Implementation of normalising flows and constrained random variable transformations
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September 2018


Stable Interface tests AD tests

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.

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 this can be implemented "automatically" using automatic differentiation, e.g. ForwardDiff.jl.


  1. link: maps a sample of a random distribution dist from its support to a value in ℝⁿ. Example:
julia> using Bijectors

julia> dist = Beta(2, 2)
Beta{Float64}=2.0, β=2.0)

julia> x = rand(dist)

julia> y = link(dist, x)
  1. invlink: the inverse of the link function. Example:
julia> z = invlink(dist, y)

julia> x  z
  1. logpdf_with_trans: finds log of the (transformed) probability density function of a distribution dist at a sample x. Example:
julia> using Bijectors

julia> dist = Dirichlet(2, 3)
Dirichlet{Float64}(alpha=[3.0, 3.0])

julia> x = rand(dist)
2-element Array{Float64,1}:

julia> logpdf_with_trans(dist, x, false) # ignoring the transformation

julia> logpdf_with_trans(dist, x, true) # considering the transformation

Bijector interface

A Bijector is a differentiable bijection with a differentiable inverse. That's basically it.

The primary application of Bijectors is the (very profitable) business of transforming (usually continuous) probability densities. If we transfrom a random variable x ~ p(x) to y = b(x) where b is a Bijector, we also get a canonical density q(y) = p(b⁻¹(y)) |det J(b⁻¹, y)| for y. Here J(b⁻¹, y) is the Jacobian of the inverse transform evaluated at y. q is also known as the push-forward of p by b in measure theory.

There's plenty of different reasons why one would want to do something like this. It can be because your p has non-zero probability (support) on a closed interval [a, b] and you want to use AD without having to worry about reaching the boundary. E.g. Beta has support [0, 1] so if we could transform p = Beta into a density q with support on ℝ, we could instead compute the derivative of logpdf(q, y) wrt. y, and then transform back x = b⁻¹(y). This is very useful for certain inference methods, e.g. Hamiltonian Monte-Carlo, where we need to take the derivative of the logpdf-computation wrt. input.

Another use-case is constructing a parameterized Bijector and consider transforming a "simple" density, e.g. MvNormal, to match a more complex density. One class of such bijectors is Normalizing Flows (NFs) which are compositions of differentiable and invertible neural networks, i.e. composition of a particular family of parameterized bijectors.[1] We'll see an example of this later on.

Basic usage

Other than the logpdf_with_trans methods, the package also provides a more composable interface through the Bijector types. Consider for example the one from above with Beta(2, 2).

julia> using Random; Random.seed!(42);

julia> using Bijectors; using Bijectors: Logit

julia> dist = Beta(2, 2)
Beta{Float64}=2.0, β=2.0)

julia> x = rand(dist)

julia> b = bijector(dist) # bijection (0, 1) → ℝ
Logit{Float64}(0.0, 1.0)

julia> y = b(x)

In this case we see that bijector(d::Distribution) returns the corresponding constrained-to-unconstrained bijection for Beta, which indeed is a Logit with a = 0.0 and b = 1.0. The resulting Logit <: Bijector has a method (b::Logit)(x) defined, allowing us to call it just like any other function. Comparing with the above example, b(x) ≈ link(dist, x). Just to convince ourselves:

julia> b(x)  link(dist, x)


What about invlink?

julia> b⁻¹ = inverse(b)
Inverse{Logit{Float64},0}(Logit{Float64}(0.0, 1.0))

julia> b⁻¹(y)

julia> b⁻¹(y)  invlink(dist, y)

Pretty neat, huh? Inverse{Logit} is also a Bijector where we've defined (ib::Inverse{<:Logit})(y) as the inverse transformation of (b::Logit)(x). Note that it's not always the case that inverse(b) isa Inverse, e.g. the inverse of Exp is simply Log so inverse(Exp()) isa Log is true.


