diff --git a/benchmark/benchmarks/tensordirichlet.jl b/benchmark/benchmarks/tensordirichlet.jl
index 855fcb7a..eaaa58ae 100644
--- a/benchmark/benchmarks/tensordirichlet.jl
+++ b/benchmark/benchmarks/tensordirichlet.jl
@@ -8,8 +8,8 @@ SUITE["tensordirichlet"] = BenchmarkGroup(
 # `prod` BenchmarkGroup ========================
 for rank in (3, 4, 5, 6)
     for d in (5, 10, 20)
-        left = TensorDirichlet(rand([d for _ in 1:rank]...))
-        right = TensorDirichlet(rand([d for _ in 1:rank]...))
+        left = TensorDirichlet(rand([d for _ in 1:rank]...) .+ 1)
+        right = TensorDirichlet(rand([d for _ in 1:rank]...) .+ 1)
         SUITE["tensordirichlet"]["prod"]["Closed(rank=$rank, d=$d)"] = @benchmarkable prod(ClosedProd(), $left, $right)
     end
 end
diff --git a/scripts/benchmark.jl b/scripts/benchmark.jl
index 4ef9c9f9..1009bcc1 100644
--- a/scripts/benchmark.jl
+++ b/scripts/benchmark.jl
@@ -10,6 +10,6 @@ if isempty(ARGS)
     export_markdown("./benchmark_logs/last.md", result)
 else
     name = first(ARGS)
-    BenchmarkTools.judge(ExponentialFamily, name; judgekwargs = Dict(:time_tolerance => 0.1, :memory_tolerance => 0.05))
-    export_markdown("benchmark_vs_$(name)_result.md", result)
+    result = BenchmarkTools.judge(ExponentialFamily, name; judgekwargs = Dict(:time_tolerance => 0.1, :memory_tolerance => 0.05))
+    export_markdown("./benchmark_logs/benchmark_vs_$(name)_result.md", result)
 end
diff --git a/src/distributions/tensor_dirichlet.jl b/src/distributions/tensor_dirichlet.jl
index b312db45..3440d890 100644
--- a/src/distributions/tensor_dirichlet.jl
+++ b/src/distributions/tensor_dirichlet.jl
@@ -4,6 +4,7 @@ import SpecialFunctions: digamma, loggamma
 import Base: eltype
 import Distributions: pdf, logpdf
 using Distributions
+using SpecialFunctions, LogExpFunctions
 
 import FillArrays: Ones, Eye
 import LoopVectorization: vmap, vmapreduce
@@ -24,8 +25,18 @@ The `a` field stores the matrix parameter of the distribution.
 - a[:,m,n] are the parameters of a Dirichlet distribution
 - a[:,m_1,n_1] and a[:,m_2,n_2] are supposed independent if (m_1,n_1) not equal to (m_2,n_2).
 """
-struct TensorDirichlet{T <: Real, N, A <: AbstractArray{T, N}} <: ContinuousTensorDistribution
+struct TensorDirichlet{T <: Real, N, A <: AbstractArray{T, N}, Ts} <: ContinuousTensorDistribution
     a::A
+    α0::Ts
+    lmnB::Ts
+    function TensorDirichlet(alpha::AbstractArray{T, N}) where {T, N}
+        if !all(x -> x > zero(x), alpha)
+            throw(ArgumentError("All elements of the alpha tensor should be positive"))
+        end
+        alpha0 = sum(alpha; dims = 1)
+        lmnB = sum(loggamma, alpha; dims = 1) - loggamma.(alpha0)
+        new{T, N, typeof(alpha), typeof(alpha0)}(alpha, alpha0, lmnB)
+    end
 end
 
 get_dirichlet_parameters(dist::TensorDirichlet{T, N, A}) where {T, N, A} = eachslice(dist.a, dims = Tuple(2:N))
@@ -35,7 +46,7 @@ BayesBase.params(::MeanParametersSpace, dist::TensorDirichlet) = (reduce(vcat, e
 getbasemeasure(::Type{TensorDirichlet}) = (x) -> sum([x[:, i] for i in CartesianIndices(Base.tail(size(x)))])
 getsufficientstatistics(::TensorDirichlet) = (x -> vmap(log, x),)
 
