Factor graphs



For as the sufferings of dimensions abound in us, so our consolation aboundeth through localisation.

Many statistical models are naturally treated as directed Bayesian networks — for example in causal inference I might naturally think about generating processes for my work as a graphical model. Factor graphs decompose the model differently than the balls-and-arrows DAGs. Specifically, the factor graph structures the graph with regard to how we would implement method-passing inference, rather than how we would statistically interpret the model. The result is harder to interpret intuitively but easier to apply practically. Also, it is more general than a DAG; factor graphs can also be applied to undirected graphical models.

There are at least two different factor graph formalisms, and I will try to explain both. The first, “classic” factor graphs, are the ones that I first encountered in the literature, and the ones that I think are most commonly used. The second, “Forney-style” factor graphs, are even less intuitive and even more practical.

Loeliger’s 2004 zoo of graphical models, including two flavours of factor graphs.

Classic factor graphs

A long history, but in a recognisable form these appear in (Frey et al. 1997; Frey 2003; Kschischang, Frey, and Loeliger 2001). This classic version is explained lots of places, but for various reasons I needed the FFG version first.

The representation that we need to “make all the inference steps easy” ends up being a bipartite graph with two types of nodes: variables and factors. Given a factorization of a function \(g\left(X_1, X_2, \ldots, X_n\right)\), \[ g\left(X_1, X_2, \ldots, X_n\right)=\prod_{j=1}^m f_j\left(S_j\right) \] where \(S_j \subseteq\left\{X_1, X_2, \ldots, X_n\right\}\), the corresponding factor graph \(G=(X, F, E)\) consists of variable vertices \(X=\left\{X_1, X_2, \ldots, X_n\right\}\), factor vertices \(F=\left\{f_1, f_2, \ldots, f_m\right\}\), and edges \(E\). The edges depend on the factorization as follows: there is an undirected edge between factor vertex \(f_j\) and variable vertex \(X_k\) if \(X_k \in S_j\).

Quibble: We often assume the function to be factored is a probability density function; more generally we wish to factorise measures.

Once the factor graph is written out (which is a slightly weird process) computations on the graph are essentially the same for every node. By contrast, in a classic I-graph DAG there are many different rules to remember for leaf nodes and branches, for colliders and forks etc.

I may return to classic factor graphs.

For now, a pretty good introduction is included in Ortiz, Evans and Davison’s tutorial on Gaussian belief propagation, or Minka’s Minka (2005).

Forney-style factor graphs (FFGs)

A tweaked formalism. Citations tell me this was introduce in a recondite article (Forney 2001) which I did not remotely understand because it was way too far into coding theory for me. It is explained and exploited much better for someone of my background in subsequent articles and used in several computational toolkits (Korl 2005; Cox, van de Laar, and de Vries 2019; H.-A. Loeliger et al. 2007; H.-A. Loeliger 2004; van de Laar et al. 2018; Akbayrak, Bocharov, and de Vries 2021).

Relative to classic factor graphs, H.-A. Loeliger (2004) advocates for FFGs for the following advantages

  • suited for hierarchical modeling (“boxes within boxes”)
  • compatible with standard block diagrams
  • simplest formulation of the summary-product message update rule
  • natural setting for Forney’s results on Fourier transforms and duality.

Mao and Kschischang (2005) argues:

Forney graphs possess a strikingly elegant duality property: by a local dualization operation, a Forney graph for a linear code may be transformed into another graph, called the dual Forney graph, which represents the dual code

The explanation in Cox, van de Laar, and de Vries (2019) gives the flavour of how Forney-style graphs work:

Figure 1 from Cox, van de Laar, and de Vries (2019): In an FFG, edges correspond to variables and nodes represent factors that encode constraints among variables. A node connects to all edges that correspond to variables that occur in its factor function. For example, node \(f_{b}\) connects to edges \(x_{1}\) and \(x_{2}\) since those variables occur in \(f_{b}\left(x_{1}, x_{2}\right)\). Variables that occur in just one factor \(\left(x_{3}\right.\) and \(x_{5}\) in this case) are represented by half-edges. While an FFG is principally an undirected graph, we usually specify a direction for the (half-)edges to indicate the generative direction of the model and to anchor the direction of messages flowing on the graph.

A Forney-style factor graph (FFG) offers a graphical representation of a factorized probabilistic model. In an FFG, edges represent variables and nodes specify relations between variables. As a simple example, consider a generative model (joint probability distribution) over variables \(x_{1}, \ldots, x_{5}\) that factors as \[ f\left(x_{1}, \ldots, x_{5}\right)=f_{a}\left(x_{1}\right) f_{b}\left(x_{1}, x_{2}\right) f_{c}\left(x_{2}, x_{3}, x_{4}\right) f_{d}\left(x_{4}, x_{5}\right), \] where \(f_{\bullet}(\cdot)\) denotes a probability density function. This factorized model can be represented graphically as an FFG, as shown in Fig. 1. Note that although an FFG is principally an undirected graph, in the case of generative models we specify a direction for the edges to indicate the “generative direction”. The edge direction simply anchors the direction of messages flowing on the graph (we speak of forward and backward messages that flow with or against the edge direction, respectively). In other words, the edge directionality is purely a notational issue and has no computational consequences.…

