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PCM20240213_BayesianMixtureModels.jl
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PCM20240213_BayesianMixtureModels.jl
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### A Pluto.jl notebook ###
# v0.19.38
using Markdown
using InteractiveUtils
# ╔═╡ fd168b2b-242e-4e80-953a-cec00ccd4daa
begin
using Distributions
using StatsPlots
using LinearAlgebra
using Random
using LaTeXStrings, Latexify
using Turing
end
# ╔═╡ 97cbc792-ca6b-11ee-21a2-a9e81216fb65
md"
===================================================================================
#### [Bayesian Mixture Models](https://turinglang.org/v0.30/tutorials/01-gaussian-mixture-model/)
##### file: PCM20240213\_BayesianMixtureModels.jl
##### Julia/Pluto.jl-code (1.10.1/19.36) by PCM *** 2024/03/11 ***
===================================================================================
"
# ╔═╡ bd54afc5-bcbd-40f9-8c11-17dca28e9448
md"
This is a personal *interpretation*, *reconstruction* and *serialization* of the official corresponding [Turing.jl-tutorial](https://turinglang.org/v0.30/tutorials/01-gaussian-mixture-model/) into steps of increasing complexity. This seems to be helpful for learners (at least for me) because the original script is highly optimized and constrained.
In contrast to the original in the Turing.jl tutorial we modified some features:
- *Unconstrained* selection of *generative centroids* in $\mathbf x \in \mathbb R^2$ instead of $\mathbf x \in \mathbb R^1$. This means that we allow $D \cdot K$ parameters $\mu$ in our model instead of only $K$ as in the original model. There generative centroids could only lay on a line of 45 degrees inclination.
- *Increasing* $N$: we estimate membership parameters $k[i]\;;\; i=1,...,N$ for $N=500$ instead of $N=60$
- *Introduction* of a new variance parameter $\sigma^2$ to allow a more flexible variance-covariance matrix $\sigma2 \cdot \mathbf I$. This has three effects. First, the variance of the data generating process need not be known as was the case in the original. Second, we allow varying degrees of cluster overlap in the data. Third, in combination with the parameter vector $\mathbf w$ clusters may have different variance-covarianve matrix $\sigma_{kk}\;;\;kk=1,...,K$.
- *Estimation* and display of posterior cluster centroids
- *Combination* of cluster-specific mixture weights $w_{kk}\;;\;kk=1,...,K$ with the variance parameter $\sigma^2$. This models different attractiveness of clusters and cluster-specific variances.
- *Using* the *PG-MCMC*-algorithm for the estimation of *all* parameters. This is due to the brittleness of the originally proposed combined *Gibbs(PG(...), HMC(...))*. The latter often ran into interrupts.
"
# ╔═╡ b1fe04fe-26c7-4cda-b957-e1bbce32b7b2
# Set a random seed.
Random.seed!(3)
# ╔═╡ 348f7db7-34a4-4ead-abb2-f6ed469d20fb
md"
---
##### 1. Generation of Bivariate Bicluster Test Data
"
# ╔═╡ 1a993c1f-766c-490b-8059-53507a4c8dc5
let N = 500
K = 2
w = [ 0.6, 0.4] # unequal weight of clusters
μ = [-4.5 +0.5; # unconstrained centroids in R^2
+3.5 -0.6]
rangeXY = 12.5
length = 100
xs = range(-rangeXY, rangeXY; length = length)
ys = range(-rangeXY, rangeXY; length = length)
#--------------------------------------------------------------------------------
mixtureModel = MixtureModel([MvNormal(μ[:, kk], 3.5 .* I) for kk in 1:K], w)
# ^== explicit column vectors |
# continuous cluster membership ========================^
#--------------------------------------------------------------------------------
x = rand(mixtureModel, N)
#--------------------------------------------------------------------------------
# plot of data points
scatter(x[1, :], x[2, :], label="data point", title=L"Data 1 (Mixed Gaussian Model with Prior $\mu[1], \mu[2]$)")
#--------------------------------------------------------------------------------
# plot of generative centroids
scatter!([(μ[1, 1], μ[2, 1]), (μ[1, 2], μ[2, 2])], color=:violetred4, markershape = :cross, label="generative centroid", markersize=8, markerstrokewidth = 3)
#--------------------------------------------------------------------------------
# plot of density contours
contour!(xs, ys, (x, y) -> logpdf(mixtureModel, [x, y]), widen=false, fill=false)
# plot of the data points
#--------------------------------------------------------------------------------
end # let
# ╔═╡ 61e33f1c-86eb-4ccb-b036-f6d9caedbf97
md"
---
##### 2. Sequence of Increasing Complex Bayesian Multivar. Mixture Models
"
# ╔═╡ bee285ac-9388-441a-9299-cc25ecf021af
md"
---
###### 2.1 Mixed Gaussian Model with Discrete Posterior Latent Membership $k[i]\;;\;i=1,...,N$
The beauty of these results are treacherous because the matrix $\mathbf \mu$ is used *twice* in the data generating process *and* as a prior in the estimation process. This is only justified as an intermediate step towards the final model where this identity is avoided.
