Causal Markov property

As explained in Causal AI

The global Markov property states that d-separation in a causal graph implies conditional independence in its joint probability distribution.

The causal Markov property connects a causal directed acyclic graph (DAG) to conditional independence. A node is independent of its non-descendants given its parents; equivalently, the joint distribution factorizes into conditional distributions given direct parents. section 3.1.6 section 3.5

The property has three equivalent formulations:

• Local Markov property: each node is conditionally independent of its non-descendants given its parents. section 4.3.1
• Markov factorization property: the joint probability distribution decomposes into parent-conditioned distributions. section 4.3.1
• Global Markov property: d-separation in the DAG implies conditional independence in the joint distribution. section 4.3.1

Ancestry and prediction

Grandfather and grandchild blood types are correlated through the father. Once the father’s blood type is known, the grandfather’s type provides no additional predictive information about the grandchild’s type. section 3.1.6

The assumption is operational: d-separations generate statistical tests for conditional independence, so failed tests can refute the proposed causal model. It also supports probabilistic parameterizations, including conditional probability tables, regressions, and neural networks. section 4.3.1 section 3.5

Figure 3.9 The causal relationships in the transportation DAG encode key assumptions about conditional independence.

Use the transportation DAG to inspect how causal structure encodes conditional-independence assumptions. section 3.1.6

How Causal AI covers it

How deep it goes

Defines local, factorization, and global forms and uses d-separation for refutation, but defers detailed testing procedures.

Examples

Uses blood-type ancestry, transportation DAGs, VAE factorization, and MDP transition functions as concrete applications.

What you can do after

Evaluate causal models by deriving implied independences, factorizing distributions, and identifying tests that can refute a DAG.

How it connects

Links causal DAGs, conditional independence, d-separation, Markov factorization, causal Markov kernels, and Markov decision processes.

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Related concepts

• Markov propertyequivalent to · The Markov property is the condition that, given the current state, the distribution of future states is independent of the process’s earlier history.
• conditional independenceproduces · Two variables are conditionally independent given specified information when their conditional joint distribution factors into the product of their conditional distributions.
• d-separationuses · D-separation is a graphical criterion for determining whether a conditioning set blocks all relevant paths between variables in a graph, implying conditional independence under the graph’s assumptions.
• causal graphical modelused by · A causal graphical model is a probabilistic or structural model whose graph represents causal relationships and whose mechanisms support reasoning about interventions.

This explainer was generated by Manning's knowledge graph from the text of Causal AI. The section links point to the parts of the book it draws on.