed path from A to B in G. We say A is "structurally dependent" with B in a DAG G if and only if there is a marginal d-connecting path from A to B in G. A marginal d-connecting path between two nodes is a path with no consecutive edges of the form * -> * <- * (that is, no colliders on the path). In other words all directed paths are marginal d-connecting but the opposite isn't true.

That seems to make sense to me.

The justification for this definition is that if A "structurally causes" B in a DAG G, then if we were to intervene on A, we would observe B change (but not vice versa) in "most" distributions that arise from causal structures consistent with G. Similarly, if A and B are "structurally dependent" in a DAG G, then in "most" distributions consistent with G, A and B would be marginally dependent (e.g. what you probably mean when you say 'correlations are there').

I'm not sure about marginal dependence.

I qualify with "most" because we cannot simultaneously represent dependences and independences by a graph, so we have to choose. People have chosen to represent independences. That is, if in a DAG G some arrow is missing, then in any distribution (causal structure) consistent with G, there is some sort of independence (missing effect). But if the arrow is not missing we cannot say anything. Maybe there is dependence, maybe there is independence. An arrow may be present in G, and there may still be independence in a distribution consistent with G. We call such distributions "unfaithful" to G. If we pick distributions consistent with G randomly, we are unlikely to hit on unfaithful ones (subset of all distributions consistent with G that is unfaithful to G has measure zero), but Nature does not pick randomly.. so unfaithful distributions are a worry.

I'm afraid I don't understand you here. If we draw an arrow from A to B, either as a causal or Bayesian net, because we've observed correlation or causation (maybe we actually randomized A for once), how can there not be a relationship in any underlying reality and there actually be an 'independence' and the graph be 'unfaithful'?

Anyway, it seems that either way, there might be something to this idea. I'll keep it in mind for the future.