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conditionals_dual.r
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conditionals_dual.r
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# Sampling functions
#
library(MASS)
library(mvtnorm)
source('ars_alpha.r')
source('ars_beta.r')
DEBUG <- FALSE
sample_z <- function(u, A, alpha, z, mu_ar, S_ar, mu_a0, R_a0, beta_a0, W_a0){
#
# Chinese Restaurant Process with auxiliary tables (Neal's algorithm 8)
#
m <- 3 # number of auxiliary tables to approximate the infinite number of empty tables
if(alpha==0) {m=0} # in case of no DP
n <- tabulate(z)
K <- length(unique(z))
D <- dim(A)[1]
R_a0_inv <- solve(R_a0)
W_a0_inv <- solve(W_a0)
# Compute likelihood for every table
logprobs <- rep(NA, K+m)
for(k in 1:K){
logp <- log(n[k]-as.numeric(z[u]==k))
logp <- logp + mvtnorm::dmvnorm(A[,u], mean=mu_ar[,k,drop=FALSE], sigma=solve(S_ar[,,k]), log=TRUE)
logprobs[k] <- logp
}
if(m>0){
# Create m auxiliary tables from the base distribution
aux.tables.mean <- t(mvrnorm(m, mu_a0, R_a0_inv))
aux.tables.covariance <- rWishart(m, max(beta_a0,D), W_a0/beta_a0)
# If last point in cluster, re-use its parameters for the auxiliary table so that
# it has some probability of staying
if (n[z[u]]==1){
aux.tables.mean[,1] <- mu_ar[,z[u], drop=FALSE]
aux.tables.covariance[,,1] <- S_ar[,,z[u]]
}
# Compute likelihood for auxilary table
for(k in 1:m){
logp <- log(alpha/m)
logp <- logp + mvtnorm::dmvnorm(A[,u], mean=aux.tables.mean[,k], sigma=as.matrix(aux.tables.covariance[,,k]), log=TRUE)
logprobs[K+k] <- logp
}
}
# normalize probabilities to avoid numerical underflow
probs <- logprobs-max(logprobs)
probs <- exp(probs)
probs <- probs/sum(probs)
# Choose table
chosen <- base::sample(1:(K+m), 1, prob=probs)
# Re-label tables
##########################################
# if auxiliary table is chosen, assign label K+i-th to this new table
if(chosen>K){
mu_ar[,K+1] <- aux.tables.mean[, chosen - K]
S_ar[,,K+1] <- solve(aux.tables.covariance[,, chosen - K])
z[u] <- K+1
}
else{
z[u] <- chosen
}
# if old cluster is empty, shift right tables to the left
# do it only if DP is used (m>0)
n <- tabulate(z)
if(any(n==0) && (m>0)){
empty <- which(n==0)
right <- (empty+1):(K+1)
mu_ar[,(empty:K)] <- mu_ar[,(empty+1):(K+1)]
S_ar[,,(empty:K)] <- S_ar[,,(empty+1):(K+1)]
z[z>empty] <- z[z>empty] - 1
}
if((max(z) != length(unique(z))) && (m>0)){
stop("Some cluster is not being used")
}
return(list(z=z,
mu_ar=mu_ar,
S_ar=S_ar))
}
sample_z_dual <- function(u, A, B, alpha, z,
mu_ar, S_ar, mu_a0, R_a0, beta_a0, W_a0,
mu_br, S_br, mu_b0, R_b0, beta_b0, W_b0){
#
# Chinese Restaurant Process with auxiliary tables (Neal's algorithm 8)
#
m <- 3 # number of auxiliary tables to approximate the infinite number of empty tables
if(alpha==0) {m=0} # in case of no DP
n <- tabulate(z)
K <- length(unique(z))
D.a <- dim(A)[1]
D.b <- dim(B)[1]
R_a0_inv <- solve(R_a0)
W_a0_inv <- solve(W_a0)
R_b0_inv <- solve(R_b0)
W_b0_inv <- solve(W_b0)
# Compute likelihood for every table
logprobs <- rep(NA, K+m)
for(k in 1:K){
logp <- log(n[k]-as.numeric(z[u]==k))
logp <- logp + mvtnorm::dmvnorm(A[,u], mean=mu_ar[,k,drop=FALSE], sigma=solve(S_ar[,,k]), log=TRUE)
logp <- logp + mvtnorm::dmvnorm(B[,u], mean=mu_br[,k,drop=FALSE], sigma=solve(S_br[,,k]), log=TRUE)
logprobs[k] <- logp
}
if(m>0){
# Create m auxiliary tables from the base distribution
aux.tables.mean.a <- t(mvrnorm(m, mu_a0, R_a0_inv))
aux.tables.covariance.a <- rWishart(m, max(beta_a0,D.a), W_a0/beta_a0)
aux.tables.mean.b <- t(mvrnorm(m, mu_b0, R_b0_inv))
aux.tables.covariance.b <- rWishart(m, max(beta_b0,D.b), W_b0/beta_b0)
