What is the difference between causal models and directed graphical models?

Question

What is the difference between causal models and directed graphical models?

or:

What is the difference between causal relationships and directed probabilistic relationships?

or, even better:

What would you put in the interface of a DirectedProbabilisticModel class, and what in a CausalModel class? Would one inherit from the other?

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Collaborative solution:

interface DirectedModel {
  bool NodesDependent(set<Node> nodes, map<Node, Distribution> context)
  map<Node, Distribution> InferredProbabilities(map<Node, Distribution> observed_probabilities,
                                          set<Node> nodes_of_interest)
}
interface CausalModel: DirectedModel {
  map<Node, Distribution> InferredProbabilities(map<Node, Distribution> observed_probabilities,
                                          map<Node, Distribution> externally_forced_probabilities,
                                          set<Node> nodes_of_interest)
}

Answer 1

If I understand this post correctly, casual models and directed graphical models (Bayesian networks) aim at different stages of the workflow. A casual model is a way of assigning dependencies such that they reflect causation. Bayesian networks provide us with inference techniques. So, one can perform estimation using something different. On the other hand, one can model Bayesian networks using different techniques than SCM.

If you dig into it deeper, please let us know, because I don't fully understand the subject of SCM (while I'd like to :).

Answer 2

directed graphical models are a way of encoding causal relationships between variables. probabilistic graphical models are a way of encoding causality in a probabilistic manner. I would recommend reading this book written by Judea Pearl who is one of the pioneers in the field (whom I see you refer to in the paper you mentioned in the comment).

a directed graph is simply a graph (nodes and edges) which is directed (edges have directions). causal models are models which tell you how variables affect each other, one way of doing that is using directed graphs. AI research has shown that deterministic causal relationships are not sufficient to encode knowledge of the world around us because it is too messy. That is why probability was added to the picture.

Answer 3

Causality by Judea Pearl is the book to read.

The difference is that one is causal and the other is merely statistical. Before dismissing me as a member of the tautology club, hear me through.

A directed probabilistic relationship (AKA a complete set of Conditional Probability Tables , AKA Bayesian Network) only contains statistical information. Meaning that anything you can infer from the Joint Probability table you can infer from the directed probabilistic relationship, nothing more, nothing less. The two are equivalent.

A causal relationship is something else entirely. A causal relationship (AKA Causal Bayesian Network) must specify what happens under any variable intervention. Intervention is when a variable is forced to a value outside of the normal influences of the model. This is equivalent to replacing the conditional probability for the forced variable (or variables, but we consider just one for simplicity) with a new table in which the variable takes it's forced value with probability one.

If this does not make sense, please follow up and I will clarify.

This section added to address Neil's questions in the comments

Neil asks:

How can you determine the direction of directed probabilistic relationships without performing interventions? In other words, doesn't the directed graphical model have causal information in it (i.e., information about probabilities conditional on interventions?)

You can determine the direction of directed probabilistic relationships by making additional non-statistical assumptions. These assumptions commonly include: assuming no hidden variables, and the really important one, assuming that the conditional independence relationships found in the joint distribution are stable (meaning they exist not by chance or cancellation). Bayesian Networks do not make these assumptions.

For details of how to recover the directions research the IC, PC, and IC* algorithms. I believe the specific details of IC are covered in: "A Theory of Inferred Causation"