A very simple example of a "bijector"/diffeomorphism, i.e. a differentiable transformation with a differentiable inverse, is the exp function:

  • The inverse of exp is log.
  • The derivative of exp at an input x is simply exp(x), hence logabsdetjac is simply x.
julia> using Bijectors
julia> transform(exp, 1.0)2.718281828459045
julia> logabsdetjac(exp, 1.0)1.0
julia> with_logabsdet_jacobian(exp, 1.0)(2.718281828459045, 1.0)

Some transformations are well-defined for different types of inputs, e.g. exp can also act elementwise on an N-dimensional Array{<:Real,N}. To specify that a transformation should act elementwise, we use the elementwise method:

julia> x = ones(2, 2)2×2 Matrix{Float64}:
 1.0  1.0
 1.0  1.0
julia> transform(elementwise(exp), x)2×2 Matrix{Float64}: 2.71828 2.71828 2.71828 2.71828
julia> logabsdetjac(elementwise(exp), x)4.0
julia> with_logabsdet_jacobian(elementwise(exp), x)([2.718281828459045 2.718281828459045; 2.718281828459045 2.718281828459045], 4.0)

These methods also work nicely for compositions of transformations:

julia> transform(elementwise(log ∘ exp), x)2×2 Matrix{Float64}:
 1.0  1.0
 1.0  1.0

Unlike exp, some transformations have parameters affecting the resulting transformation they represent, e.g. Logit has two parameters a and b representing the lower- and upper-bound, respectively, of its domain:

julia> using Bijectors: Logit
julia> f = Logit(0.0, 1.0)Bijectors.Logit{Float64, Float64}(0.0, 1.0)
julia> f(rand()) # takes us from `(0, 1)` to `(-∞, ∞)`0.9098920879818054

User-facing methods

Without mutation:

transform(b, x)

Transform x using b, treating x as a single input.

logabsdetjac(b, x)

Return log(abs(det(J(b, x)))), where J(b, x) is the jacobian of b at x.


With mutation:

transform!(b, x[, y])

Transform x using b, storing the result in y.

If y is not provided, x is used as the output.

logabsdetjac!(b, x[, logjac])

Compute log(abs(det(J(b, x)))) and store the result in logjac, where J(b, x) is the jacobian of b at x.

with_logabsdet_jacobian!(b, x[, y, logjac])

Compute transform(b, x) and logabsdetjac(b, x), storing the result in y and logjac, respetively.

If y is not provided, then x will be used in its place.

Defaults to calling with_logabsdet_jacobian(b, x) and updating y and logjac with the result.

Implementing a transformation

Any callable can be made into a bijector by providing an implementation of ChangeOfVariables.with_logabsdet_jacobian(b, x).

You can also optionally implement transform and logabsdetjac to avoid redundant computations. This is usually only worth it if you expect transform or logabsdetjac to be used heavily without the other.

Similarly with the mutable versions with_logabsdet_jacobian!, transform!, and logabsdetjac!.

Working with Distributions.jl


Returns the constrained-to-unconstrained bijector for distribution d.

transformed(d::Distribution, b::Bijector)

Couples distribution d with the bijector b by returning a TransformedDistribution.

If no bijector is provided, i.e. transformed(d) is called, then transformed(d, bijector(d)) is returned.



Alias for Base.Fix1(broadcast, f).

In the case where f::ComposedFunction, the result is Base.Fix1(broadcast, f.outer) ∘ Base.Fix1(broadcast, f.inner) rather than Base.Fix1(broadcast, f).


Returns true or false depending on whether or not evaluation of b has a closed-form implementation.

Most transformations have closed-form evaluations, but there are cases where this is not the case. For example the inverse evaluation of PlanarLayer requires an iterative procedure to evaluate.



Abstract type for a transformation.


A subtype of Transform of should at least implement transform(b, x).

If the Transform is also invertible:

  • Required:
    • Either of the following:
      • transform(::Inverse{<:MyTransform}, x): the transform for its inverse.
      • InverseFunctions.inverse(b::MyTransform): returns an existing Transform.
    • logabsdetjac: computes the log-abs-det jacobian factor.
  • Optional:
    • with_logabsdet_jacobian: transform and logabsdetjac combined. Useful in cases where we can exploit shared computation in the two.

