／var／logmarcus chiu

Markov Condition／Assumption／Property - First／Second／Nth-Order - (Pairwise - Local - Global - Markov-Blanket) - Causal Markov (CMC) Condition／Assumption／Property

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Markov Property in Stochastic Processes

  • Markov Property refers to the memory-less property of a stochastic process
    • First-Order Markov Property - probability of an observation at time 𝑡 only depends on the observation at time 𝑡-1
    • Second-Order Markov Property - probability of an observation at time 𝑡 depend on both 𝑡-1 and 𝑡-2
    • Nth-Order Markov Property - probability of an observation at time 𝑡 depend on all {𝑡-1, …, 𝑡-𝑛}

Markov Property in Bayesian Networks

  • Markov Condition/Assumption that every node in a Bayesian Network is conditionally independent of its non-descendents, given its parents
  • Causal Markov Condition (CMC) states that, conditional on the set of all its direct causes, a node is independent of all variables which are not direct causes or direct effects of that node

The 2 conditions are equivalent, iff the structure of a Bayesian Network accurately depicts causality. However, a network may accurately embody the Markov Condition without depicting causality, in which case it should not be assumed to embody the Causal Markov Condition

see: Causation vs Dependence vs Correlation vs Relationships vs Association vs Laws

Markov Property in Markov Networks

Given an undirected graph 𝐆=(𝑉,𝐸), a set of random variables 𝐗=(𝑋𝑣)𝑣∈𝑉 indexed by 𝑉:

Variant

Definition

Strongness

Pairwise Markov Property

Pairwise Markovianity

Any two non-adjacent variables are conditionally independent given all other variables:

𝑋𝓊⫫ 𝑋𝓋 | 𝑋𝑉 \{𝓊,𝓋}

WEAKER
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STRONGER

Local Markov Property

Local Markovianity

A variable is conditionally independent of all other variables given its neighbors

𝑋𝓋⫫ 𝑋𝑉 \neighbors-of(𝓋) | 𝑋neighbors-of(𝓋)

Markov Blanket

Markov Blanket for a node in a graphical model contains all the variables that shield the node from the rest of the network

Global Markov Property

Global Markovianity

Any two subsets of variables are conditionally independent given a separating subset

𝑋𝐴⫫ 𝑋𝐵 | 𝑋𝑆

where every path from a node in 𝐴 to a node in 𝐵 passes through 𝑆

The Global Markov Property is stronger than the Local Markov Property, which in turn is stronger than the Pairwise Markov Property. However, the above three Markov Properties are equivalent for a positive probability