Helper nodes and message types

StandaloneDistribution

The node out ~ d for a distribution value d of any family, given on its interface distribution as a constant: what a model writes as x ~ prior, with prior a distribution rather than a family and its parameters. Its interfaces are out and distribution, and it runs under DefaultAlgorithm.

Rules. The message towards out is d itself, from the marginal of distribution, a PointMass holding d; its log scale is zero. There is no rule towards distribution, which must be a constant. The average energy is the cross entropy E_q[-log d(out)] = KL(q ‖ d) + H(q), defined for every pair (q, d) for which Distributions has a kldivergence.

Examples

julia> prior = Beta(2.0, 3.0);

julia> getresult(@call_message_update_rule(node = StandaloneDistribution, target = :out, q = (distribution = PointMass(prior),))) === prior
true

See also Uninformative.

Uninformative()

The uninformative node, a factor of one over its single interface out, and the value of its message. The rule towards out takes no inputs and returns Uninformative() itself; it declares no log scale, so the message's log scale is undefined. Uninformative() is the identity of prod under the default product rule, so it changes nothing it is multiplied with. The average energy is zero.

A model uses it to leave a variable without a prior, x ~ Uninformative(), where every other factor on x then decides its marginal. Uninformative() is not a distribution and has no mean or density, so a variable needs some other factor for its marginal to be one.

Examples

julia> prior = NormalMeanVariance(0.0, 1.0);

julia> prod(GenericProd(), Uninformative(), prior) === prior
true

julia> prod(GenericProd(), Uninformative(), Uninformative())
Uninformative()

See also StandaloneDistribution.

GammaShapeLikelihood(p::Real, γ::Real)

The likelihood of a Gamma distribution's shape α, the unnormalised density

\[f(α) = \exp(γ α - p \log Γ(α)), \quad α > 0,\]

which is the message towards the shape of a GammaShapeRate node, with p = 1 and γ = E[log β] + E[log out], and towards a shape (:a, k) of a GammaMixture, with p the component's responsibility. p and γ are promoted to one type.

It has logpdf, support (the positive half line) and params, (p, γ), but no mean or normaliser. The product of two of them is a GammaShapeLikelihood again, with the parameters added; with any other distribution the default product is a lazy ProductOf.

Examples

julia> left, right = GammaShapeLikelihood(1.0, 2.0), GammaShapeLikelihood(0.5, -1.0);

julia> product = prod(GenericProd(), left, right);

julia> product isa GammaShapeLikelihood && all(params(product) .≈ (1.5, 1.0))
true

julia> logpdf(left, 1.0) ≈ 2.0
true