# Causal Bayesian network, causal diagram, structural causal model and marginal structural model: what do they exactly mean?

In the Book of Why, Judea Pearl gives a comprehensive overview of the causal diagrams (or causal graphs), but to me, the terminology is not clear yet. In the book, he presents Bayesian network in the context of artificial intelligence before introducing the causal Bayesian network.

Question 1: What is the difference between causal diagrams and causal Bayesian network?

Additionally, he defines the structual causal model (SCM) and justifies its need in order to handle counterfactuals:

The response function is the key ingredient that gives SCMs the power to handle counterfactuals. It is implicit in Rubin’s potential outcome paradigm but a major point of difference between SCMs and Bayesian networks, including causal Bayesian networks. In a probabilistic Bayesian network, the arrows into Y mean that the probability of Y is governed by the conditional probability tables for Y, given observations of its parent variables. The same is true for causal Bayesian networks, except that the conditional probability tables specify the probability of Y given interventions on the parent variables. Both models specify probabilities for Y, not a specific value of Y. In a structural causal model, there are no conditional probability tables. The arrows simply mean Y is a function of its parents, as well as the exogenous variable $$U_Y$$: $$Y = f_Y(X, A, B, C,…, U_Y)$$ (...) To turn a noncausal Bayesian network into a causal model—or, more precisely, to make it capable of answering counterfactual queries—we need a dose-response relationship at each node.

While I understand the need of a model that use a dose-response relationship in order to do counterfactuals, I do not see the difference between the SCMs, defined above by Pearl, and the causal structural model, defined in this book, by Hernan and Robins.

Question 2: Is there a fundamental difference between these two models?

• Core difference would be vanilla Bayesian networks do not handle interventions out of box. Addition of do-calculus provides one of the ingredient of a causal model, i.e., as Pearl states, making them to answer counterfactual queries. PS: do-operator is the core difference. Sep 23, 2021 at 12:54

I will give my answer based on Pearl's other book (Causality)

First, some terminology: there are 3 types of queries: observational, interventional, and counterfactual.

• For observational queries, you only need a joint distribution
• For interventional queries, you also need a directed graph (e.g. a Bayesian Network(BN), and especially a Causal Bayesian Network(CBN).) As you quoted, CBNs are required to be able to see how the variables influence each other (hence a graph is used). On a high level, the takeaway should be this simple: you need a graph. The CBN is a BN where you interpret the probabilities in another way.
• For counterfactual queries, you also need to know the quantitative relationship between the different variables. So here you need a graph and a parametrization that describes these functional relationships.

A causal diagram is a directed acyclic graph (DAG)

This provides all we need to answer your first question Answer to question 1: TL;DR: SCM = causal diagram + functions for each edge The causal diagram is a graph that describes what variables relate to each other, whereas an SCM additionally gives a quantitative description of these relationships,

P.S.: I cannot find "causal structural models" in the linked book with the search function. My guess based on Google searches (the term arises in a paper co-authored by Pearl) that they are the same