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Factor graphs

A factor graph is a graphical representation of how a joint probability distribution factorizes into a product of local functions. ReactiveMP.jl uses factor graphs as the underlying structure for all inference computations.

Variables and factors

A factor graph has two kinds of nodes:

  • Variable nodes — represent the random quantities in your model (latent variables, observed data, or constants).
  • Factor nodes — represent the local functions (conditional distributions, likelihoods, deterministic transforms) that connect variables together.

An edge between a factor node and a variable node means that the factor involves that variable.

Consider a simple model with three variables x, y, and z and two factors f and g:

  (x) ── [f] ── (y) ── [g] ── (z)

This graph represents the factorization:

\[p(x, y, z) = f(x, y) \cdot g(y, z)\]

Each factor is a local function: f only involves x and y, and g only involves y and z. Messages can therefore be computed locally at each factor, using only the information from neighboring nodes.

Stochastic and deterministic factors

ReactiveMP.jl distinguishes two kinds of factor nodes:

  • Stochastic nodes represent probability distributions, e.g. p(x | μ, σ). They are used for likelihood terms, priors, and latent variable relationships.
  • Deterministic nodes represent hard functional constraints, e.g. z = x + y. They do not add probability mass — they enforce an exact relationship.

This distinction matters for how messages are computed and how the variational free energy objective is structured. See isdeterministic and isstochastic.

How ReactiveMP.jl represents factor nodes

Every factor in ReactiveMP.jl is a Julia type registered with the @node macro. The macro declares the node's name, its type (Stochastic or Deterministic), and the fixed set of edges (interfaces) it connects to:

struct MyFactor end

@node MyFactor Stochastic [ out, x, y ]
#     ^^^^^^^^ ^^^^^^^^^^   ^^^^^^^^^^
#     tag      type          edges (first = output by convention)

After registration, MyFactor can be used as a factor node in a model. The inference engine then dispatches message update rules defined with @rule for that node type.

Note

The @node macro only registers the factor's structure. Message update rules must be added separately using @rule and @marginalrule. See Message update rules for details.

ReactiveMP.jl ships with many predefined nodes for common distributions and operations — Gaussian, Gamma, Beta, Bernoulli, arithmetic operations, and more. Custom nodes can be registered using the same @node macro.

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