Beliefs

definition overview

A belief is what travels along an edge of a factor graph and what sits on a variable: a representation of a probability distribution, or of a likelihood, over that variable’s space. Four concrete types exist, each with its own note: Trivial Belief, Dirac Belief, Gaussian Belief and Sample Belief. Their algebra is one operation, combine, the product of densities.

Sources: original to this vault (design); code: open_model.jl (AbstractBelief, DiracBelief, SampleBelief, TrivialBelief), beliefs.jl (GaussianBelief), messages.jl (combine, belief_distance, belief_logdensity)

Theory (CT-ML wiki): Giry Monad · Markov Category · Bayesian Inversion · Distribution Monad

What a belief is

Categorically, a belief on a variable $X$ is an element of $\mathcal{P}X$, a state $1\to X$ in a Markov category (for continuous spaces, the Giry monad). It is the prior $\pi$ of a Bayesian lens and the posterior that inference returns. On an edge it may also be a likelihood: a non-negative function that need not integrate to one, which is why some beliefs are allowed to be improper.

All belief types subtype LenticulumCore.AbstractBelief. The core package defines the three that need no linear algebra. GaussianBelief lives in the top-level Lenticulum package, which no lib/ package may depend on — the reason the equilibrium and diffusion factors cannot yet return Gaussian messages (DEQ as a Relation §5, The Diffusion Factor §5).

The four types

type	represents	where	isexact	typical source
Trivial Belief	no information (the unit)	LenticulumCore	true	an unset message, the prior of a prior
Dirac Belief	a point mass, ${\delta }_x$	LenticulumCore	true	an observation (a hard clamp), a point inference
Gaussian Belief	$\mathcal{N}$ in canonical form $(\eta ,\Lambda )$, possibly improper	Lenticulum	false (the default)	Gaussian factors, Laplace approximations
Sample Belief	a weighted particle set	LenticulumCore	false	samplers, the fallback

isexact says whether a computed belief is exact rather than approximate. It defaults to false so that silence never implies exactness; only the Trivial and Dirac beliefs override it. A Gaussian can be exact (on a linear-Gaussian tree it is), but the flag does not claim it.

The algebra: combine

Pooling two beliefs about one variable is the pointwise product of their densities, the sum–product step at a variable node (Messages are Inversions). Mycelium.combine implements exactly the cases it can do honestly:

combine	result	why
Trivial with anything	the other	the unit
Gaussian with Gaussian	add $(\eta ,\Lambda )$	exact, associative, commutative
Dirac with Gaussian, Sample or Trivial	the Dirac	a hard clamp is the $\Lambda \to \infty$ limit and dominates
two equal Diracs	that Dirac	idempotent
two different Diracs	error	contradictory hard clamps are a modelling error
anything else (e.g. two Sample beliefs)	error	needs densities (belief_logdensity) and importance reweighting

So beliefs form a partial commutative monoid under combine, with the Trivial belief as unit and the Dirac beliefs as absorbing elements. Gaussians are the one family closed under it, which is why Gaussian belief propagation is exact on trees (The Linear Gaussian Chain).

Other operations

• belief_distance(a, b) — how much a message changed, the convergence test of propagate!. It returns Inf whenever it cannot tell, so that a schedule never declares convergence it cannot verify.
• belief_logdensity(b, x) — $\log p_b(x)$; implemented for Gaussians only.
• variable_entropy(b) — the entropy in the Bethe Free Energy; for Gaussians it can be negative.

Everything else one might do with beliefs (addition, mixture, logic, projection, tempering) is catalogued, with what each needs, in Belief Algebra.

Related: Messages are Inversions, Factor Graphs, Channels and Polarity, Inversions and Bayesian Lenses, Probabilistic Types

tab: Julia
**Docs:** [Lenticulum API](https://mathstruct.org/Lenticulum.jl/dev/packages/lenticulum/) · [Mycelium API](https://mathstruct.org/Lenticulum.jl/dev/packages/mycelium/)
```julia
using Lenticulum, LenticulumCore, Mycelium
using LenticulumCore: DiracBelief, TrivialBelief, SampleBelief
a = Gaussian(1.0, 4.0); b = Gaussian(3.0, 4.0)       # N(1, 4) and N(3, 4), stored as (η, Λ)
c = combine(a, b)                                     # pooling = adding canonical parameters
(belief_mean(c), belief_cov(c))                       # ≈ ([2.0], [2.0;;]): precision-weighted mean, halved variance
combine(TrivialBelief(), a) === a                     # true: the unit
u = uninformative(1); combine(u, a).η == a.η          # true: (0, 0) is the same unit in canonical form
combine(DiracBelief([0.5]), a)                        # the Dirac: a hard clamp dominates (Λ → ∞)
lik = GaussianBelief([2.0, 0.0], [1.0 0.0; 0.0 0.0])  # constrains x₁ only: an improper likelihood message
isproper(lik)                                         # false — fine for a message, not for a mean
belief_mean(combine(lik, Gaussian([0.0, 0.0], [1.0 0.0; 0.0 1.0])))   # ≈ [1.0, 0.0]: proper after a prior
```