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 1.9.3
Commit bed2cd540a1 (2023-08-24 14:43 UTC)
Build Info:
Official https://julialang.org/ release
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 16 virtual cores
Environment:
JULIA_CPU_THREADS = 16
JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/7aa0085e-79a4-45f3-a5bd-9743c91cf3da
Package Information:
Status `/cache/build/default-amdci4-5/julialang/turingtutorials/tutorials/01-gaussian-mixture-model/Project.toml`
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[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`