# Turing.jl programmatically set number of clusters in a mixture model

**URL:** <https://discourse.julialang.org/t/turing-jl-programmatically-set-number-of-clusters-in-a-mixture-model/126170>\
**Category:** Probabilistic Programming\
**Tags:** question, package, turing\
**Created:** [February 22, 2025, 1:23am UTC](https://discourse.julialang.org/t/turing-jl-programmatically-set-number-of-clusters-in-a-mixture-model/126170 "2025-02-22T01:23:37Z")\
**Posts on this page:** 7\
**Page:** 1

<div class="post-metadata">

**Author:** ![apattyn](https://avatars.discourse-cdn.com/v4/letter/a/ee7513/32.png) [@apattyn](https://discourse.julialang.org/u/apattyn)\
**Post date:** [February 22, 2025, 1:23am UTC](https://discourse.julialang.org/t/turing-jl-programmatically-set-number-of-clusters-in-a-mixture-model/126170/1 "2025-02-22T01:23:37Z")

</div>

I am trying to model some data using a mixture model with Turing.jl, but I want to programmatically set the number of Gaussian in the mixture without having to edit the model. For example if I want ten Gaussians I don’t want to manually add 10  
`mu ~ Normal()` statements.

Currently my working ‘manual’ model looks like:

```julia
using Turing, Distributions

@model function gmm()
   try
       ω ~ Dirichlet(3, 1.0)

       μ1 ~ MvNormal(zeros(2), I)
       Σ1 ~ InverseWishart(2, [1 0;0 1])
  
       μ2 ~ MvNormal(zeros(2), I)
       Σ2 ~ InverseWishart(2, [1 0;0 1])

       data ~ MixtureModel(
           [
               Product(fill(Uniform(-10,10), 2)),
               MvNormal(μ1, Σ1),
               MvNormal(μ2, Σ2)
           ],
           ω
       )
   catch me
       if me isa PosDefException
           Turing.@addlogprob! -Inf
       end
   end
end

```

I saw others recommend using something like the following:

```julia
using Turing, LinearAlgebra, Distributions

@model function test_mixture_model(;K = 3)

    try
        ω ~ Dirichlet(K, 1.0)

        μ ~ filldist(MvNormal(5 .*ones(2), 5*I), K-1)
    
        obs ~ MixtureModel(
            [
                id == 1 ? Product(fill(Uniform(-10, 10), 2)) : MvNormal(μ[id-1], I) for id in 1:K
            ],
            ω
        )

    catch me 
        if me isa PosDefException
            Turing.@addlogprob! -Inf
        end
    end
end

model = test_mixture_model()

ϕ = MixtureModel([Product(fill(Uniform(-10,10), 2)), MvNormal(zeros(2), I), MvNormal([7,7], I)])
data_synth = rand(ϕ,1_000)
model_cond = model | (; obs = data_synth)
chain = sample(model_cond, NUTS(; adtype=AutoForwardDiff()), 1000)

```

However, I encountered an issue with all the `μ`s becoming tied together, i.e. they contain similar values even though the ground truth has the clusters far apart. Then If I try to add `Σ` as a parameter they don’t show up as inferred values.

```julia
Summary Statistics
  parameters mean std mcse ess_bulk ess_tail rhat ess_per_sec 
      Symbol Float64 Float64 Float64 Float64 Float64 Float64 Float64 

        ω[1] 0.3448 0.2338 0.0073 1037.2750 695.0540 1.0017 2160.9896
        ω[2] 0.3301 0.2335 0.0078 864.5741 679.1482 1.0019 1801.1959
        ω[3] 0.3251 0.2315 0.0077 845.4391 809.9245 1.0006 1761.3315
     μ[1, 1] 4.9690 2.0700 0.0555 1356.5816 901.3844 1.0024 2826.2116
     μ[2, 1] 4.9583 2.1976 0.0582 1411.3363 785.8980 1.0046 2940.2841
     μ[1, 2] 5.0194 2.2868 0.0686 1111.6677 861.5815 0.9997 2315.9744
     μ[2, 2] 5.0070 2.1828 0.0701 967.7597 786.9313 0.9993 2016.1660

```

To summarize, I would like to have a mixture model where the user can select the number of Gaussian to include. Additionally I would like the mean and covariance of those Gaussian to be inferred by the turing from the data. Is there a recommend way of doing this?

