This website is deprecated. Please visit our new website here.
Skip to content

Unsupervised Learning using Bayesian Mixture Models

The following tutorial illustrates the use of Turing for clustering data using a Bayesian mixture model. The aim of this task is to infer a latent grouping (hidden structure) from unlabelled data.

Synthetic Data

We generate a synthetic dataset of $N = 60$ two-dimensional points $x_i \in \mathbb{R}^2$ drawn from a Gaussian mixture model. For simplicity, we use $K = 2$ clusters with

  • equal weights, i.e., we use mixture weights $w = [0.5, 0.5]$, and
  • isotropic Gaussian distributions of the points in each cluster.

More concretely, we use the Gaussian distributions $\mathcal{N}([\mu_k, \mu_k]^\mathsf{T}, I)$ with parameters $\mu_1 = -3.5$ and $\mu_2 = 0.5$.

using Distributions
using FillArrays
using StatsPlots

using LinearAlgebra
using Random

# Set a random seed.
Random.seed!(3)

# Define Gaussian mixture model.
w = [0.5, 0.5]
μ = [-3.5, 0.5]
mixturemodel = MixtureModel([MvNormal(Fill(μₖ, 2), I) for μₖ in μ], w)

# We draw the data points.
N = 60
x = rand(mixturemodel, N);

The following plot shows the dataset.

scatter(x[1, :], x[2, :]; legend=false, title="Synthetic Dataset")

Gaussian Mixture Model in Turing

We are interested in recovering the grouping from the dataset. More precisely, we want to infer the mixture weights, the parameters $\mu_1$ and $\mu_2$, and the assignment of each datum to a cluster for the generative Gaussian mixture model.

In a Bayesian Gaussian mixture model with $K$ components each data point $x_i$ ($i = 1,\ldots,N$) is generated according to the following generative process. First we draw the model parameters, i.e., in our example we draw parameters $\mu_k$ for the mean of the isotropic normal distributions and the mixture weights $w$ of the $K$ clusters. We use standard normal distributions as priors for $\mu_k$ and a Dirichlet distribution with parameters $\alpha_1 = \cdots = \alpha_K = 1$ as prior for $w$: $$ \begin{aligned} \mu_k &\sim \mathcal{N}(0, 1) \qquad (k = 1,\ldots,K)\ w &\sim \operatorname{Dirichlet}(\alpha_1, \ldots, \alpha_K) \end{aligned} $$ After having constructed all the necessary model parameters, we can generate an observation by first selecting one of the clusters $$ z_i \sim \operatorname{Categorical}(w) \qquad (i = 1,\ldots,N), $$ and then drawing the datum accordingly, i.e., in our example drawing $$ x_i \sim \mathcal{N}([\mu_{z_i}, \mu_{z_i}]^\mathsf{T}, I) \qquad (i=1,\ldots,N). $$ For more details on Gaussian mixture models, we refer to Christopher M. Bishop, Pattern Recognition and Machine Learning, Section 9.

We specify the model with Turing.

using Turing

@model function gaussian_mixture_model(x)
    # Draw the parameters for each of the K=2 clusters from a standard normal distribution.
    K = 2
    μ ~ MvNormal(Zeros(K), I)

    # Draw the weights for the K clusters from a Dirichlet distribution with parameters αₖ = 1.
    w ~ Dirichlet(K, 1.0)
    # Alternatively, one could use a fixed set of weights.
    # w = fill(1/K, K)

    # Construct categorical distribution of assignments.
    distribution_assignments = Categorical(w)

    # Construct multivariate normal distributions of each cluster.
    D, N = size(x)
    distribution_clusters = [MvNormal(Fill(μₖ, D), I) for μₖ in μ]

    # Draw assignments for each datum and generate it from the multivariate normal distribution.
    k = Vector{Int}(undef, N)
    for i in 1:N
        k[i] ~ distribution_assignments
        x[:, i] ~ distribution_clusters[k[i]]
    end

    return k
end

model = gaussian_mixture_model(x);