For the above methods, there are mutating versions which can optionally be implemented:


Abstract type of a bijector, i.e. differentiable bijection with differentiable inverse.


A Transform representing the inverse transform of b.


CorrBijector <: Bijector

A bijector implementation of Stan's parametrization method for Correlation matrix:

Basically, a unconstrained strictly upper triangular matrix y is transformed to a correlation matrix by following readable but not that efficient form:

K = size(y, 1)
z = tanh.(y)

for j=1:K, i=1:K
    if i>j
        w[i,j] = 0
    elseif 1==i==j
        w[i,j] = 1
    elseif 1<i==j
        w[i,j] = prod(sqrt(1 .- z[1:i-1, j].^2))
    elseif 1==i<j
        w[i,j] = z[i,j]
    elseif 1<i<j
        w[i,j] = z[i,j] * prod(sqrt(1 .- z[1:i-1, j].^2))

It is easy to see that every column is a unit vector, for example:

w3' w3 ==
w[1,3]^2 + w[2,3]^2 + w[3,3]^2 ==
z[1,3]^2 + (z[2,3] * sqrt(1 - z[1,3]^2))^2 + (sqrt(1-z[1,3]^2) * sqrt(1-z[2,3]^2))^2 ==
z[1,3]^2 + z[2,3]^2 * (1-z[1,3]^2) + (1-z[1,3]^2) * (1-z[2,3]^2) ==
z[1,3]^2 + z[2,3]^2 - z[2,3]^2 * z[1,3]^2 + 1 -z[1,3]^2 - z[2,3]^2 + z[1,3]^2 * z[2,3]^2 ==

And diagonal elements are positive, so w is a cholesky factor for a positive matrix.

x = w' * w

Consider block matrix representation for x

x = [w1'; w2'; ... wn'] * [w1 w2 ... wn] == 
[w1'w1 w1'w2 ... w1'wn;
 w2'w1 w2'w2 ... w2'wn;

The diagonal elements are given by wk'wk = 1, thus x is a correlation matrix.

Every step is invertible, so this is a bijection(bijector).

Note: The implementation doesn't follow their "manageable expression" directly, because their equation seems wrong (7/30/2020). Insteadly it follows definition above the "manageable expression" directly, which is also described in above doc.

LeakyReLU{T}(α::T) <: Bijector

Defines the invertible mapping

x ↦ x if x ≥ 0 else αx

where α > 0.

Stacked(bs, ranges)

A Bijector which stacks bijectors together which can then be applied to a vector where bs[i]::Bijector is applied to x[ranges[i]]::UnitRange{Int}.


  • bs can be either a Tuple or an AbstractArray of 0- and/or 1-dimensional bijectors
    • If bs is a Tuple, implementations are type-stable using generated functions
    • If bs is an AbstractArray, implementations are not type-stable and use iterative methods
  • ranges needs to be an iterable consisting of UnitRange{Int}
    • length(bs) == length(ranges) needs to be true.


b1 = Logit(0.0, 1.0)
b2 = identity
b = stack(b1, b2)
b([0.0, 1.0]) == [b1(0.0), 1.0]  # => true
RationalQuadraticSpline{T} <: Bijector

Implementation of the Rational Quadratic Spline flow [1].

  • Outside of the interval [minimum(widths), maximum(widths)], this mapping is given by the identity map.
  • Inside the interval it's given by a monotonic spline (i.e. monotonic polynomials connected at intermediate points) with endpoints fixed so as to continuously transform into the identity map.

For the sake of efficiency, there are separate implementations for 0-dimensional and 1-dimensional inputs.


There are two constructors for RationalQuadraticSpline:

  • RationalQuadraticSpline(widths, heights, derivatives): it is assumed that widths,

heights, and derivatives satisfy the constraints that makes this a valid bijector, i.e.