Also, we can compose bijectors:

julia> id_y = (b  b⁻¹)
Composed{Tuple{Inverse{Logit{Float64},0},Logit{Float64}},0}((Inverse{Logit{Float64},0}(Logit{Float64}(0.0, 1.0)), Logit{Float64}(0.0, 1.0)))

julia> id_y(y)  y

And since Composed isa Bijector:

julia> id_x = inverse(id_y)
Composed{Tuple{Inverse{Logit{Float64},0},Logit{Float64}},0}((Inverse{Logit{Float64},0}(Logit{Float64}(0.0, 1.0)), Logit{Float64}(0.0, 1.0)))

julia> id_x(x)  x

logpdf of TransformedDistribution

Thus far, we've seen that we can replicate the functionality provided by link and invlink. To replicate logpdf_with_trans we instead provide a TransformedDistribution <: Distribution implementing the Distribution interface from Distributions.jl:

julia> using Bijectors: TransformedDistribution

julia> td = transformed(dist)
dist: Beta{Float64}=2.0, β=2.0)
transform: Logit{Float64}(0.0, 1.0)

julia> td isa UnivariateDistribution

julia> logpdf(td, y)

julia> logpdf_with_trans(dist, x, true)

When computing logpdf(td, y) where td is the transformed distribution corresponding to Beta(2, 2), it makes more semantic sense to compute the pdf of the transformed variable y rather than using the "un-transformed" variable x to do so, as we do in logpdf_with_trans. With that being said, we can also do

julia> logpdf_forward(td, x)

logabsdetjac and with_logabsdet_jacobian

In the computation of both logpdf and logpdf_forward we need to compute log(abs(det(jacobian(inverse(b), y)))) and log(abs(det(jacobian(b, x)))), respectively. This computation is available using the logabsdetjac method

julia> logabsdetjac(b⁻¹, y)

julia> logabsdetjac(b, x)

Notice that

julia> logabsdetjac(b, x)  -logabsdetjac(b⁻¹, y)

which is always the case for a differentiable bijection with differentiable inverse. Therefore if you want to compute logabsdetjac(b⁻¹, y) and we know that logabsdetjac(b, b⁻¹(y)) is actually more efficient, we'll return -logabsdetjac(b, b⁻¹(y)) instead. For some bijectors it might be easy to compute, say, the forward pass b(x), but expensive to compute b⁻¹(y). Because of this you might want to avoid doing anything "backwards", i.e. using b⁻¹. This is where with_logabsdet_jacobian comes to good use:

julia> with_logabsdet_jacobian(b, x)
(-0.5369949942509267, 1.4575353795716655)


julia> with_logabsdet_jacobian(inverse(b), y)
(0.3688868996596376, -1.4575353795716655)

In fact, the purpose of with_logabsdet_jacobian is to just do the right thing, not necessarily "forward". In this function we'll have access to both the original value x and the transformed value y, so we can compute logabsdetjac(b, x) in either direction. Furthermore, in a lot of cases we can re-use a lot of the computation from b(x) in the computation of logabsdetjac(b, x), or vice-versa. with_logabsdet_jacobian(b, x) will take advantage of such opportunities (if implemented).

Sampling from TransformedDistribution

At this point we've only shown that we can replicate the existing functionality. But we said TransformedDistribution isa Distribution, so we also have rand:

julia> y = rand(td)              # ∈ ℝ

julia> x = inverse(td.transform)(y)  # transform back to interval [0, 1]

This can be quite convenient if you have computations assuming input to be on the real line.

Univariate ADVI example

But the real utility of TransformedDistribution becomes more apparent when using transformed(dist, b) for any bijector b. To get the transformed distribution corresponding to the Beta(2, 2), we called transformed(dist) before. This is simply an alias for transformed(dist, bijector(dist)). Remember bijector(dist) returns the constrained-to-constrained bijector for that particular Distribution. But we can of course construct a TransformedDistribution using different bijectors with the same dist. This is particularly useful in something called Automatic Differentiation Variational Inference (ADVI).[2] An important part of ADVI is to approximate a constrained distribution, e.g. Beta, as follows:

  1. Sample x from a Normal with parameters μ and σ, i.e. x ~ Normal(μ, σ).
  2. Transform x to y s.t. y ∈ support(Beta), with the transform being a differentiable bijection with a differentiable inverse (a "bijector")

This then defines a probability density with same support as Beta! Of course, it's unlikely that it will be the same density, but it's an approximation. Creating such a distribution becomes trivial with Bijector and TransformedDistribution:

julia> dist = Beta(2, 2)
Beta{Float64}=2.0, β=2.0)

julia> b = bijector(dist)              # (0, 1) → ℝ
Logit{Float64}(0.0, 1.0)

julia> b⁻¹ = inverse(b)                    # ℝ → (0, 1)
Inverse{Logit{Float64},0}(Logit{Float64}(0.0, 1.0))

julia> td = transformed(Normal(), b⁻¹) # x ∼ 𝓝(0, 1) then b(x) ∈ (0, 1)
dist: Normal{Float64}=0.0, σ=1.0)
transform: Inverse{Logit{Float64},0}(Logit{Float64}(0.0, 1.0))

julia> x = rand(td)                    # ∈ (0, 1)

