DiracBelief(x)is a point mass : certainty about a value. It is what an observed (hard-clamped) channel carries and what every point inference returns.
Sources: code:
open_model.jl,messages.jl; AutoBayes arXiv:2503.18608, Appendix A, Example 4 (a cup collapses the posterior)Theory (CT-ML wiki): Markov Category · Bayesian Inversion · Copy-Discard Category
Where it comes from
- Observations. An
Observed()channel has precision by default (Channels and Polarity); its value enters as a Dirac and is imposed by projection, never as a penalty. Categorically this is the cup of Open Models and Latent Channels: data clamped onto a wire. - Point inference. Solvers that return a single configuration — RED-Diff, the implicit diffusion solver (Inference Signatures #1–#4), DEQ root-finding — return Diracs. Here the Dirac is an approximation of a posterior, not certainty.
- Deterministic maps. In a Markov category the deterministic morphisms are exactly those that send Diracs to Diracs.
How it behaves
- It dominates
combine. Combined with a Gaussian, a sample set or the trivial belief, the Dirac wins: it is the limit of a Gaussian, and infinite precision outweighs any finite one. Two different Diracs on one variable are an error (“contradictory hard clamps”). - It has no entropy: . A factor whose inference returns a Dirac cannot contribute a meaningful entropy to the Bethe Free Energy (The Implicit Diffusion Factor as a Statistical Game §2).
- It cannot negotiate. Because it dominates, a factor that outputs a Dirac overrides its neighbours instead of pooling with them (The Diffusion Factor §4.2). The fix is a finite precision: a Gaussian Belief.
belief_distance between two Diracs is the Euclidean distance of their values, and Diracs over
numbers or arrays can be damped (mixed) during message passing.
Related: Beliefs, Gaussian Belief, Trivial Belief, Channels and Polarity, Messages are Inversions