-BayesBase.mean(dist::TensorDirichlet) = dist.a ./ sum(dist.a; dims = 1)
+BayesBase.mean(dist::TensorDirichlet) = dist.a ./ dist.α0
 function BayesBase.cov(dist::TensorDirichlet{T}) where {T}
     s = size(dist.a)
     news = (first(s), first(s), Base.tail(s)...)
@@ -45,12 +56,11 @@ function BayesBase.cov(dist::TensorDirichlet{T}) where {T}
     end
     return v
 end
-
-function BayesBase.var(dist::TensorDirichlet)
-    v = similar(dist.a)
-    for i in CartesianIndices(Base.tail(size(dist.a)))
-        v[:, i] .= var(Dirichlet(dist.a[:, i]))
-    end
+function BayesBase.var(dist::TensorDirichlet{T, N, A, Ts}) where {T, N, A, Ts}
+    α = dist.a
+    α0 = dist.α0
+    c = inv.(α0 .^ 2 .* (α0 .+ 1))
+    v = α .* (α0 .- α) .* c
     return v
 end
 BayesBase.std(dist::TensorDirichlet) = map(d -> std(Dirichlet(d)), extract_collection(dist))
@@ -108,21 +118,10 @@ function BayesBase.rand!(rng::AbstractRNG, dist::TensorDirichlet{A}, container::
 end
 
 function BayesBase.logpdf(dist::TensorDirichlet{R, N, A}, x::AbstractArray{T, N}) where {R, A, T <: Real, N}
-    out = zero(eltype(x))
-    for i in CartesianIndices(extract_collection(dist))
-        out += logpdf(Dirichlet(dist.a[:, i]), @view x[:, i])
-    end
-    return out
-end
-
-function _dirichlet_logpdf(α::AbstractVector{T}, x::AbstractVector{T}) where {T}
-    α0 = sum(α)
-    lmB = loggamma(α0) - sum(loggamma.(α))
-    if length(α) != length(x) || sum(x) != 1 || any(x -> x < 0, x)
-        return xlogy(one(eltype(α)), zero(eltype(x))) - lmB
-    end
-    s = sum(xlogy(αi - 1, xi) for (αi, xi) in zip(α, x))
-    return s - lmB
+    α = dist.a
+    α0 = dist.α0
+    s = sum(xlogy.(α .- 1, x); dims = 1)
+    return sum(s .- dist.lmnB)
 end
 
 BayesBase.pdf(dist::TensorDirichlet, x::Array{T, N}) where {T <: Real, N} = exp(logpdf(dist, x))
@@ -130,19 +129,7 @@ BayesBase.pdf(dist::TensorDirichlet, x::Array{T, N}) where {T <: Real, N} = exp(
 BayesBase.default_prod_rule(::Type{<:TensorDirichlet}, ::Type{<:TensorDirichlet}) = PreserveTypeProd(Distribution)
 
 function BayesBase.prod(::PreserveTypeProd{Distribution}, left::TensorDirichlet, right::TensorDirichlet)
-    paramL = extract_collection(left)
-    paramR = extract_collection(right)
-    Ones = ones(size(left.a))
-    Ones = extract_collection(TensorDirichlet(Ones))
-    param = copy(Ones)
-    for i in eachindex(paramL)
-        param[i] .= paramL[i] .+ paramR[i] .- Ones[i]
-    end
-    out = similar(left.a)
-    for i in CartesianIndices(param)
-        out[:, i] = param[i]
-    end
-    return TensorDirichlet(out)
+    return TensorDirichlet(left.a .+ right.a .- 1)
 end
 
 function BayesBase.insupport(ef::ExponentialFamilyDistribution{TensorDirichlet}, x)
diff --git a/test/distributions/tensor_dirichlet_test.jl b/test/distributions/tensor_dirichlet_test.jl
index 2775fd16..7e99f6ab 100644
--- a/test/distributions/tensor_dirichlet_test.jl
+++ b/test/distributions/tensor_dirichlet_test.jl
@@ -25,11 +25,8 @@ end
             end
         end
     end
-
-
 end
 