---

<div class="post-metadata">

**Author:** ![eteppo](https://avatars.discourse-cdn.com/v4/letter/e/90db22/32.png) [@eteppo](https://discourse.julialang.org/u/eteppo)\
**Post date:** [February 22, 2025, 6:42pm UTC](https://discourse.julialang.org/t/turing-jl-programmatically-set-number-of-clusters-in-a-mixture-model/126170/2 "2025-02-22T18:42:26Z")

</div>

As a quick amateur response, you have probably checked this tutorial [Gaussian Mixture Models – Turing.jl](https://turinglang.org/docs/tutorials/gaussian-mixture-models/)? I noticed your second approach looks similar to it, just need to make another `filldist` containing the covariance matrices and loop over it along the means. Mixture models can also suffer from a label switching problem and they solve this using `Bijectors.ordered`. Finally, it might help to get help if you add code on how you added Σ as a parameter without success.

---

<div class="post-metadata">

**Author:** ![apattyn](https://avatars.discourse-cdn.com/v4/letter/a/ee7513/32.png) [@apattyn](https://discourse.julialang.org/u/apattyn)\
**Post date:** [February 23, 2025, 3:32pm UTC](https://discourse.julialang.org/t/turing-jl-programmatically-set-number-of-clusters-in-a-mixture-model/126170/3 "2025-02-23T15:32:23Z")

</div>

EDIT: Updated how I index mu and Sigma, but it doesn’t seem to help with inferring the parameters.

When I add the covariance to the example the updated model looks like:

```julia
@model function test_mixture_model(;K = 3)

    try
        ω ~ Dirichlet(K, 1.0)

        μ ~ filldist(MvNormal(5 .*ones(2), 5*I), K-1)
        Σ ~ filldist(InverseWishart(2, [1 0;0 1]), K-1)
    
        obs ~ MixtureModel(
            [
                id == 1 ? Product(fill(Uniform(-10, 10), 2)) : MvNormal(μ[:,id-1], Σ[:,:,id-1]) for id in 1:K
            ],
            ω
        )

        println("hello")

    catch me 
        if me isa PosDefException
            Turing.@addlogprob! -Inf
        end
    end
end

```

Then by running:

```julia
model = test_mixture_model()

ϕ = MixtureModel([Product(fill(Uniform(-10,10), 2)), MvNormal(zeros(2), I), MvNormal([7,7], I)])
data_synth = rand(ϕ,1_000)
model_cond = model | (; obs = data_synth)
chain = sample(model_cond, NUTS(; adtype=AutoForwardDiff()), 1000)

```

I get:

```julia
julia> chain = sample(model_cond, NUTS(; adtype=AutoForwardDiff()), 1000)

┌ Info: Found initial step size
└ ϵ = 1.6
Chains MCMC chain (1000×19×1 Array{Float64, 3}):

Iterations = 501:1:1500
Number of chains = 1
Samples per chain = 1000
Wall duration = 7.25 seconds
Compute duration = 7.25 seconds
parameters = ω[1], ω[2], ω[3], μ[1, 1], μ[2, 1], μ[1, 2], μ[2, 2]
internals = lp, n_steps, is_accept, acceptance_rate, log_density, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size

Summary Statistics
  parameters mean std mcse ess_bulk ess_tail rhat ess_per_sec 
      Symbol Float64 Float64 Float64 Float64 Float64 Float64 Float64 

        ω[1] 0.3270 0.2311 0.0066 1129.8189 515.3923 0.9997 155.7512
        ω[2] 0.3299 0.2365 0.0069 1058.7846 738.7187 1.0032 145.9587
        ω[3] 0.3431 0.2415 0.0069 1149.3102 721.1299 1.0005 158.4381
     μ[1, 1] 5.0655 2.2270 0.0613 1319.3656 789.2345 1.0007 181.8811
     μ[2, 1] 5.0261 2.1903 0.0578 1485.8065 757.1258 0.9995 204.8258
     μ[1, 2] 5.0966 2.0562 0.0670 939.1151 713.4876 0.9990 129.4617
     μ[2, 2] 4.9521 2.1826 0.0628 1209.1306 843.9087 0.9993 166.6847

Quantiles
  parameters 2.5% 25.0% 50.0% 75.0% 97.5% 
      Symbol Float64 Float64 Float64 Float64 Float64 

        ω[1] 0.0160 0.1326 0.2861 0.4718 0.8387
        ω[2] 0.0152 0.1284 0.2739 0.4994 0.8348
        ω[3] 0.0168 0.1329 0.3077 0.5112 0.8547
     μ[1, 1] 0.7715 3.5757 5.0487 6.6641 9.2216
     μ[2, 1] 0.6594 3.6700 4.9968 6.3735 9.4630
     μ[1, 2] 1.0718 3.7447 5.1711 6.4348 9.0233
     μ[2, 2] 0.8104 3.4796 4.9707 6.4078 9.2173

