diff --git a/docs/Project.toml b/docs/Project.toml
index 3d6a9ee4e..fbb4973b6 100644
--- a/docs/Project.toml
+++ b/docs/Project.toml
@@ -1,4 +1,5 @@
 [deps]
+Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f"
 Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
 GR = "28b8d3ca-fb5f-59d9-8090-bfdbd6d07a71"
 
diff --git a/docs/src/fit.md b/docs/src/fit.md
index 24ddbad5c..b04341667 100644
--- a/docs/src/fit.md
+++ b/docs/src/fit.md
@@ -74,6 +74,23 @@ fit_mle(Categorical, x)        # equivalent to fit_mle(Categorical, max(x), x)
 fit_mle(Categorical, x, w)
 ```
 
+It is also possible to directly input a distribution `fit_mle(d::Distribution, x[, w])`. This form avoids the extra arguments:
+
+```julia
+fit_mle(Binomial(n, 0.1), x) 
+# equivalent to fit_mle(Binomial, ntrials(Binomial(n, 0.1)), x), here the parameter 0.1 is not used
+
+fit_mle(Categorical(p), x) 
+# equivalent to fit_mle(Categorical, ncategories(Categorical(p)), x), here the only the length of p is used not its values
+
+d = product_distribution([Exponential(0.5), Normal(11.3, 3.2)])
+fit_mle(d, x) 
+# equivalent to product_distribution([fit_mle(Exponential, x[1,:]), fit_mle(Normal, x[2, :])]). Again parameters of d are not used.
+```
+
+Note that for standard distributions, the values of the distribution parameters `d` are not used in `fit_mle` only the “structure” of `d` is passed into `fit_mle`.
+However, for complex Maximum Likelihood estimation requiring optimization, e.g., EM algorithm, one could use `D` as an initial guess.
+
 ## Sufficient Statistics
 
 For many distributions, the estimation can be based on (sum of) sufficient statistics computed from a dataset. To simplify implementation, for such distributions, we implement `suffstats` method instead of `fit_mle` directly:
diff --git a/docs/src/mixture.md b/docs/src/mixture.md
index e6b24b103..2ae7ab904 100644
--- a/docs/src/mixture.md
+++ b/docs/src/mixture.md
@@ -106,4 +106,4 @@ rand!(::AbstractMixtureModel, ::AbstractArray)
 ## Estimation
 
 There are several methods for the estimation of mixture models from data, and this problem remains an open research topic.
-This package does not provide facilities for estimating mixture models. One can resort to other packages, *e.g.* [*GaussianMixtures.jl*](https://github.com/davidavdav/GaussianMixtures.jl), for this purpose.
+This package does not provide facilities for estimating mixture models. One can resort to other packages, *e.g.* [*GaussianMixtures.jl*](https://github.com/davidavdav/GaussianMixtures.jl) or [*ExpectationMaximization.jl*](https://github.com/dmetivie/ExpectationMaximization.jl), for this purpose.
diff --git a/src/genericfit.jl b/src/genericfit.jl
index 2b4b5e67e..f5cb7c72b 100644
--- a/src/genericfit.jl
+++ b/src/genericfit.jl
@@ -30,5 +30,7 @@ fit_mle(dt::Type{D}, x::AbstractArray, w::AbstractArray) where {D<:UnivariateDis
 fit_mle(dt::Type{D}, x::AbstractMatrix) where {D<:MultivariateDistribution} = fit_mle(D, suffstats(D, x))
 fit_mle(dt::Type{D}, x::AbstractMatrix, w::AbstractArray) where {D<:MultivariateDistribution} = fit_mle(D, suffstats(D, x, w))
 
+fit_mle(dist::Distribution, args...) = fit_mle(typeof(dist), args...)
