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CAVI_Bayesian_GMM.R
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# Using the CAVI algorithm on a (Bayesian) GMM example | |
library(mvtnorm) # for multivariate normal density | |
library(extraDistr) # for Categorical distribution | |
library(pracma) # for 2d-integral, sqrtm | |
library(matlib) # for solving linear equations | |
library(ggplot2) # for plotting | |
# Hyper Parameters | |
sigma = 3 # Variance of mu | |
K = 5 # No. of clusters | |
n = 1000 # No. of observations | |
############### | |
# 1D Gaussian # | |
############### | |
set.seed(247) | |
mu = rnorm(K, mean=0, sd=sigma) # Means | |
cs = rcat(n, rep(1/K, K)) # C's | |
x = rnorm(n, mean=mu[cs], sd=1) # X's | |
# plot the data | |
df = data.frame(x=x, mu=as.factor(cs)) | |
ggplot(df, aes(x=x, color=mu, fill=mu)) + geom_histogram(alpha=0.5) | |
# CAVI algorithm | |
mk = rnorm(K) # random start values | |
sk2 = rgamma(K, 5) # random (positive) start values | |
phis = rdirichlet(n, c(1,1,1,1,1)) # random phis | |
# rowSums(phis) # sanity check | |
ELBO = function(mk, sk2, phis) { | |
t = sk2+mk^2 | |
a = -(1/(2*sigma^2))*sum(t) | |
b = 2*sum(sweep(sweep(phis, MARGIN=2, mk, '*'), MARGIN=1, x, '*'))-0.5*sum(sweep(phis, MARGIN=2, t, '*')) | |
c = -0.5*sum(log(2*pi*sk2)) | |
d = sum(phis*log(phis)) | |
return(a+b+c+d) | |
} | |
iter = 30 | |
elbos = rep(NA, iter+1) | |
elbos[1] = ELBO(mk,sk2,phis) | |
for (i in 1:iter) { | |
phis.new = matrix(nrow=n, ncol=K) | |
for (j in 1:n) { | |
phis.new[j,] = exp(x[j]*mk-0.5*(sk2+mk^2)) | |
phis.new = phis.new/rowSums(phis.new) | |
} | |
phis = phis.new | |
mk.new = rep(NA, K) | |
sk2.new = rep(NA, K) | |
for (k in 1:K) { | |
sk2.new[k] = 1/(1/sigma^2+sum(phis[,k])) | |
mk.new[k] = sk2.new[k]*sum(phis[,k]*x) | |
} | |
sk2 = sk2.new | |
mk = mk.new | |
elbos[i+1] = ELBO(mk,sk2,phis) | |
cat("Iteration: ", i, "ELBO-diff: ", abs(elbos[i+1]-elbos[i]), "\n") | |
if (abs(elbos[i+1]-elbos[i])<0.1) break | |
} | |
ggplot(data=data.frame(Iter=seq(1,i,1), ELBO=elbos[1:i]), aes(x=Iter,y=ELBO)) + | |
geom_line(color="#2E9FDF") | |
# CAVI finds clusters center almost perfectly | |
mk | |
mu | |
# There is very little doubt over the centers | |
sk2 | |
ggplot(df, aes(x=x, color=mu, fill=mu)) + | |
geom_histogram(alpha=0.5) + | |
geom_vline(data=data.frame(x=mk), aes(xintercept=x, color=as.factor(c(2,4,1,3,5))), | |
linetype="dashed", size=1) # + | |
# geom_vline(data=data.frame(x=mk), aes(xintercept=x+2*sqrt(sk2), | |
# color=as.factor(c(2,4,1,3,5))), size=1, linetype="dashed") + | |
# geom_vline(data=data.frame(x=mk), aes(xintercept=x-2*sqrt(sk2), | |
# color=as.factor(c(2,4,1,3,5))), size=1, linetype="dashed") | |
############### | |
# 2D Gaussian # | |
############### | |
I = diag(c(1,1)) | |
mu = rmvnorm(K, mean=c(0,0), sigma=sigma^2*I) # Means | |
cs = rcat(n, rep(1/K, K)) # C's | |
x = mu[cs,]+rmvnorm(n, mean=c(0,0), sigma=I) # X's | |
# plot the data | |
df = data.frame(x=x, mu=as.factor(cs)) | |
ggplot(df, aes(x=x[,1], y=x[,2], color=mu, fill=mu)) + geom_point() | |
# CAVI algorithm | |
mk = rmvnorm(K, mean=c(0,0), sigma=I) # random start values | |
sk2 = rgamma(K, 5) # random (positive) start values | |
phis = rdirichlet(n, c(1,1,1,1,1)) # random phis | |
# rowSums(phis) # sanity check | |
plotClusters = function() { | |
ggplot(df, aes(x=x[,1], y=x[,2], color=mu)) + | |
geom_point(alpha=0.3) + | |
geom_point(data=data.frame(x1=mk[,1], x2=mk[,2], mu=as.factor(c(2,1,5,3,4))), | |
aes(x=x1, y=x2, color=mu), size = 3, colour = "black") + | |
geom_point(data=data.frame(x1=mk[,1], x2=mk[,2], mu=as.factor(c(2,1,5,3,4))), | |
aes(x=x1, y=x2, color=mu), size=2) | |
} | |
plotClusters() | |
iter = 30 | |
for (i in 1:iter) { | |
phis.new = matrix(nrow=n, ncol=K) | |
for (j in 1:n) { | |
for (k in 1:K) { | |
phis.new[j,k] = exp(t(x[j,])%*%mk[k,]-0.5*(2*sk2[k]+t(mk[k,])%*%mk[k,])) | |
} | |
phis.new = phis.new/rowSums(phis.new) | |
} | |
phis = phis.new | |
mk.new = matrix(rep(NA, K*2), ncol=2) | |
sk2.new = rep(NA, K) | |
for (k in 1:K) { | |
sk2.new[k] = 1/(1/sigma^2+sum(phis[,k])) | |
mk.new[k,] = sk2.new[k]*colSums(phis[,k]*x) | |
} | |
sk2 = sk2.new | |
mk = mk.new | |
} | |
# CAVI finds clusters center almost perfectly | |
mk | |
mu | |
plotClusters() | |
# D. Refaeli © |
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