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The terminology here is a mess. "Structural equation" is about as vague as "architectural bridge" and "Bayesian network" is not intrinsically Bayesian. Even better, God-of-causality Judea Pearl says that the two schools of models are almost identical.

So, what are the important differences?

(Amazing to me, the Wikipedia page for SEMs doesn't even include the word "network" as of this writing.)

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    $\begingroup$ Here's a succinct explanation from Judea Pearl himself: causality.cs.ucla.edu/blog/index.php/2012/12/07/… $\endgroup$ – dmp Jan 4 '17 at 23:25
  • $\begingroup$ @dmp, thanks, that appears to be the new version of my previously broken link above on 'Judea Pearl' -- fixed $\endgroup$ – zkurtz Jan 5 '17 at 15:28
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As far as I can tell, Bayesian Networks do not claim to be able to estimate causal effects in non-directed acyclic graphs, whereas SEM does. That's a generalization in favor of SEM... if you believe it.

An example of this might be measuring cognitive decline among people where cognition is a latent effect estimated using a survey instrument like 3MSE, but some people may display decreased cognition as a function of pain meds usage. Their pain meds may have been a consequence of injuring themselves due to cognitive decline (falling for example). And so, in a cross sectional analysis, you would see a graph that has a circular shape. SEM analysts like to tackle problems like that. I steer clear.

In the Bayes network world, you have very general methods of assessing conditional independence/dependence of nodes. One can use a fully parametric approach with any number of distributions, or go about the Bayesian nonparametric approaches I've heard about. SEM estimated using ML are (usually) assumed to be normal, which means that conditional independence is equivalent to having zero covariance for 2 nodes in the graph. I personally believe that's a rather strong assumption and would have very little robustness to model misspecification.

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  • $\begingroup$ That might be a difference in what practitioners call their analysis, but nothing forces a system of structural equations to be parametric. @zkurtz: There is a long and technically detailed discussion of what SEMs are in Pearl's Causality. If you don't have the book I could attempt to post a brief summary and track down the example he refers to in the link you posted. $\endgroup$ – CloseToC Jun 13 '14 at 16:35
  • $\begingroup$ While it's true that the covariance estimates are consistent for non-normal probability models, the main issue is the interpretation of 0 covariance as conditional independence. In general, that can only be said of normally distributed variables. $\endgroup$ – AdamO Jun 13 '14 at 17:07
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I don't really understand this, but see here:

Structural equation models and Bayesian networks appear so intimately connected that it could be easy to forget the differences. The structural equation model is an algebraic object. As long as the causal graph remains acyclic, algebraic manipulations are interpreted as interventions on the causal system. The Bayesian network is a generative statistical model representing a class of joint probability distributions, and, as such, does not support algebraic manipulations. However, the symbolic representation of its Markov factorization is an algebraic object, essentially equivalent to the structural equation model.

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  • $\begingroup$ Specifically, I wonder what they mean by "algebraic manipulations" in this context. $\endgroup$ – zkurtz Mar 5 '17 at 16:00

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