```

---

<div class="post-metadata">

**Author:** ![apattyn](https://avatars.discourse-cdn.com/v4/letter/a/ee7513/32.png) [@apattyn](https://discourse.julialang.org/u/apattyn)\
**Post date:** [February 23, 2025, 3:43pm UTC](https://discourse.julialang.org/t/turing-jl-programmatically-set-number-of-clusters-in-a-mixture-model/126170/4 "2025-02-23T15:43:14Z")

</div>

I am not too worried about using `Bijectors.ordered` for this set of code, since I am just trying to fit an arbitrary number of Gaussian’s to my dataset to approximate the pdf. So I am not concerned with which cluster contains which data. Although if I update my code to use `MCMCThreads()` I’ll have to modify things a bit.

But currently even if I use `filldist` my model parameters never match what the ground truth is. I am not sure if I need to index the parameters in a particular way.

---

<div class="post-metadata">

**Author:** ![apattyn](https://avatars.discourse-cdn.com/v4/letter/a/ee7513/32.png) [@apattyn](https://discourse.julialang.org/u/apattyn)\
**Post date:** [March 1, 2025, 1:25am UTC](https://discourse.julialang.org/t/turing-jl-programmatically-set-number-of-clusters-in-a-mixture-model/126170/5 "2025-03-01T01:25:25Z")

</div>

I think I see what your referring to in the GMM example (last section - “inferring assignments”). That does help move things in the right direction.

## Updated Model

```julia
@model function test_mixture_model(;K = 3)

    try
        ω ~ Dirichlet(K, 1.0)

        μ ~ MvNormal(zeros(2*(K-1)), I)
        
    
        obs ~ MixtureModel(
            [
                id == 1 ? Product(fill(Uniform(-10, 10), 2)) : MvNormal(μ[(id-1)*2-1:(id-1)*2], I) for id in 1:K
            ],
            ω
        )

    catch me 
        if me isa PosDefException
            Turing.@addlogprob! -Inf
        end
    end
end

```

Still working on getting the covariance to work…

---

<div class="post-metadata">

**Author:** ![apattyn](https://avatars.discourse-cdn.com/v4/letter/a/ee7513/32.png) [@apattyn](https://discourse.julialang.org/u/apattyn)\
**Post date:** [March 1, 2025, 1:37am UTC](https://discourse.julialang.org/t/turing-jl-programmatically-set-number-of-clusters-in-a-mixture-model/126170/6 "2025-03-01T01:37:25Z")

</div>

So I was getting some weird results before, i.e. incorrect parameter values and seemly having some parameters not be inferred. But the updated model below seems to work, but I had to update my `sample` call to have `10_000` samples for the means to converge.

This new model does complain about not being able to find initial values:

```julia
┌ Warning: failed to find valid initial parameters in 10 tries; consider providing explicit initial parameters using the `initial_params` keyword
└ @ Turing.Inference ~/.julia/packages/Turing/r3Hmj/src/mcmc/hmc.jl:192

```

Didn’t have this issue with the ‘manual’ model, so I’ll have to do some reading…

## Updated Model with Covariance

```julia
@model function test_mixture_model(;K = 3)

    try
        ω ~ Dirichlet(K, 1.0)

        μ ~ MvNormal(zeros(2*(K-1)), I)
        Sigma ~ MvNormal(zeros(3*(K-1)), I)
        
    
        obs ~ MixtureModel(
            [
                id == 1 ? Product(fill(Uniform(-10, 10), 2)) : MvNormal(μ[(id-1)*2-1:(id-1)*2], [Sigma[(id-1)*3-2] Sigma[(id-1)*3-1]; Sigma[(id-1)*3-1] Sigma[(id-1)*3]]) for id in 1:K
            ],
            ω
        )

    catch me 
        if me isa PosDefException
            Turing.@addlogprob! -Inf
        end
    end
end

```

---

<div class="post-metadata">

**Author:** ![RoyiAvital](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/royiavital/32/571_2.png) [@RoyiAvital](https://discourse.julialang.org/u/RoyiAvital)\
**Post date:** [May 10, 2025, 12:52pm UTC](https://discourse.julialang.org/t/turing-jl-programmatically-set-number-of-clusters-in-a-mixture-model/126170/7 "2025-05-10T12:52:05Z")

</div>

It is way far from my capabilities in Julia.  
But maybe the simple solution here is Meta Programming to automatically generate as much as needed lines of the code?