Trivial Belief

definition

TrivialBelief() is the belief that carries no information: the unit of [[Beliefs|combine]]. It is also, literally, the unique belief on the one-point space $1$.

Sources: code: open_model.jl, messages.jl

Theory (CT-ML wiki): Terminal Object · Markov Category · Monoid

Two readings, one object

1. The unit of pooling. combine(TrivialBelief(), b) == b for every belief b. An edge whose message has not been computed yet carries it, so that a variable’s marginal is the product of the messages that do exist.
2. The belief on the one-point space. $\mathcal{P}1=\{\ast \}$: there is exactly one distribution on a point. A factor with no input ($X\cong 1$, a prior) receives this as its prior argument; in AutoBayes’ terms, a prior is an open model $1\to X$.

In canonical Gaussian form it is $(\eta ,\Lambda )=(0,0)$, uninformative(n), the improper flat density, which is why the two readings agree: adding $(0,0)$ changes nothing.

belief_distance of two Trivial beliefs is $0$, and isexact(TrivialBelief()) == true.

Related: Beliefs, Gaussian Belief, Dirac Belief, Open Models and Latent Channels