Undirected Graphical Models are usually known as "Markov Networks" and Directed Graphical Models are known as "Bayesian Networks". This is naming is not very clear to me. It might be because of some historical background, though I am just curious to know if there is any concrete reason for this naming.
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1$\begingroup$ My impression, from Koller/Friedman's book, is that the 'Bayesian network' name may come from the fact that such models are generalizations of what is called a 'Naive Bayes model'. (I.e. mostly historical, as you guess.) For 'Markov Random Fields', I think this is also a historical term derived somehow from the statistical physics literature, in particular from work about the so-called 'Ising Model' (note that 'field' can mean very different thing in physics than in, say, some areas of math). I'm not 100% about either of these though, hence why this is only a comment. $\endgroup$– Chill2MachtCommented Dec 10, 2017 at 16:14
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2 Answers
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An undirected graph is not necessarily a Markov Network (Markov Random Field), and likewise, a acylic directed graph (ADG), more commonly known as, directed acyclic graph (DAG), is not necessarily a Bayesian Belief Network (BBN), if not at all.
Formally, a BBN is a pair(G,P) and satisfies the Markov condition.
- P is a joint probability distribution over U, P(U)
- U is a set of variables, U = { X1, X2, ..., XN }
- G is a DAG defined by the pair (V,E)
- V is a set of vertices with 1-to-1 correspondence with variables in U
- E is a set of directed edges
- Markov condition states that a node is conditionally independent of its nondescendants given its parents
A Directed Graph can be cyclic or acyclic, in a BBN, G is a DAG, so you have to be precise in usage of terms.
Also, G is only one of the three things that contributes to defining a BBN. So, you should not say, a DAG is a BBN or a BBN is a DAG.
Furthermore, a DAG is used to represent other things, not necessarily variables and dependencies (as in a BBN).
Whoever names an undirected graph a Markov Network, or a directed graph a BBN, probably doesn't have the full picture.
Research Andrey Markov on etymology of MRF, Markov Network, Markov blanket, Markov condition, etc... BBNs build on Markov's work, reusing as much as is applicable, just like quantum BN's build on and reuse the work of BBNs.
answered Aug 22, 2014 at 17:28
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It's probably not the best name. In a Bayes Net, a directed edge from A->B represents the conditional probability $P(B|A)$. If A represents an unknown parameter, then the network is parameterized with a prior on the parameter, and running inference amounts to using Bayes' Rule. In a Markov Network, you don't (/ can't?) parameterize the model with the prior. The potentials on the nodes aren't generally singleton marginals.
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$\begingroup$ Why do you say "In a Markov Network, you don't (/ can't?) parameterize the model with the prior."? $\endgroup$– DanielCommented Jul 23, 2014 at 0:11
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$\begingroup$ In a Bayes Net of A->B, there is a parameter $\psi(A)$ which happens to be equal to $p(A)$, which is the prior. In a Markov Network of A-B-C, there are parameters on the cliques $\psi(A, B), \psi(B,C)$. Neither of these are P(A). You can also add a parameters $\psi(A), \psi(B), \psi(C)$, but they are not generally $P(A)$, etc. $\endgroup$– JeffCommented Jul 23, 2014 at 2:10