"
# ╔═╡ 6ede6e55-3d43-41c3-aff7-a53036ae92bd
#------------------------------------------------------------------------------------
# μ[:, k]: here, prior
# k[i] are the only *latent* parameters
#------------------------------------------------------------------------------------
let N = 500
K = 2
D = 2
w = [ 0.6, 0.4] # unequal weight of clusters
μ = [-4.5 +0.5; # unconstrained centroids in R^2
+3.5 -0.6]
x = Array{Float64}(undef, (D, N))
# ===============================================================================
# Data Generation
#--------------------------------------------------------------------------------
# stretched standard deviation ====\/
mixtureModel0 = MixtureModel([MvNormal(μ[:, k], 3.5 .* I) for k in 1:K], w)
# ^== explicit column vectors |
# continuous cluster membership ===============^
x = rand(mixtureModel0, N)
# ===============================================================================
# Mixture Model
#--------------------------------------------------------------------------------
@model function bayesianMixtureModel(x)
k = Vector{Int64}(undef, N)
α = [2.0, 1.0]
#----------------------------------------------------------------------------
σ2 ~ Gamma(α...) # α is 'splatted'
mixtureModel = [MvNormal(μ[:, kk], σ2 .* I) for kk in 1:K]
# ^<=== discrete membership
# ^==== explicit column vectors
for i in 1:N
k[i] ~ Categorical(w) # k[i], vector is important
# ^======================== discrete cluster membership
x[:,i] ~ mixtureModel[k[i]] # likelihood
end # for i
end # function bayesianMixtureModel
#--------------------------------------------------------------------------------
model = bayesianMixtureModel(x)
#--------------------------------------------------------------------------------
chains =
let nSamples = 100
# sampler = Prior()
# sampler = Gibbs(PG(10, :k, :σ2))
sampler = Gibbs(PG(20, :k), HMC(0.03, 10, :μ, :σ2))
sample(model, sampler, nSamples)
end # let
#--------------------------------------------------------------------------------
assignments = [mean(chains, "k[$i]") for i in 1:N]
# ===============================================================================
# plot of data points
scatter(x[1, :], x[2, :], title=L"Data 2 (Post. Latent Membership $k[i]$ and $σ2$)", label="data point", zcolor=assignments)
#--------------------------------------------------------------------------------
# plot of the generative centroids
scatter!([(μ[1, 1], μ[2, 1]), (μ[1, 2], μ[2, 2])], color=:violetred4, markershape = :cross, label="generative centroid", markersize=8, markerstrokewidth = 3)
#--------------------------------------------------------------------------------
end # let
# ╔═╡ cb56e40f-fc73-4e84-a97f-967b3bf95f31
md"
---
###### 2.2 Bayesian Mixed Gaussian Model with Post.Latent Membership $k[i]$, cluster-specific mixture weights $w[kk]$, and centroids $\mu[kk]$
"
# ╔═╡ d02585c8-0dab-4c35-a7cc-b7c88a9dd242
begin
# ===============================================================================
# Data Generation
#--------------------------------------------------------------------------------
N = 500
K = 2
D = 2
w = [ 0.6, 0.4] # unequal weight of clusters
μ0 = [-4.5 +0.5; # unconstrained centroids in R^2
+3.5 -0.6]
rangeXY = 12.5
length = 100
xs = range(-rangeXY, rangeXY; length = length)
ys = range(-rangeXY, rangeXY; length = length)
x = Array{Float64}(undef, (D, N))
# ===============================================================================
# Data Generation
#--------------------------------------------------------------------------------
# stretched standard deviation ====\/
mixtureModel0 = MixtureModel([MvNormal(μ0[:, k], 3.5 .* I) for k in 1:K], w)
# ^== explicit column vectors |
# continuous cluster membership ===============^
x = rand(mixtureModel0, N)
#--------------------------------------------------------------------------------
scatter(x[1, :], x[2, :]; label="data point", title="Data 3 (Mixed Multivariate Gaussian Data)")
#--------------------------------------------------------------------------------
# plot of generative centroids
scatter!([(μ0[1, 1], μ0[2, 1]), (μ0[1, 2], μ0[2, 2])], color=:violetred4, markershape = :cross, label="generative centroid", markersize=8, markerstrokewidth = 3)
#--------------------------------------------------------------------------------
# plot of density contours
contour!(xs, ys, (x, y) -> logpdf(mixtureModel0, [x, y]), widen=false, fill=false)