We run a MCMC simulation to obtain an approximation of the posterior distribution of the parameters $\mu$ and $w$ and assignments $k$. We use a Gibbs sampler that combines a particle Gibbs sampler for the discrete parameters (assignments $k$) and a Hamiltonion Monte Carlo sampler for the continuous parameters ($\mu$ and $w$). We generate multiple chains in parallel using multi-threading.

sampler = Gibbs(PG(100, :k), HMC(0.05, 10, :μ, :w))
nsamples = 100
nchains = 3
chains = sample(model, sampler, MCMCThreads(), nsamples, nchains);

Inferred Mixture Model

After sampling we can visualize the trace and density of the parameters of interest.

We consider the samples of the location parameters $\mu_1$ and $\mu_2$ for the two clusters.

plot(chains[["μ[1]", "μ[2]"]]; colordim=:parameter, legend=true)

It can happen that the modes of $\mu_1$ and $\mu_2$ switch between chains. For more information see the Stan documentation for potential solutions.

We also inspect the samples of the mixture weights $w$.

plot(chains[["w[1]", "w[2]"]]; colordim=:parameter, legend=true)

In the following, we just use the first chain to ensure the validity of our inference.

chain = chains[:, :, 1];

As the distributions of the samples for the parameters $\mu_1$, $\mu_2$, $w_1$, and $w_2$ are unimodal, we can safely visualize the density region of our model using the average values.

# Model with mean of samples as parameters.
μ_mean = [mean(chain, "μ[$i]") for i in 1:2]
w_mean = [mean(chain, "w[$i]") for i in 1:2]
mixturemodel_mean = MixtureModel([MvNormal(Fill(μₖ, 2), I) for μₖ in μ_mean], w_mean)

contour(
    range(-7.5, 3; length=1_000),
    range(-6.5, 3; length=1_000),
    (x, y) -> logpdf(mixturemodel_mean, [x, y]);
    widen=false,
)
scatter!(x[1, :], x[2, :]; legend=false, title="Synthetic Dataset")

Inferred Assignments

Finally, we can inspect the assignments of the data points inferred using Turing. As we can see, the dataset is partitioned into two distinct groups.

assignments = [mean(chain, "k[$i]") for i in 1:N]
scatter(
    x[1, :],
    x[2, :];
    legend=false,
    title="Assignments on Synthetic Dataset",
    zcolor=assignments,
)

Appendix

These tutorials are a part of the TuringTutorials repository, found at: https://github.com/TuringLang/TuringTutorials.

To locally run this tutorial, do the following commands:

using TuringTutorials
TuringTutorials.weave("01-gaussian-mixture-model", "01_gaussian-mixture-model.jmd")

Computer Information:

Julia Version 1.9.2
Commit e4ee485e909 (2023-07-05 09:39 UTC)
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 128 × AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-14.0.6 (ORCJIT, znver2)
  Threads: 1 on 128 virtual cores
Environment:
  JULIA_CPU_THREADS = 128
  JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/7aa0085e-79a4-45f3-a5bd-9743c91cf3da

Package Information:

Status `/cache/build/exclusive-amdci1-0/julialang/turingtutorials/tutorials/01-gaussian-mixture-model/Project.toml`
  [31c24e10] Distributions v0.25.100
  [1a297f60] FillArrays v1.5.0
  [f3b207a7] StatsPlots v0.15.6
  [fce5fe82] Turing v0.28.1
  [37e2e46d] LinearAlgebra
  [9a3f8284] Random

And the full manifest:

Status `/cache/build/exclusive-amdci1-0/julialang/turingtutorials/tutorials/01-gaussian-mixture-model/Manifest.toml`
  [47edcb42] ADTypes v0.1.6
  [621f4979] AbstractFFTs v1.5.0
  [80f14c24] AbstractMCMC v4.4.2
⌅ [7a57a42e] AbstractPPL v0.5.4
  [1520ce14] AbstractTrees v0.4.4
  [79e6a3ab] Adapt v3.6.2
  [0bf59076] AdvancedHMC v0.5.3
  [5b7e9947] AdvancedMH v0.7.5
  [576499cb] AdvancedPS v0.4.3
  [b5ca4192] AdvancedVI v0.2.4
  [dce04be8] ArgCheck v2.3.0
  [7d9fca2a] Arpack v0.5.4
  [4fba245c] ArrayInterface v7.4.11
  [a9b6321e] Atomix v0.1.0
  [13072b0f] AxisAlgorithms v1.0.1
  [39de3d68] AxisArrays v0.4.7
  [198e06fe] BangBang v0.3.39
  [9718e550] Baselet v0.1.1
⌅ [76274a88] Bijectors v0.12.8
  [d1d4a3ce] BitFlags v0.1.7
  [fa961155] CEnum v0.4.2
  [49dc2e85] Calculus v0.5.1
  [082447d4] ChainRules v1.53.0
  [d360d2e6] ChainRulesCore v1.16.0
  [9e997f8a] ChangesOfVariables v0.1.8
  [aaaa29a8] Clustering v0.15.4
  [944b1d66] CodecZlib v0.7.2
  [35d6a980] ColorSchemes v3.23.0
  [3da002f7] ColorTypes v0.11.4
  [c3611d14] ColorVectorSpace v0.10.0
  [5ae59095] Colors v0.12.10
  [861a8166] Combinatorics v1.0.2
  [38540f10] CommonSolve v0.2.4
  [bbf7d656] CommonSubexpressions v0.3.0
  [34da2185] Compat v4.9.0
  [a33af91c] CompositionsBase v0.1.2
  [f0e56b4a] ConcurrentUtilities v2.2.1
  [88cd18e8] ConsoleProgressMonitor v0.1.2
  [187b0558] ConstructionBase v1.5.3
  [d38c429a] Contour v0.6.2
  [a8cc5b0e] Crayons v4.1.1
  [9a962f9c] DataAPI v1.15.0
  [864edb3b] DataStructures v0.18.15
  [e2d170a0] DataValueInterfaces v1.0.0
  [244e2a9f] DefineSingletons v0.1.2
  [8bb1440f] DelimitedFiles v1.9.1
  [b429d917] DensityInterface v0.4.0
  [163ba53b] DiffResults v1.1.0
  [b552c78f] DiffRules v1.15.1
  [b4f34e82] Distances v0.10.9
  [31c24e10] Distributions v0.25.100
  [ced4e74d] DistributionsAD v0.6.52
  [ffbed154] DocStringExtensions v0.9.3
  [fa6b7ba4] DualNumbers v0.6.8
⌃ [366bfd00] DynamicPPL v0.23.0
  [cad2338a] EllipticalSliceSampling v1.1.0
  [4e289a0a] EnumX v1.0.4
  [460bff9d] ExceptionUnwrapping v0.1.9
  [e2ba6199] ExprTools v0.1.10
  [c87230d0] FFMPEG v0.4.1
  [7a1cc6ca] FFTW v1.7.1
  [1a297f60] FillArrays v1.5.0
  [53c48c17] FixedPointNumbers v0.8.4
  [59287772] Formatting v0.4.2
  [f6369f11] ForwardDiff v0.10.36
  [069b7b12] FunctionWrappers v1.1.3
  [77dc65aa] FunctionWrappersWrappers v0.1.3
  [d9f16b24] Functors v0.4.5
  [46192b85] GPUArraysCore v0.1.5
  [28b8d3ca] GR v0.72.9
  [42e2da0e] Grisu v1.0.2
  [cd3eb016] HTTP v1.9.14
  [34004b35] HypergeometricFunctions v0.3.23