-
 @testitem "TensorDirichlet: var" begin
     include("distributions_setuptests.jl")
 
@@ -50,7 +47,6 @@ end
             end
         end
     end
-
 end
 
 @testitem "TensorDirichlet: mean" begin
@@ -73,7 +69,6 @@ end
             end
         end
     end
-
 end
 
 @testitem "TensorDirichlet: std" begin
@@ -96,7 +91,6 @@ end
             end
         end
     end
-
 end
 
 @testitem "TensorDirichlet: cov" begin
@@ -112,10 +106,10 @@ end
 
                 temp = cov.(mat_of_dir)
                 old_shape = size(alpha)
-                new_shape = (first(old_shape),first(old_shape),Base.tail(old_shape)...)
+                new_shape = (first(old_shape), first(old_shape), Base.tail(old_shape)...)
                 mat_cov = ones(new_shape)
                 for i in CartesianIndices(Base.tail(size(alpha)))
-                    mat_cov[:,:, i] = temp[i]
+                    mat_cov[:, :, i] = temp[i]
                 end
                 @test cov(distribution) ≈ mat_cov
             end
@@ -258,20 +252,20 @@ end
                 alpha2 = rand([d for _ in 1:rank]...) .+ 1
                 distribution1 = TensorDirichlet(alpha1)
                 distribution2 = TensorDirichlet(alpha2)
-            
+
                 mat_of_dir_1 = Dirichlet.(eachslice(alpha1, dims = Tuple(2:rank)))
                 mat_of_dir_2 = Dirichlet.(eachslice(alpha2, dims = Tuple(2:rank)))
-                dim = rank-1
+                dim = rank - 1
 
-                prod_temp = Array{Dirichlet,dim}(undef, Base.tail(size(alpha1)))
+                prod_temp = Array{Dirichlet, dim}(undef, Base.tail(size(alpha1)))
                 for i in CartesianIndices(Base.tail(size(alpha1)))
-                    prod_temp[i] = prod(PreserveTypeProd(Distribution),mat_of_dir_1[i],mat_of_dir_2[i])
+                    prod_temp[i] = prod(PreserveTypeProd(Distribution), mat_of_dir_1[i], mat_of_dir_2[i])
                 end
                 mat_prod = similar(alpha1)
                 for i in CartesianIndices(Base.tail(size(alpha1)))
-                    mat_prod[:,i] = prod_temp[i].alpha
+                    mat_prod[:, i] = prod_temp[i].alpha
                 end
-                @test @inferred(prod(PreserveTypeProd(Distribution),distribution1,distribution2)) ≈ TensorDirichlet(mat_prod)
+                @test @inferred(prod(PreserveTypeProd(Distribution), distribution1, distribution2)) ≈ TensorDirichlet(mat_prod)
             end
         end
     end
@@ -300,7 +294,6 @@ end
             end
         end
     end
-    
 end
 
 @testitem "TensorDirichlet: vague" begin
@@ -330,11 +323,10 @@ end
     for rank in (3, 5)
         for d in (2, 5, 10)
             for _ in 1:10
-                
                 alpha = rand([d for _ in 1:rank]...)
                 distribution = TensorDirichlet(alpha)
                 (naturalParam,) = unpack_parameters(TensorDirichlet, alpha)
-                
+
                 mat_logPartition = sum(logPartitionDirichlet.(eachslice(alpha, dims = Tuple(2:rank))))
                 mat_grad = grad.(eachslice(alpha, dims = Tuple(2:rank)))
                 mat_info = sum(info.(eachslice(alpha, dims = Tuple(2:rank))))
@@ -350,9 +342,9 @@ end
 @testitem "TensorDirichlet: logpdf" begin
     include("distributions_setuptests.jl")
 
-    for rank in (3, 5)
-        for d in (2, 5, 10)
-            for _ in 1:10
+    for rank in (3, 4, 5, 6)
+        for d in (2, 4, 5, 10)
+            for i in 1:10
                 alpha = rand([d for _ in 1:rank]...)
 
                 distribution = TensorDirichlet(alpha)