(Barber 2012; Lauritzen 1996) are rigorous abstract introductions. (Murphy 2012) has a minimal introduction intermixed with some related models, with a more ML, more Bayesian formalism. For use in causality, (Pearl 2009; Spirtes, Glymour, and Scheines 2001) are readable.
People recommend me (Koller and Friedman 2009) which is probably the most detailed an comprehensive, but I found it was hard to see the forest for the trees in this one. YMMV.
What’s special here is how we handle independence relations and reasoning about them. In one sense there is nothing special about graphical models; it’s just a graph of which variables are conditionally independent of which others. On the other hand, that graph is a powerful analytic tool, telling you what is confounded with what, and when. Moreover, you can use conditional independence tests to construct that graph even without necessarily constructing the whole model (e.g. (Zhang et al. 2012)).
Graphs of conditional, directed independence are a convenient formalism for many models. These are also called Bayes nets (not to be confused with Bayesian inference.)
Once you have the graph, you can infer more detailed relations than mere conditional dependence or otherwise; this is precisely that hierarchical models emphasise.
These can even be causal graphical models, and when we can infer those we are extracting Science (ONO) from observational data. This is really interesting; see causal graphical models
BayesNets is a Julia package for reasoning over directed graphical models.
Undirected, a.k.a. Markov graphs
a.k.a Markov random fields, Markov random networks. (other types?)
I would like to know about spatial Poisson random fields, Markov random fields, Bernoulli (or is it Boolean?) random fields, esp for discrete multivariate sequences. Gibbs and Boltzman distribution inference.
Wasserman’s explanation of the use case here is good: Estimating Undirected Graphs Under Weak Assumptions
A unifying formalism for the directed and undirected graphical models How does that work then?
A factor graph is a bipartite graph representing the factorization of a function. In probability theory and its applications, factor graphs are used to represent factorization of a probability distribution function, enabling efficient computations, such as the computation of marginal distributions through the sum-product algorithm.
Pedagogically useful, although probably not industrial-grade, David Barber’s discrete graphical model code (Julia).
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