Sample Belief

definition

SampleBelief(samples, weights) represents a distribution by weighted particles: the general-purpose fallback when no closed form is available, and the natural output of a sampler.

Sources: code: open_model.jl, messages.jl; Doucet, de Freitas & Gordon (eds.), Sequential Monte Carlo Methods in Practice (Springer 2001)

Theory (CT-ML wiki): Distribution Monad · Giry Monad

What it is

A finite, weighted sum of point masses, ${\sum }_i{w}_i{\delta }_{x_i}$ (equal weights when weights is nothing): an element of the finite distribution monad used as an approximation of a continuous distribution. It is an approximation by construction (isexact is false), and two different particle sets can represent the same distribution.

What it cannot do yet

• Pooling. combine of two sample beliefs needs the density of at least one of them (importance reweighting), and belief_logdensity is not defined for particles, so combine throws. A Dirac or the trivial belief still combines with it.
• Convergence checks. belief_distance returns Inf, so a schedule involving sample messages runs to its iteration limit rather than claiming convergence it cannot verify.
• A point. The diffusion factor’s assemble_state ignores sample beliefs, because a particle set has no single value to clamp to.

It is the representation a sampling inference would return (DPS, Langevin; Inference Signatures #6), the only one that can carry several branches of a multivalued relation at once.

Related: Beliefs, Dirac Belief, Gaussian Belief, Inference Signatures