It's worth noting that support(Beta) is the closed interval [0, 1], while the constrained-to-unconstrained bijection, Logit in this case, is only well-defined as a map (0, 1) → ℝ for the open interval (0, 1). This is of course not an implementation detail. is itself open, thus no continuous bijection exists from a closed interval to . But since the boundaries of a closed interval has what's known as measure zero, this doesn't end up affecting the resulting density with support on the entire real line. In practice, this means that

td = transformed(Beta())


will never result in 0 or 1 though any sample arbitrarily close to either 0 or 1 is possible. Disclaimer: numerical accuracy is limited, so you might still see 0 and 1 if you're lucky.

Multivariate ADVI example

We can also do multivariate ADVI using the Stacked bijector. Stacked gives us a way to combine univariate and/or multivariate bijectors into a singe multivariate bijector. Say you have a vector x of length 2 and you want to transform the first entry using Exp and the second entry using Log. Stacked gives you an easy and efficient way of representing such a bijector.

julia> Random.seed!(42);

julia> using Bijectors: Exp, Log, SimplexBijector

julia> # Original distributions
       dists = (
           Dirichlet(2, 3)

julia> # Construct the corresponding ranges
       ranges = [];

julia> idx = 1;

julia> for i = 1:length(dists)
           d = dists[i]
           push!(ranges, idx:idx + length(d) - 1)

           global idx
           idx += length(d)

julia> ranges
3-element Array{Any,1}:

julia> # Base distribution; mean-field normal
       num_params = ranges[end][end]

julia> d = MvNormal(zeros(num_params), ones(num_params))
dim: 4
μ: [0.0, 0.0, 0.0, 0.0]
Σ: [1.0 0.0 0.0 0.0; 0.0 1.0 0.0 0.0; 0.0 0.0 1.0 0.0; 0.0 0.0 0.0 1.0]

julia> # Construct the transform
       bs = bijector.(dists)     # constrained-to-unconstrained bijectors for dists
(Logit{Float64}(0.0, 1.0), Log{0}(), SimplexBijector{true}())

julia> ibs = inverse.(bs)            # invert, so we get unconstrained-to-constrained
(Inverse{Logit{Float64},0}(Logit{Float64}(0.0, 1.0)), Exp{0}(), Inverse{SimplexBijector{true},1}(SimplexBijector{true}()))

julia> sb = Stacked(ibs, ranges) # => Stacked <: Bijector
Stacked{Tuple{Inverse{Logit{Float64},0},Exp{0},Inverse{SimplexBijector{true},1}},3}((Inverse{Logit{Float64},0}(Logit{Float64}(0.0, 1.0)), Exp{0}(), Inverse{SimplexBijector{true},1}(SimplexBijector{true}())), (1:1, 2:2, 3:4))

julia> # Mean-field normal with unconstrained-to-constrained stacked bijector
       td = transformed(d, sb);

julia> y = rand(td)
4-element Array{Float64,1}:

julia> 0.0  y[1]  1.0   # => true

julia> 0.0 < y[2]         # => true

julia> sum(y[3:4])  1.0  # => true

Normalizing flows

A very interesting application is that of normalizing flows.[1] Usually this is done by sampling from a multivariate normal distribution, and then transforming this to a target distribution using invertible neural networks. Currently there are two such transforms available in Bijectors.jl: PlanarLayer and RadialLayer. Let's create a flow with a single PlanarLayer:

julia> d = MvNormal(zeros(2), ones(2));

julia> b = PlanarLayer(2)
PlanarLayer{Array{Float64,2},Array{Float64,1}}([1.77786; -1.1449], [-0.468606; 0.156143], [-2.64199])

julia> flow = transformed(d, b)
dist: DiagNormal(
dim: 2
μ: [0.0, 0.0]
Σ: [1.0 0.0; 0.0 1.0]

transform: PlanarLayer{Array{Float64,2},Array{Float64,1}}([1.77786; -1.1449], [-0.468606; 0.156143], [-2.64199])

julia> flow isa MultivariateDistribution

That's it. Now we can sample from it using rand and compute the logpdf, like any other Distribution.

