diff --git a/Project.toml b/Project.toml
index f069172..47459d2 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,7 +1,7 @@
 name = "GPLikelihoods"
 uuid = "6031954c-0455-49d7-b3b9-3e1c99afaf40"
 authors = ["JuliaGaussianProcesses Team"]
-version = "0.2.7"
+version = "0.3.0"
 
 [deps]
 Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f"
diff --git a/src/GPLikelihoods.jl b/src/GPLikelihoods.jl
index 1640577..34890ca 100644
--- a/src/GPLikelihoods.jl
+++ b/src/GPLikelihoods.jl
@@ -16,6 +16,7 @@ export BernoulliLikelihood,
     GammaLikelihood
 export Link,
     ChainLink,
+    BijectiveSimplexLink,
     ExpLink,
     LogLink,
     InvLink,
diff --git a/src/likelihoods/categorical.jl b/src/likelihoods/categorical.jl
index 7126ec1..10c5c36 100644
--- a/src/likelihoods/categorical.jl
+++ b/src/likelihoods/categorical.jl
@@ -1,22 +1,41 @@
 """
-    CategoricalLikelihood(l=softmax)
+    CategoricalLikelihood(l=BijectiveSimplexLink(softmax))
 
 Categorical likelihood is to be used if we assume that the 
-uncertainity associated with the data follows a Categorical distribution.
+uncertainty associated with the data follows a [Categorical distribution](https://en.wikipedia.org/wiki/Categorical_distribution).
+
+Assuming a distribution with `n` categories:
+
+## `n-1` inputs (bijective link)
+
+One can work with a bijective transformation by wrapping a link (like `softmax`)
+into a [`BijectiveSimplexLink`](@ref) and only needs `n-1` inputs:
 ```math
     p(y|f_1, f_2, \\dots, f_{n-1}) = \\operatorname{Categorical}(y | l(f_1, f_2, \\dots, f_{n-1}, 0))
 ```
-Given an `AbstractVector` ``[f_1, f_2, ..., f_{n-1}]``, returns a `Categorical` distribution,
-with probabilities given by ``l(f_1, f_2, ..., f_{n-1}, 0)``.
+The default constructor is a bijective link around `softmax`.
+
+## `n` inputs (non-bijective link)
+
+One can also pass directly the inputs without concatenating a `0`:
+```math
+    p(y|f_1, f_2, \\dots, f_n) = \\operatorname{Categorical}(y | l(f_1, f_2, \\dots, f_n))
+```
+This variant is over-parametrized, as there are `n-1` independent parameters 
+embedded in a `n` dimensional parameter space.
+For more details, see the end of the section of this [Wikipedia link](https://en.wikipedia.org/wiki/Exponential_family#Table_of_distributions)
+where it corresponds to Variant 1 and 2.
 """
 struct CategoricalLikelihood{Tl<:AbstractLink} <: AbstractLikelihood
     invlink::Tl
 end
 
-CategoricalLikelihood(l=softmax) = CategoricalLikelihood(link(l))
+CategoricalLikelihood(l=BijectiveSimplexLink(softmax)) = CategoricalLikelihood(link(l))
 
-(l::CategoricalLikelihood)(f::AbstractVector{<:Real}) = Categorical(l.invlink(vcat(f, 0)))
+function (l::CategoricalLikelihood)(f::AbstractVector{<:Real})
+    return Categorical(l.invlink(f))
+end
 
 function (l::CategoricalLikelihood)(fs::AbstractVector)
-    return Product(Categorical.(l.invlink.(vcat.(fs, 0))))
+    return Product(Categorical.(l.invlink.(fs)))
 end
diff --git a/src/links.jl b/src/links.jl
index 1e86a99..e7e66c9 100644
--- a/src/links.jl
+++ b/src/links.jl
@@ -30,6 +30,23 @@ link(l::AbstractLink) = l
 
 Base.inv(l::Link) = Link(InverseFunctions.inverse(l.f))
 
+"""
+    BijectiveSimplexLink(link)
+
+Wrapper to preprocess the inputs by adding a `0` at the end before passing it to 
+the link `link`.
+This is a necessary step to work with simplices.
+For example with the [`SoftMaxLink`](@ref), to obtain a `n`-simplex leading to
+`n+1` categories for the [`CategoricalLikelihood`](@ref),
+one needs to pass `n+1` latent GP.
+However, by wrapping the link into a `BijectiveSimplexLink`, only `n` latent are needed. 
+"""
+struct BijectiveSimplexLink{L} <: AbstractLink
+    link::L
+end
+
+(l::BijectiveSimplexLink)(f::AbstractVector{<:Real}) = l.link(vcat(f, 0))
+
 # alias
 const LogLink = Link{typeof(log)}
 const ExpLink = Link{typeof(exp)}
diff --git a/test/likelihoods/categorical.jl b/test/likelihoods/categorical.jl
index b129a24..8ed48be 100644
--- a/test/likelihoods/categorical.jl
+++ b/test/likelihoods/categorical.jl
@@ -1,9 +1,13 @@
 @testset "CategoricalLikelihood" begin
-    for args in ((), (softmax,), (SoftMaxLink(),))
-        @test CategoricalLikelihood(args...) isa CategoricalLikelihood{SoftMaxLink}
-    end
+    @test CategoricalLikelihood() isa
+        CategoricalLikelihood{<:GPLikelihoods.BijectiveSimplexLink}
+
+    @test CategoricalLikelihood(softmax) isa CategoricalLikelihood{SoftMaxLink}
+    @test CategoricalLikelihood(SoftMaxLink()) isa CategoricalLikelihood{SoftMaxLink}
 
-    lik = CategoricalLikelihood()
     OUT_DIM = 4
-    test_interface(lik, Categorical, OUT_DIM)
+    lik_bijective = CategoricalLikelihood()
+    test_interface(lik_bijective, Categorical, OUT_DIM)
+    lik_nonbijective = CategoricalLikelihood(softmax)
+    test_interface(lik_nonbijective, Categorical, OUT_DIM)
 end
diff --git a/test/links.jl b/test/links.jl
index b527a39..67c1af3 100644
--- a/test/links.jl
+++ b/test/links.jl
@@ -10,6 +10,11 @@
     l = GPLikelihoods.link(ExpLink())
     @test l == ExpLink()
 
+    ## SimplexBijective link
+    l = SoftMaxLink()
+    sbl = BijectiveSimplexLink(l)
+    @test sbl(xs) == l(vcat(xs, 0))
+
     # Log
     l = LogLink()
     @test l(x) == log(x)