  [22cec73e] InitialValues v0.3.1
  [505f98c9] InplaceOps v0.3.0
  [a98d9a8b] Interpolations v0.14.7
  [8197267c] IntervalSets v0.7.7
  [3587e190] InverseFunctions v0.1.12
  [41ab1584] InvertedIndices v1.3.0
  [92d709cd] IrrationalConstants v0.2.2
  [c8e1da08] IterTools v1.8.0
  [82899510] IteratorInterfaceExtensions v1.0.0
  [1019f520] JLFzf v0.1.5
  [692b3bcd] JLLWrappers v1.4.1
  [682c06a0] JSON v0.21.4
  [63c18a36] KernelAbstractions v0.9.8
  [5ab0869b] KernelDensity v0.6.7
  [929cbde3] LLVM v6.1.0
  [8ac3fa9e] LRUCache v1.4.1
  [b964fa9f] LaTeXStrings v1.3.0
  [23fbe1c1] Latexify v0.16.1
  [50d2b5c4] Lazy v0.15.1
  [1d6d02ad] LeftChildRightSiblingTrees v0.2.0
  [6f1fad26] Libtask v0.8.6
  [6fdf6af0] LogDensityProblems v2.1.1
  [996a588d] LogDensityProblemsAD v1.6.1
  [2ab3a3ac] LogExpFunctions v0.3.24
  [e6f89c97] LoggingExtras v1.0.0
  [c7f686f2] MCMCChains v6.0.3
  [be115224] MCMCDiagnosticTools v0.3.5
  [e80e1ace] MLJModelInterface v1.8.0
  [1914dd2f] MacroTools v0.5.10
  [dbb5928d] MappedArrays v0.4.2
  [739be429] MbedTLS v1.1.7
  [442fdcdd] Measures v0.3.2
  [128add7d] MicroCollections v0.1.4
  [e1d29d7a] Missings v1.1.0
  [6f286f6a] MultivariateStats v0.10.2
  [872c559c] NNlib v0.9.4
  [77ba4419] NaNMath v1.0.2
  [86f7a689] NamedArrays v0.9.8
  [c020b1a1] NaturalSort v1.0.0
  [b8a86587] NearestNeighbors v0.4.13
  [510215fc] Observables v0.5.4
  [6fe1bfb0] OffsetArrays v1.12.10
  [4d8831e6] OpenSSL v1.4.1
  [3bd65402] Optimisers v0.2.19
  [bac558e1] OrderedCollections v1.6.2
  [90014a1f] PDMats v0.11.17
  [69de0a69] Parsers v2.7.2
  [b98c9c47] Pipe v1.3.0
  [ccf2f8ad] PlotThemes v3.1.0
  [995b91a9] PlotUtils v1.3.5
  [91a5bcdd] Plots v1.38.17
  [aea7be01] PrecompileTools v1.1.2
  [21216c6a] Preferences v1.4.0
  [08abe8d2] PrettyTables v2.2.7
  [33c8b6b6] ProgressLogging v0.1.4
  [92933f4c] ProgressMeter v1.7.2
  [1fd47b50] QuadGK v2.8.2
  [74087812] Random123 v1.6.1
  [e6cf234a] RandomNumbers v1.5.3
  [b3c3ace0] RangeArrays v0.3.2
  [c84ed2f1] Ratios v0.4.5
  [c1ae055f] RealDot v0.1.0
  [3cdcf5f2] RecipesBase v1.3.4
  [01d81517] RecipesPipeline v0.6.12
  [731186ca] RecursiveArrayTools v2.38.7
  [189a3867] Reexport v1.2.2
  [05181044] RelocatableFolders v1.0.0
  [ae029012] Requires v1.3.0
  [79098fc4] Rmath v0.7.1
  [f2b01f46] Roots v2.0.17
  [7e49a35a] RuntimeGeneratedFunctions v0.5.12
  [0bca4576] SciMLBase v1.94.0
  [c0aeaf25] SciMLOperators v0.3.6
  [30f210dd] ScientificTypesBase v3.0.0
  [6c6a2e73] Scratch v1.2.0