julia> y = rand(flow)
2-element Array{Float64,1}:

julia> logpdf(flow, y)         # uses inverse of `b`

julia> x = rand(flow.dist)
2-element Array{Float64,1}:

julia> logpdf_forward(flow, x) # more efficent and accurate

Similarily to the multivariate ADVI example, we could use Stacked to get a bounded flow:

julia> d = MvNormal(zeros(2), ones(2));

julia> ibs = inverse.(bijector.((InverseGamma(2, 3), Beta())));

julia> sb = stack(ibs...) # == Stacked(ibs) == Stacked(ibs, [i:i for i = 1:length(ibs)]
Stacked{Tuple{Exp{0},Inverse{Logit{Float64},0}},2}((Exp{0}(), Inverse{Logit{Float64},0}(Logit{Float64}(0.0, 1.0))), (1:1, 2:2))

julia> b = sb  PlanarLayer(2)
Composed{Tuple{PlanarLayer{Array{Float64,2},Array{Float64,1}},Stacked{Tuple{Exp{0},Inverse{Logit{Float64},0}},2}},1}((PlanarLayer{Array{Float64,2},Array{Float64,1}}([1.49138; 0.367563], [-0.886205; 0.684565], [-1.59058]), Stacked{Tuple{Exp{0},Inverse{Logit{Float64},0}},2}((Exp{0}(), Inverse{Logit{Float64},0}(Logit{Float64}(0.0, 1.0))), (1:1, 2:2))))

julia> td = transformed(d, b);

julia> y = rand(td)
2-element Array{Float64,1}:

julia> 0 < y[1]

julia> 0  y[2]  1

Want to fit the flow?

julia> using Tracker

julia> b = PlanarLayer(2, param)                  # construct parameters using `param`
PlanarLayer{TrackedArray{…,Array{Float64,2}},TrackedArray{…,Array{Float64,1}}}([-1.05099; 0.502079] (tracked), [-0.216248; -0.706424] (tracked), [-4.33747] (tracked))

julia> flow = transformed(d, b)
dist: DiagNormal(
dim: 2
μ: [0.0, 0.0]
Σ: [1.0 0.0; 0.0 1.0]

transform: PlanarLayer{TrackedArray{…,Array{Float64,2}},TrackedArray{…,Array{Float64,1}}}([-1.05099; 0.502079] (tracked), [-0.216248; -0.706424] (tracked), [-4.33747] (tracked))

julia> rand(flow)
Tracked 2-element Array{Float64,1}:

julia> x = rand(flow.dist)
2-element Array{Float64,1}:

julia> Tracker.back!(logpdf_forward(flow, x), 1.0) # backprob

julia> Tracker.grad(b.w)
2×1 Array{Float64,2}:

We can easily create more complex flows by simply doing PlanarLayer(10) ∘ PlanarLayer(10) ∘ RadialLayer(10) and so on.

In those cases, it might be useful to use Flux.jl's Flux.params to extract the parameters:

julia> using Flux

julia> Flux.params(flow)
Params([[-1.05099; 0.502079] (tracked), [-0.216248; -0.706424] (tracked), [-4.33747] (tracked)])

Another useful function is the forward(d::Distribution) method. It is similar to with_logabsdet_jacobian(b::Bijector, x) in the sense that it does a forward pass of the entire process "sample then transform" and returns all the most useful quantities in process using the most efficent computation path.

julia> x, y, logjac, logpdf_y = forward(flow) # sample + transform and returns all the useful quantities in one pass
(x = [-0.839739, 0.169613], y = [-0.810354, 0.963392] (tracked), logabsdetjac = -0.0017416108706436628 (tracked), logpdf = -2.203100286792651 (tracked))

This method is for example useful when computing quantities such as the expected lower bound (ELBO) between this transformed distribution and some other joint density. If no analytical expression is available, we have to approximate the ELBO by a Monte Carlo estimate. But one term in the ELBO is the entropy of the base density, which we do know analytically in this case. Using the analytical expression for the entropy and then using a monte carlo estimate for the rest of the terms in the ELBO gives an estimate with lower variance than if we used the monte carlo estimate for the entire expectation.


  1. Rezende, D. J., & Mohamed, S. (2015). Variational Inference With Normalizing Flows. arXiv:1505.05770.
  2. Kucukelbir, A., Tran, D., Ranganath, R., Gelman, A., & Blei, D. M. (2016). Automatic Differentiation Variational Inference. arXiv:1603.00788.