Figure 2 from Cox, van de Laar, and de Vries (2019): Visualization of the message passing schedule corresponding with observed variable \(x_{5}=\hat{x}_{5}\). The observation is indicated by terminating edge \(x_{5}\) by a small solid node that technically represents the factor \(\delta\left(x_{5}-\hat{x}_{5}\right) .\) Messages are represented by numbered arrows, and the message sequence is chosen such that there are only backward dependencies. Dashed boxes mark the parts of the graph that are covered by the respective messages coming out of those boxes. The marginal posterior distribution \(f\left(x_{2} \mid x_{5}=\hat{x}_{5}\right)\) is obtained by taking the product of the messages that flow on edge \(x_{2}\) and normalizing.

The FFG representation of a probabilistic model helps to automate probabilistic inference tasks. As an example, consider we observe \(x_{5}=\hat{x}_{5}\) and are interested in calculating the marginal posterior probability distribution of \(x_{2}\) given this observation.

In the FFG context, observing the realization of a variable leads to the introduction of an extra factor in the model which “clamps” the variable to its observed value. In our example where \(x_{5}\) is observed at value \(\hat{x}_{5}\), we extend the generative model to \(f\left(x_{1}, \ldots, x_{5}\right) \cdot \delta\left(x_{5}-\hat{x}_{5}\right).\) Following the notation introduced in Reller (2013), we denote such “clamping” factors in the FFG by solid black nodes. The FFG of the extended model is illustrated in Fig. 2

Clamping ends up being important in FFGs. Another place clamping arises is that, since a variable can only appear in two factors, if we want a variable to appear in more than two, we add extra factors in, each of which constrains the variables touching it to be equal to one another. This seems weird; but it seems to be what we would do in a classic factor graph anyway, in the sense that we would add an extra message passing step in for each extra factor, so this is not actually crazy.

Computing the marginal posterior distribution of \(x_{2}\) under the observation \(x_{5}=\hat{x}_{5}\) involves integrating the extended model over all variables except \(x_{2},\) and renormalizing: \[ f\left(x_{2} \mid x_{5}=\hat{x}_{5}\right) \propto \int \ldots \int f\left(x_{1}, \ldots, x_{5}\right) \cdot \delta\left(x_{5}-\hat{x}_{5}\right) \mathrm{d} x_{1} \mathrm{~d} x_{3} \mathrm{~d} x_{4} \mathrm{~d} x_{5} \] \[ =\overbrace{\int \underbrace{f_{a}\left(x_{1}\right)}_{1} f_{b}\left(x_{1}, x_{2}\right) \mathrm{d} x_{1}}^{2} \overbrace{\iint f_{c}\left(x_{2}, x_{3}, x_{4}\right) \underbrace{\left(\int f_{d}\left(x_{4}, x_{5}\right) \cdot \delta\left(x_{5}-\hat{x}_{5}\right) \mathrm{d} x_{5}\right)}_{3} \mathrm{~d} x_{3} \mathrm{~d} x_{4}}^{(4)} . \] The nested integrals result from substituting the [original] factorization and rearranging the integrals according to the distributive law. Rearranging large integrals of this type as a product of nested sub-integrals can be automated by exploiting the FFG representation of the corresponding model. The sub-integrals indicated by circled numbers correspond to integrals over parts of the model (indicated by dashed boxes in Fig. 2), and their solutions can be interpreted as messages flowing on the FFG. Therefore, this procedure is known as message passing (or summary propagation). The messages are ordered (“scheduled”) in such a way that there are only backward dependencies, i.e., each message can be calculated from preceding messages in the schedule. Crucially, these schedules can be generated automatically, for example by performing a depth-first search on the FFG.

What local means here might not be intuitive. What we mean is that, in this high dimensional integral, only some dimensions are involved in each sub-step, thanks to the multiplicative factorisation of the overall function. This is nothing to do with locality over the extent of the functions.

Generic messages

Question: These introductory texts discuss sum-product message passing, which is essentially about solving integrals. It might seem like I want to pass some other kind of update, e.g. maximum likelihood? Does this factor graph still help us? Yes, but I am not sure how wide is the class of things for which this holds true. TBD

Fourier transforms in

What does it benefit to take a fourier transform of a graph? (Kschischang, Frey, and Loeliger 2001; Forney 2001; Mao and Kschischang 2005)

In Bayesian brain models

See de Vries and Friston (2017) and van de Laar et al. (2018) for a connection to predictive coding.

Causal inference in

How does do-calculus work in factor graphs?

References

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Akbayrak, Semih, Ivan Bocharov, and Bert de Vries. 2021. Extended Variational Message Passing for Automated Approximate Bayesian Inference.” Entropy 23 (7): 815.
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Cox, Marco, Thijs van de Laar, and Bert de Vries. 2019. A Factor Graph Approach to Automated Design of Bayesian Signal Processing Algorithms.” International Journal of Approximate Reasoning 104 (January): 185–204.
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