Bijectors.jl

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.

Bijectors.link — Function

link(d::Distribution, x)

Transforms the input x using the constrained-to-unconstrained bijector for distribution d.

See also: link.

Example

julia> using Bijectors

julia> d = LogNormal()    # support is (0, Inf)
LogNormal{Float64}(μ=0.0, σ=1.0)

julia> link(LogNormal(), 1.0)   # uses a log transform
0.0

julia> invlink(LogNormal(), 0.0)
1.0

source

Bijectors.logpdf_with_trans — Function

logpdf_with_trans(d::Distribution, x, transform::Bool)

If transform is false, logpdf_with_trans calculates the log probability density function (logpdf) of distribution d at x.

If transform is true, x is transformed using the constrained-to-unconstrained bijector for distribution d, and then the logpdf of the resulting value is calculated with respect to the unconstrained (transformed) distribution. Equivalently, if x is distributed according to d and y = link(d, x) is distributed according to td = transformed(d), then logpdf_with_trans(d, x, true) = logpdf(td, y). This is accomplished by subtracting the log Jacobian of the transformation.

Example

julia> using Bijectors

julia> logpdf_with_trans(LogNormal(), ℯ, false)
-2.4189385332046727

julia> logpdf(LogNormal(), ℯ)  # Same as above
-2.4189385332046727

julia> logpdf_with_trans(LogNormal(), ℯ, true)
-1.4189385332046727

julia> # If x ~ LogNormal(), then log(x) ~ Normal()
       logpdf(Normal(), 1.0)   
-1.4189385332046727

julia> # The difference between the two is due to the Jacobian
       logabsdetjac(bijector(LogNormal()), ℯ)
-1

source

The distributions supported are:

RealDistribution: Union{Cauchy, Gumbel, Laplace, Logistic, NoncentralT, Normal, NormalCanon, TDist},
PositiveDistribution: Union{BetaPrime, Chi, Chisq, Erlang, Exponential, FDist, Frechet, Gamma, InverseGamma, InverseGaussian, Kolmogorov, LogNormal, NoncentralChisq, NoncentralF, Rayleigh, Weibull},
UnitDistribution: Union{Beta, KSOneSided, NoncentralBeta},
SimplexDistribution: Union{Dirichlet},
PDMatDistribution: Union{InverseWishart, Wishart}, and
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.

Operation	Method	Automatic
`b ↦ b⁻¹`	`inverse(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_* p`	`q = transformed(p, b)`	✓
`y ∼ q`	`y = rand(q)`	✓
`p ↦ b` such that `support(b_* p) = ℝᵈ`	`bijector(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.