+
 fit(dt::Type{D}, x) where {D<:Distribution} = fit_mle(D, x)
 fit(dt::Type{D}, args...) where {D<:Distribution} = fit_mle(D, args...)
diff --git a/src/multivariate/product.jl b/src/multivariate/product.jl
index 3bb0f0d9e..32366fed3 100644
--- a/src/multivariate/product.jl
+++ b/src/multivariate/product.jl
@@ -36,6 +36,8 @@ function Base.eltype(::Type{<:Product{S,T}}) where {S<:ValueSupport,
     return eltype(T)
 end
 
+params(g::Product) = params.(g.v)
+
 _rand!(rng::AbstractRNG, d::Product, x::AbstractVector{<:Real}) =
     map!(Base.Fix1(rand, rng), x, d.v)
 function _logpdf(d::Product, x::AbstractVector{<:Real})
@@ -60,3 +62,17 @@ maximum(d::Product) = map(maximum, d.v)
 function product_distribution(dists::V) where {S<:ValueSupport,T<:UnivariateDistribution{S},V<:AbstractVector{T}}
     return Product{S,T,V}(dists)
 end
+
+#### Fitting
+
+"""
+    fit_mle(g::Product, x::AbstractMatrix)
+    fit_mle(g::Product, x::AbstractMatrix, γ::AbstractVector)
+
+The `fit_mle` for a multivariate Product distributions `g` is the `product_distribution` of `fit_mle` of each components of `g`.
+"""
+function fit_mle(g::Product, x::AbstractMatrix, args...)
+    d = size(x, 1)
+    length(g) == d || throw(DimensionMismatch("The dimensions of g and x are inconsistent."))
+    return product_distribution([fit_mle(g.v[s], y, args...) for (s, y) in enumerate(eachrow(x))])
+end
diff --git a/src/product.jl b/src/product.jl
index 7a4904ae7..e862abc32 100644
--- a/src/product.jl
+++ b/src/product.jl
@@ -74,6 +74,8 @@ end
 
 size(d::ProductDistribution) = d.size
 
+params(d::ArrayOfUnivariateDistribution) = params.(d.dists)
+
 mean(d::ProductDistribution) = reshape(mapreduce(vec ∘ mean, vcat, d.dists), size(d))
 var(d::ProductDistribution) = reshape(mapreduce(vec ∘ var, vcat, d.dists), size(d))
 cov(d::ProductDistribution) = Diagonal(vec(var(d)))
@@ -244,3 +246,22 @@ function product_distribution(dists::AbstractVector{<:Normal})
     σ2 = map(var, dists)
     return MvNormal(µ, Diagonal(σ2))
 end
+
+#### Fitting
+promote_sample(::Type{dT}, x::AbstractArray{T}) where {T<:Real, dT<:Real} = T <: dT ? x : convert.(dT, x)
+
+"""
+    fit_mle(dists::ArrayOfUnivariateDistribution, x::AbstractArray)
+    fit_mle(dists::ArrayOfUnivariateDistribution, x::AbstractArray, γ::AbstractVector)
+
+The `fit_mle` for a `ArrayOfUnivariateDistribution` distributions `dists` is the `product_distribution` of `fit_mle` of each components of `dists`.
+"""
+function fit_mle(dists::VectorOfUnivariateDistribution, x::AbstractMatrix{<:Real}, args...)
+    length(dists) == size(x, 1) || throw(DimensionMismatch("The dimensions of dists and x are inconsistent."))
+    return product_distribution([fit_mle(d, promote_sample(eltype(d), x[s, :]), args...) for (s, d) in enumerate(dists.dists)])
+end
+
+function fit_mle(dists::ArrayOfUnivariateDistribution, x::AbstractArray, args...)
+    size(dists) == size(first(x)) || throw(DimensionMismatch("The dimensions of dists and x are inconsistent."))