  • widths: monotonically increasing and length(widths) == K,
  • heights: monotonically increasing and length(heights) == K,
  • derivatives: non-negative and derivatives[1] == derivatives[end] == 1.
  • RationalQuadraticSpline(widths, heights, derivatives, B): other than than the lengths, no assumptions are made on parameters. Therefore we will transform the parameters s.t.:
  • widths_new ∈ [-B, B]ᴷ⁺¹, where K == length(widths),
  • heights_new ∈ [-B, B]ᴷ⁺¹, where K == length(heights),
  • derivatives_new ∈ (0, ∞)ᴷ⁺¹ with derivatives_new[1] == derivates_new[end] == 1, where (K - 1) == length(derivatives).



julia> using StableRNGs: StableRNG; rng = StableRNG(42);  # For reproducibility.

julia> using Bijectors: RationalQuadraticSpline

julia> K = 3; B = 2;

julia> # Monotonic spline on '[-B, B]' with `K` intermediate knots/"connection points".
       b = RationalQuadraticSpline(randn(rng, K), randn(rng, K), randn(rng, K - 1), B);

julia> b(0.5) # inside of `[-B, B]` → transformed

julia> b(5.) # outside of `[-B, B]` → not transformed

julia> b = RationalQuadraticSpline(b.widths, b.heights, b.derivatives);

julia> b(0.5) # inside of `[-B, B]` → transformed

julia> d = 2; K = 3; B = 2;

julia> b = RationalQuadraticSpline(randn(rng, d, K), randn(rng, d, K), randn(rng, d, K - 1), B);

julia> b([-1., 1.])
2-element Vector{Float64}:

julia> b([-5., 5.])
2-element Vector{Float64}:

julia> b([-1., 5.])
2-element Vector{Float64}:


[1] Durkan, C., Bekasov, A., Murray, I., & Papamakarios, G., Neural Spline Flows, CoRR, arXiv:1906.04032 [stat.ML], (2019).

Coupling{F, M}(θ::F, mask::M)

Implements a coupling-layer as defined in [1].


julia> using Bijectors: Shift, Coupling, PartitionMask, coupling, couple

julia> m = PartitionMask(3, [1], [2]); # <= going to use x[2] to parameterize transform of x[1]

julia> cl = Coupling(Shift, m); # <= will do `y[1:1] = x[1:1] + x[2:2]`;

julia> x = [1., 2., 3.];

julia> cl(x)
3-element Vector{Float64}:

julia> inverse(cl)(cl(x))
3-element Vector{Float64}:

julia> coupling(cl) # get the `Bijector` map `θ -> b(⋅, θ)`

julia> couple(cl, x) # get the `Bijector` resulting from `x`

julia> with_logabsdet_jacobian(cl, x)
([3.0, 2.0, 3.0], 0.0)


[1] Kobyzev, I., Prince, S., & Brubaker, M. A., Normalizing flows: introduction and ideas, CoRR, (), (2019).


A bijector mapping ordered vectors in ℝᵈ to unordered vectors in ℝᵈ.

See also

  • Stan's documentation
    • Note that this transformation and its inverse are the opposite of in this reference.
NamedTransform <: AbstractNamedTransform

Wraps a NamedTuple of key -> Bijector pairs, implementing evaluation, inversion, etc.


julia> using Bijectors: NamedTransform, Scale

julia> b = NamedTransform((a = Scale(2.0), b = exp));

julia> x = (a = 1., b = 0., c = 42.);

julia> b(x)
(a = 2.0, b = 1.0, c = 42.0)

julia> (a = 2 * x.a, b = exp(x.b), c = x.c)
(a = 2.0, b = 1.0, c = 42.0)
NamedCoupling{target, deps, F} <: AbstractNamedTransform

Implements a coupling layer for named bijectors.

See also: Coupling


julia> using Bijectors: NamedCoupling, Scale

julia> b = NamedCoupling(:b, (:a, :c), (a, c) -> Scale(a + c));

julia> x = (a = 1., b = 2., c = 3.);

julia> b(x)
(a = 1.0, b = 8.0, c = 3.0)

julia> (a = x.a, b = (x.a + x.c) * x.b, c = x.c)
(a = 1.0, b = 8.0, c = 3.0)