# If last point in cluster, re-use its parameters for the auxiliary table so that
# it has some probability of staying
if (n[z[u]]==1){
aux.tables.mean.a[,1] <- mu_ar[,z[u], drop=FALSE]
aux.tables.covariance.a[,,1] <- S_ar[,,z[u]]
aux.tables.mean.b[,1] <- mu_br[,z[u], drop=FALSE]
aux.tables.covariance.b[,,1] <- S_br[,,z[u]]
}
# Compute likelihood fro auxiliary tables
for(k in 1:m){
logp <- log(alpha/m)
logp <- logp + mvtnorm::dmvnorm(A[,u], mean=aux.tables.mean.a[,k], sigma=as.matrix(aux.tables.covariance.a[,,k]), log=TRUE)
logp <- logp + mvtnorm::dmvnorm(B[,u], mean=aux.tables.mean.b[,k], sigma=as.matrix(aux.tables.covariance.b[,,k]), log=TRUE)
logprobs[K+k] <- logp
}
}
# normalize probabilities to avoid numerical underflow
probs <- logprobs-max(logprobs)
probs <- exp(probs)
probs <- probs/sum(probs)
# Choose table
chosen <- base::sample(1:(K+m), 1, prob=probs)
# Re-label tables
##########################################
# if auxiliary table is chosen, assign label K+i-th to this new table
if(chosen>K){
mu_ar[,K+1] <- aux.tables.mean.a[, chosen - K]
S_ar[,,K+1] <- solve(aux.tables.covariance.a[,, chosen - K])
mu_br[,K+1] <- aux.tables.mean.b[, chosen - K]
S_br[,,K+1] <- solve(aux.tables.covariance.b[,, chosen - K])
z[u] <- K+1
}
else{
z[u] <- chosen
}
# if old cluster is empty, shift right tables to the left
# do it only if DP is used (m>0)
n <- tabulate(z)
if(any(n==0) && (m>0)){
empty <- which(n==0)
right <- (empty+1):(K+1)
mu_ar[,(empty:K)] <- mu_ar[,(empty+1):(K+1)]
S_ar[,,(empty:K)] <- S_ar[,,(empty+1):(K+1)]
mu_br[,(empty:K)] <- mu_br[,(empty+1):(K+1)]
S_br[,,(empty:K)] <- S_br[,,(empty+1):(K+1)]
z[z>empty] <- z[z>empty] - 1
}
if((max(z) != length(unique(z))) && (m>0)){
stop("Some cluster is not being used")
}
return(list(z=z,
mu_ar=mu_ar,
S_ar=S_ar,
mu_br=mu_br,
S_br=S_br))
}
sample_b <- function(P, y, z, intercept, mu_ar, S_ar, s_y=10){
nthreads <- dim(P)[2]
P <- as.matrix(P)
Lambda_post <- diag(S_ar[,,z]) + P%*%diag(rep(s_y, nthreads))%*%t(P)
# Note: be aware of computationally singular matrices (use ginv if necessary)
Sigma_post <- solve(Lambda_post)
mu_post <- Sigma_post%*%(as.matrix(S_ar[z]*mu_ar[z]) + s_y*(P%*%as.matrix(y-intercept)))
return(t(mvrnorm(mu=mu_post, Sigma=Sigma_post)))
}
sample_intercept <- function(P, y, z, B, mu_ar, S_ar, s_y){
b <- t(B)
offsets <- (y-t(P)%*%b)
nthreads <- dim(P)[2]
P <- as.matrix(P)
lambda_post <- S_ar + nthreads*s_y
sigma_post <- solve(lambda_post)
mu_post <- sigma_post*(mu_ar*S_ar + sum(offsets)*s_y)
return(rnorm(1, mean=mu_post, sd=sqrt(sigma_post)))
}
sample_noise_inv <- function(P, y, B, variance_y){
# Sample the noise (its precision) inherent in threads length (y_t = p^t b + N(0, 1/s_y))
# params gamma(1, var_y
b <- t(B)
nthreads <- dim(P)[2]
gamma_dof_post <- nthreads + 1
gamma_s_post <- solve(variance_y + sum((y - t(P)%*%b)^2))
return(rgamma(1, gamma_dof_post/2, scale=2*gamma_s_post))
}
########################################################
# Component parameters
########################################################
sample_mu_ar <- function(A_r, S_ar, mu_a0, R_a0){
#
# Samples a mean for cluster r
# from its conditional probability
#
n <- dim(A_r)[2]
# If nobody in the cluster sample from prior
if (n == 0){
return(mvrnorm(1, mu_a0, solve(R_a0)))
}
ar_mean <- rowMeans(A_r)
Lambda_post <- R_a0 + n*S_ar
Sigma_post <- solve(Lambda_post)
mu_post <- Sigma_post %*% ((R_a0 %*% mu_a0) + n*(S_ar%*%ar_mean))
return(mvrnorm(1, mu_post, Sigma_post))
}
sample_S_ar <- function(A_r, mu_ar_k, beta_a0, W_a0){
#
# Sample attributes covariance matrix of cluster r
# from its conditional probability
#
n <- dim(A_r)[2]
D <- dim(A_r)[1]