  [91c51154] SentinelArrays v1.4.0
  [efcf1570] Setfield v1.1.1
  [992d4aef] Showoff v1.0.3
  [777ac1f9] SimpleBufferStream v1.1.0
  [ce78b400] SimpleUnPack v1.1.0
  [a2af1166] SortingAlgorithms v1.1.1
  [276daf66] SpecialFunctions v2.3.0
  [171d559e] SplittablesBase v0.1.15
  [90137ffa] StaticArrays v1.6.2
  [1e83bf80] StaticArraysCore v1.4.2
  [64bff920] StatisticalTraits v3.2.0
  [82ae8749] StatsAPI v1.6.0
  [2913bbd2] StatsBase v0.34.0
  [4c63d2b9] StatsFuns v1.3.0
  [f3b207a7] StatsPlots v0.15.6
  [892a3eda] StringManipulation v0.3.0
  [09ab397b] StructArrays v0.6.15
  [2efcf032] SymbolicIndexingInterface v0.2.2
  [ab02a1b2] TableOperations v1.2.0
  [3783bdb8] TableTraits v1.0.1
  [bd369af6] Tables v1.10.1
  [62fd8b95] TensorCore v0.1.1
  [5d786b92] TerminalLoggers v0.1.7
  [9f7883ad] Tracker v0.2.26
  [3bb67fe8] TranscodingStreams v0.9.13
  [28d57a85] Transducers v0.4.78
  [410a4b4d] Tricks v0.1.7
  [781d530d] TruncatedStacktraces v1.4.0
  [fce5fe82] Turing v0.28.1
  [5c2747f8] URIs v1.5.0
  [1cfade01] UnicodeFun v0.4.1
  [1986cc42] Unitful v1.16.2
  [45397f5d] UnitfulLatexify v1.6.3
  [013be700] UnsafeAtomics v0.2.1
  [d80eeb9a] UnsafeAtomicsLLVM v0.1.3
  [41fe7b60] Unzip v0.2.0
  [cc8bc4a8] Widgets v0.6.6
  [efce3f68] WoodburyMatrices v0.5.5
  [700de1a5] ZygoteRules v0.2.3
⌅ [68821587] Arpack_jll v3.5.1+1
  [6e34b625] Bzip2_jll v1.0.8+0
  [83423d85] Cairo_jll v1.16.1+1
  [2e619515] Expat_jll v2.5.0+0
⌃ [b22a6f82] FFMPEG_jll v4.4.2+2
  [f5851436] FFTW_jll v3.3.10+0
  [a3f928ae] Fontconfig_jll v2.13.93+0
  [d7e528f0] FreeType2_jll v2.13.1+0
  [559328eb] FriBidi_jll v1.0.10+0
  [0656b61e] GLFW_jll v3.3.8+0
  [d2c73de3] GR_jll v0.72.9+1
  [78b55507] Gettext_jll v0.21.0+0
  [7746bdde] Glib_jll v2.74.0+2
  [3b182d85] Graphite2_jll v1.3.14+0
  [2e76f6c2] HarfBuzz_jll v2.8.1+1
  [1d5cc7b8] IntelOpenMP_jll v2023.2.0+0
  [aacddb02] JpegTurbo_jll v2.1.91+0
  [c1c5ebd0] LAME_jll v3.100.1+0
  [88015f11] LERC_jll v3.0.0+1
  [dad2f222] LLVMExtra_jll v0.0.23+0
  [1d63c593] LLVMOpenMP_jll v15.0.4+0
  [dd4b983a] LZO_jll v2.10.1+0
⌅ [e9f186c6] Libffi_jll v3.2.2+1
  [d4300ac3] Libgcrypt_jll v1.8.7+0
  [7e76a0d4] Libglvnd_jll v1.6.0+0
  [7add5ba3] Libgpg_error_jll v1.42.0+0
  [94ce4f54] Libiconv_jll v1.16.1+2
  [4b2f31a3] Libmount_jll v2.35.0+0
  [89763e89] Libtiff_jll v4.5.1+1
  [38a345b3] Libuuid_jll v2.36.0+0
  [856f044c] MKL_jll v2023.2.0+0
  [e7412a2a] Ogg_jll v1.3.5+1
⌅ [458c3c95] OpenSSL_jll v1.1.22+0
  [efe28fd5] OpenSpecFun_jll v0.5.5+0
  [91d4177d] Opus_jll v1.3.2+0