# plot of the data points
#--------------------------------------------------------------------------------
end # begin
# ╔═╡ 39d56e33-5092-4ba7-a283-e74ac7ef56a2
x
# ╔═╡ 6d3fe7ee-d5e5-4110-86a5-9d4751a3afaf
begin
# ===============================================================================
# Mixture Model
#--------------------------------------------------------------------------------
@model function bayesianMixtureModel(x)
μ0 = Array{Float64}(undef, (D, K))
μ = Array{Float64}(undef, (D, K))
k = Vector{Int64}(undef, N)
μ0 = [-0.0 +0.0; # μ1 = [μ1[1], μ1[2]] as priors
+0.0 -0.0]
α = [2.0, 1.0] # hyperparameter for Dirichlet
#----------------------------------------------------------------------------
σ2 ~ Gamma(α...) # α is 'splatted'
w ~ Dirichlet(α)
for kk in 1:K
# sampling priors μ1 = [μ1[1], μ1[2]]
μ[kk] ~ MvNormal(μ0[:, kk], σ2 .* I)
# ^======= unknown standard deviation
end # for kk
#
mixtureModel = [MvNormal(μ[kk][:], σ2 .* I) for kk in 1:K]
# ^<=== discrete membership
# ^==== explicit column vectors
for i in 1:N
k[i] ~ Categorical(w) # k[i], vector is important
# ^======================== discrete cluster membership
x[:, i] ~ mixtureModel[k[i]] # likelihood
end # for i
end # function bayesianMixtureModel
#--------------------------------------------------------------------------------
model = bayesianMixtureModel(x)
#--------------------------------------------------------------------------------
chains =
let nSamples = 80
# sampler = Prior()
# sampler = Gibbs(PG(50, :k, :μ, :w, :σ2)) # <========= seems ok
sampler = Gibbs(PG(50, :k), HMC(0.03, 10, :μ, :w, :σ2)) # <===== brittle
sample(model, sampler, nSamples)
end # let
#--------------------------------------------------------------------------------
end # begin
# ╔═╡ 01337a9d-57c8-486d-99af-ee42e909fb04
# group(chains, :k)
# group(chains, :μ)
#--------------------------------------------------------------------------------
plot(chains[["μ[1][1]", "μ[1][2]", "μ[2][1]", "μ[2][2]"]], colordim = :parameter, legend=true)
# ╔═╡ b95680a8-c045-4b03-9e65-66ed13bedf5d
describe(chains[["μ[1][1]", "μ[1][2]", "μ[2][1]", "μ[2][2]"]])
# ╔═╡ b5a94a85-4d62-45a8-b6fc-e53ab882c9df
# group(chains, :w)
# ╔═╡ 0cffe1f5-491b-4664-b051-db5bc60fcf4f
plot(chains[["w[1]", "w[2]"]])
# ╔═╡ b052e912-012e-43ac-a9a0-493bca4137a1
describe(chains[["w[1]", "w[2]"]])
# ╔═╡ 592152f0-407a-46bc-ae93-f8fc964f2fdf
plot(chains[["σ2"]])
# ╔═╡ 7451e9c1-3300-4cae-8e40-8aecab8f08e0
describe(chains[["σ2"]])
# ╔═╡ f4c03d6c-c283-497b-ae3f-a09ad033b630
begin
#--------------------------------------------------------------------------------
assignments = [mean(chains, "k[$i]") for i in 1:N]
#--------------------------------------------------------------------------------
μMean = Array{Float64}(undef, (D, K))
for kk in 1:K
μMean[:,kk] = [mean(chains, "μ[$kk][$i]") for i in 1:D]
end # for kk
# ===============================================================================
scatter(x[1, :], x[2, :]; label="data point", title=L"Data 3 (Post. Membership $k[i], w[kk]$ and $σ2$)", zcolor=assignments)
#--------------------------------------------------------------------------------
# plot of generative centroids
scatter!([(μ0[1, 1], μ0[2, 1]), (μ0[1, 2], μ0[2, 2])], color=:violetred4, markershape = :cross, label="generative centroid", markersize=8, markerstrokewidth = 3)
#--------------------------------------------------------------------------------
# plot of the posterior cluster centroids
scatter!([(μMean[1,1], μMean[2,1]), (μMean[1,2], μMean[2,2])], color=:violet, label="posterior cluster centroid", markersize=6)
#--------------------------------------------------------------------------------
end # begin
# ╔═╡ 0c9b7028-b77f-46dc-b3b6-79b7d5bee5aa
md"
---
##### References
- **Bishop, C.M.**; *Pattern Recognition and Machine Learning*, Heidelberg: Springer, 2006 (8th printing, 2009)
- **Turing.jl-team**; [Unsupervised Learning using Bayesian Mixture Models](https://turinglang.org/v0.30/tutorials/01-gaussian-mixture-model/); last visit 2024/03/09
"
# ╔═╡ 0e7be074-1d2e-40a3-a398-69ac9dc4fa7e
md"
====================================================================================
This is a **draft** under the Attribution-NonCommercial-ShareAlike 4.0 International **(CC BY-NC-SA 4.0)** license. Comments, improvement and issue reports are welcome: **claus.moebus(@)uol.de**
====================================================================================
"
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