+    return product_distribution([fit_mle(d, promote_sample(eltype(d), [x[i][s] for i in eachindex(x)]), args...) for (s, d) in enumerate(dists.dists)])
+end
\ No newline at end of file
diff --git a/src/univariate/discrete/binomial.jl b/src/univariate/discrete/binomial.jl
index f4102cbb8..3aa24545c 100644
--- a/src/univariate/discrete/binomial.jl
+++ b/src/univariate/discrete/binomial.jl
@@ -196,6 +196,7 @@ suffstats(::Type{T}, data::BinomData, w::AbstractArray{<:Real}) where {T<:Binomi
 
 fit_mle(::Type{T}, ss::BinomialStats) where {T<:Binomial} = T(ss.n, ss.ns / (ss.ne * ss.n))
 
+fit_mle(d::T, x::AbstractArray{<:Integer}) where {T<:Binomial} = fit_mle(T, suffstats(T, ntrials(d), x))
 fit_mle(::Type{T}, n::Integer, x::AbstractArray{<:Integer}) where {T<:Binomial}= fit_mle(T, suffstats(T, n, x))
 fit_mle(::Type{T}, n::Integer, x::AbstractArray{<:Integer}, w::AbstractArray{<:Real}) where {T<:Binomial} = fit_mle(T, suffstats(T, n, x, w))
 fit_mle(::Type{T}, data::BinomData) where {T<:Binomial} = fit_mle(T, suffstats(T, data))
diff --git a/src/univariate/discrete/categorical.jl b/src/univariate/discrete/categorical.jl
index 1dc08960c..a49810d45 100644
--- a/src/univariate/discrete/categorical.jl
+++ b/src/univariate/discrete/categorical.jl
@@ -156,6 +156,7 @@ function fit_mle(::Type{<:Categorical}, ss::CategoricalStats)
     Categorical(normalize!(ss.h, 1))
 end
 
+fit_mle(d::T, x::AbstractArray{<:Integer}) where {T<:Categorical} = fit_mle(T, ncategories(d), x)
 function fit_mle(::Type{<:Categorical}, k::Integer, x::AbstractArray{T}) where T<:Integer
     Categorical(normalize!(add_categorical_counts!(zeros(k), x), 1), check_args=false)
 end
diff --git a/test/fit.jl b/test/fit.jl
index 4483dd1ec..d587d0d87 100644
--- a/test/fit.jl
+++ b/test/fit.jl
@@ -115,6 +115,11 @@ end
             @test d isa D
             @test ntrials(d) == 100
             @test succprob(d) ≈ 0.3 atol=0.01
+
+            d2 = @inferred fit_mle(D(100, 0.5), x)
+            @test d2 isa D
+            @test ntrials(d2) == 100
+            @test succprob(d2) ≈ 0.3 atol = 0.01
         end
     end
 end
@@ -141,6 +146,10 @@ end
         @test isa(d2, Categorical)
         @test probs(d2) == probs(d)
 
+        d3 = fit_mle(Categorical(p), x)
+        @test isa(d3, Categorical)
+        @test probs(d3) == probs(d)
+
         ss = suffstats(Categorical, (3, x), w)
         h = Float64[sum(w[x .== i]) for i = 1 : 3]
         @test isa(ss, Distributions.CategoricalStats)
@@ -414,6 +423,23 @@ end
     end
 end
 
+@testset "Testing fit_mle for ProductDistribution" begin
+    dists = [product_distribution([Exponential(0.5), Normal(11.3, 3.2)]), product_distribution([Exponential(0.5) Normal(11.3, 3.2)
+                                                                                                Bernoulli(0.2)  Normal{Float32}(0f0,0f0)])]
+    for func in funcs, dist in dists 
+        x = rand(dist, N)
+        w = func[1](N)
+
+        d = fit_mle(dist, x)
+        @test isa(d, typeof(dist))
+        @test isapprox(collect.(params(d)), collect.(params(dist)), atol=0.1)
+
+        d = fit_mle(dist, x, w)
+        @test isa(d, typeof(dist))
+        @test isapprox(collect.(params(d)), collect.(params(dist)), atol=0.1)
+    end
+end
+
 @testset "Testing fit for InverseGaussian" begin
     for func in funcs, dist in (InverseGaussian, InverseGaussian{Float64})
         x = rand(dist(3.9, 2.1), n0)