This package implements a set of functions for transforming constrained random variables (e.g. simplexes, intervals) to Euclidean space. The 3 main functions implemented in this package are the link, invlink and logpdf_with_trans for a number of distributions. The distributions supported are:

  1. RealDistribution: Union{Cauchy, Gumbel, Laplace, Logistic, NoncentralT, Normal, NormalCanon, TDist},
  2. PositiveDistribution: Union{BetaPrime, Chi, Chisq, Erlang, Exponential, FDist, Frechet, Gamma, InverseGamma, InverseGaussian, Kolmogorov, LogNormal, NoncentralChisq, NoncentralF, Rayleigh, Weibull},
  3. UnitDistribution: Union{Beta, KSOneSided, NoncentralBeta},
  4. SimplexDistribution: Union{Dirichlet},
  5. PDMatDistribution: Union{InverseWishart, Wishart}, and
  6. TransformDistribution: Union{T, Truncated{T}} where T<:ContinuousUnivariateDistribution.

All exported names from the Distributions.jl package are reexported from Bijectors.

Bijectors.jl also provides a nice interface for working with these maps: composition, inversion, etc. The following table lists mathematical operations for a bijector and the corresponding code in Bijectors.jl.

b ↦ b⁻¹inverse(b)
(b₁, b₂) ↦ (b₁ ∘ b₂)b₁ ∘ b₂
(b₁, b₂) ↦ [b₁, b₂]stack(b₁, b₂)
x ↦ b(x)b(x)×
y ↦ b⁻¹(y)inverse(b)(y)×
x ↦ log|det J(b, x)|logabsdetjac(b, x)AD
x ↦ b(x), log|det J(b, x)|with_logabsdet_jacobian(b, x)
p ↦ q := b_* pq = transformed(p, b)
y ∼ qy = rand(q)
p ↦ b such that support(b_* p) = ℝᵈbijector(p)
(x ∼ p, b(x), log|det J(b, x)|, log q(y))forward(q)

In this table, b denotes a Bijector, J(b, x) denotes the Jacobian of b evaluated at x, b_* denotes the push-forward of p by b, and x ∼ p denotes x sampled from the distribution with density p.

The "Automatic" column in the table refers to whether or not you are required to implement the feature for a custom Bijector. "AD" refers to the fact that it can be implemented "automatically" using automatic differentiation.