  [30392449] Pixman_jll v0.42.2+0
  [c0090381] Qt6Base_jll v6.4.2+3
  [f50d1b31] Rmath_jll v0.4.0+0
  [a2964d1f] Wayland_jll v1.21.0+0
  [2381bf8a] Wayland_protocols_jll v1.25.0+0
  [02c8fc9c] XML2_jll v2.10.3+0
  [aed1982a] XSLT_jll v1.1.34+0
  [ffd25f8a] XZ_jll v5.4.4+0
  [4f6342f7] Xorg_libX11_jll v1.8.6+0
  [0c0b7dd1] Xorg_libXau_jll v1.0.11+0
  [935fb764] Xorg_libXcursor_jll v1.2.0+4
  [a3789734] Xorg_libXdmcp_jll v1.1.4+0
  [1082639a] Xorg_libXext_jll v1.3.4+4
  [d091e8ba] Xorg_libXfixes_jll v5.0.3+4
  [a51aa0fd] Xorg_libXi_jll v1.7.10+4
  [d1454406] Xorg_libXinerama_jll v1.1.4+4
  [ec84b674] Xorg_libXrandr_jll v1.5.2+4
  [ea2f1a96] Xorg_libXrender_jll v0.9.10+4
  [14d82f49] Xorg_libpthread_stubs_jll v0.1.1+0
  [c7cfdc94] Xorg_libxcb_jll v1.15.0+0
  [cc61e674] Xorg_libxkbfile_jll v1.1.2+0
  [12413925] Xorg_xcb_util_image_jll v0.4.0+1
  [2def613f] Xorg_xcb_util_jll v0.4.0+1
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.0+1
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.9+1
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.1+1
  [35661453] Xorg_xkbcomp_jll v1.4.6+0
  [33bec58e] Xorg_xkeyboard_config_jll v2.39.0+0
  [c5fb5394] Xorg_xtrans_jll v1.5.0+0
  [3161d3a3] Zstd_jll v1.5.5+0
⌅ [214eeab7] fzf_jll v0.29.0+0
  [a4ae2306] libaom_jll v3.4.0+0
  [0ac62f75] libass_jll v0.15.1+0
  [f638f0a6] libfdk_aac_jll v2.0.2+0
  [b53b4c65] libpng_jll v1.6.38+0
  [f27f6e37] libvorbis_jll v1.3.7+1
  [1270edf5] x264_jll v2021.5.5+0
  [dfaa095f] x265_jll v3.5.0+0
  [d8fb68d0] xkbcommon_jll v1.4.1+0
  [0dad84c5] ArgTools v1.1.1
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8ba89e20] Distributed
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL v0.6.3
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.9.2
  [de0858da] Printf
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays
  [10745b16] Statistics v1.9.0
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll v1.0.5+0
  [deac9b47] LibCURL_jll v7.84.0+0
  [29816b5a] LibSSH2_jll v1.10.2+0
  [c8ffd9c3] MbedTLS_jll v2.28.2+0
  [14a3606d] MozillaCACerts_jll v2022.10.11
  [4536629a] OpenBLAS_jll v0.3.21+4
  [05823500] OpenLibm_jll v0.8.1+0
  [efcefdf7] PCRE2_jll v10.42.0+0
  [bea87d4a] SuiteSparse_jll v5.10.1+6
  [83775a58] Zlib_jll v1.2.13+0
  [8e850b90] libblastrampoline_jll v5.8.0+0
  [8e850ede] nghttp2_jll v1.48.0+0
  [3f19e933] p7zip_jll v17.4.0+0
Info Packages marked with ⌃ and ⌅ have new versions available, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`