# If nobody in the cluster sample from prior
if (n == 0){
# unfortunately wishart implementation does not accept dof > F-1
# but dof >= F
# This will only affect when the cluster is empty.
df <- max(beta_a0, D)
return(rWishart(1, df, solve(W_a0)/beta_a0))
}
# If only one point, scatter is just this product
if(n==1){
scatter_matrix <- (A_r-mu_ar_k)%*%t(A_r-mu_ar_k)
}
if(n > 1){
scatter_matrix <- cov(t(A_r-c(mu_ar_k)))*(dim(A_r)[2]-1)
}
wishart_dof_post <- beta_a0 + n
wishart_S_post <- solve(beta_a0*W_a0 + scatter_matrix)
return(rWishart(1, wishart_dof_post, wishart_S_post)[,,1])
}
########################################################
# Hyperparameters
########################################################
# Cluster means
#################
sample_mu_a0 <- function(Lambda_a, mu_a, mu_ar, R_a0){
#
# Samples mean of gaussian hyperprior placed over clusters centroids
#
K <- dim(mu_ar)[2]
mean_ar = rowMeans(mu_ar)
Lambda_post = Lambda_a + K*R_a0
Sigma_post = solve(Lambda_post)
mu_post = Sigma_post %*% (Lambda_a %*% mu_a) + K*(R_a0 %*% mean_ar)
return(mvrnorm(1, mu_post, Sigma_post))
}
sample_R_a0 <-function(Sigma_a, mu_ar, mu_a0){
#
# Samples precision of gaussian hyperprior placed over clusters centroids
#
K <- dim(mu_ar)[2]
D <- dim(mu_ar)[1]
# Can't use the trick cov(t(mu_ar-mu_a0))*(K-1) when there is only one cluster
scatter_matrix <- 0
for(i in 1:K){
scatter_matrix <- scatter_matrix + (mu_ar[,i]-mu_a0)%*%t(mu_ar[,i]-mu_a0)
}
wishart_dof_post = D + K
wishart_S_post = solve(D*Sigma_a + scatter_matrix)
return(as.matrix(rWishart(1, wishart_dof_post, wishart_S_post)[,,1]))
}
# Cluster covariances
#####################
sample_W_a0 <- function(Lambda_a, S_ar, beta_a0){
#
# Sample base covariance of attributes (hyperparameter)
#
K <- dim(S_ar)[3]
D <- dim(S_ar)[1]
# TODO: why apply does not work properly?
#scatter_matrix <- apply(S_ar, 3, sum)
scatter_matrix <- 0
for (i in 1:dim(S_ar)[3]){
scatter_matrix <- scatter_matrix + S_ar[,,i]
}
wishart_dof_post = D + K*beta_a0
wishart_S_post = solve(D * Lambda_a + beta_a0*scatter_matrix)
return(as.matrix(rWishart(1, wishart_dof_post, wishart_S_post)[,,1]))
}
sample_beta_a0 <- function(S, W, init=4){
# Sample the degrees of freedom of the Wishart
# given a scale matrix W and some observed precision matrices S
ars.sample_beta_a0(S, W, init=init)
# beta_a0 <- NULL
# attempt <- 1
# while( is.null(beta_a0) && attempt <= 3 ) {
# attempt <- attempt + 1
# try(
# beta_a0 <- ars.sample_beta_a0(S, W, init=init)
# )
# }
# return(beta_a0)
}