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bayesian_cure_rate_model.R
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bayesian_cure_rate_model.R
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library("Rcpp")
library("RcppArmadillo")
sourceCpp("cll_reparameterization_fix.cpp")
library('coda')
library('doParallel')
library('foreach')
#myData must contain the columns "Covariate1", "Covariate2" and "Y"
#We also need D1, D0, and I_sim
#dercomp takes on {1,2,3,4,5,6,7}->{g,lambda,a1,a2,b0,b1,b2} depending on which derivative must be computed
derv_complete_log_likelihood_fotis<-function(g, lambda, a1, a2, b0, b1, b2, I_sim){
ct = exp(exp(-1))
x1<-myData[,"Covariate1"]
x2<-myData[,"Covariate2"]
tau<-myData[,"Y"]
theta1<-exp(b0+b1*x1 + b2*x2)
p00 <- (1 + g*theta1*ct^(g*theta1))^(-1/g)
FW<- 1 - exp(-(a1*tau)^a2)
fW<- (a2/tau)*exp(-(a1*tau)^a2)*(a1*tau)^a2
Sp <- (1 + g *theta1* ct^(g*theta1) * FW^lambda)^(-1/g)
fp <- lambda*ct^(g*theta1)*theta1*(1 + g *theta1* ct^(g*theta1) * FW^lambda)^(-1/g-1)*FW^(lambda-1)*fW
fp0 <- (1 + g * theta1*ct^(g*theta1) * FW^lambda)^(-1/g-1)
DFa1 <- a2*tau*((a1*tau)^(a2-1))*exp(-(a1*tau)^a2)
DFa2 <- log(a1*tau)*((a1*tau)^a2)*exp(-(a1*tau)^a2)
Dfa1 <- ((a1*tau)^(a2-1))*a2^2*exp(-(a1*tau)^a2)*(1-(a1*tau)^a2)
Dfa2 <- (1/tau)*exp(-(a1*tau)^a2)*(a1*tau)^a2*(1+a2*log(a1*tau)-a2*(a1*tau)^a2*log(a1*tau))
Dp0g<- p00*(1/g)*(log(1 + g*theta1*ct^(g*theta1))/g-(theta1*ct^(g*theta1))*(1+g*exp(-1)*theta1)/(1 + g*theta1*ct^(theta1*g)))
Dp0b0<- -p00*ct^(g*theta1)*((1+g*exp(-1)*theta1)/(1 + g*theta1*ct^(theta1*g)))*theta1
Dp0b1<- -p00*ct^(g*theta1)*((1+g*exp(-1)*theta1)/(1 + g*theta1*ct^(theta1*g)))*theta1*x1
Dp0b2<- -p00*ct^(g*theta1)*((1+g*exp(-1)*theta1)/(1 + g*theta1*ct^(theta1*g)))*theta1*x2
Dlogfp0lam <- (-1/g-1)*(g*theta1*ct^(g*theta1)*FW^lambda*log(FW))/(1 + g *theta1* ct^(g*theta1) * FW^lambda)
Dlogfp0gam <-(1/g^2)*log((1 + g *theta1* ct^(g*theta1) * FW^lambda))+(-1/g-1)*theta1* ct^(g*theta1) * FW^lambda*(1+g*exp(-1)*theta1)/(1 + g*theta1*ct^(theta1*g)*FW^lambda)
Dlogfp0a1 <- (-1/g-1)*(lambda*g*theta1*ct^(g*theta1)*FW^(lambda-1)*DFa1)/(1+g*theta1*ct^(g*theta1)*FW^(lambda))
Dlogfp0a2 <- (-1/g-1)*(lambda*g*theta1*ct^(g*theta1)*FW^(lambda-1)*DFa2)/(1+g*theta1*ct^(g*theta1)*FW^(lambda))
Dlogfp0theta <- (-1/g-1)*(g*ct^(g*theta1)*FW^(lambda))*(1+g*exp(-1)*theta1)/(1+g*theta1*ct^(g*theta1)*FW^(lambda))
Dfplam <- fp*(lambda^(-1)+log(FW)+Dlogfp0lam)
Dfpg <- fp*(theta1*exp(-1)+Dlogfp0gam)
Dfpa1 <- fp*((lambda-1)*DFa1/FW+Dfa1/fW+Dlogfp0a1)
Dfpa2 <- fp*((lambda-1)*DFa2/FW+Dfa2/fW+Dlogfp0a2)
Dfpb0 <- fp*(theta1^(-1)+g*exp(-1)+Dlogfp0theta)*theta1
Dfpb1 <- fp*(theta1^(-1)+g*exp(-1)+Dlogfp0theta)*theta1*x1
Dfpb2 <- fp*(theta1^(-1)+g*exp(-1)+Dlogfp0theta)*theta1*x2
DSpg <- Sp*
(1/g^2*log((1 + g *theta1* ct^(g*theta1) * FW^lambda))-((theta1* ct^(g*theta1) * FW^lambda)/(g))*(1+g*theta1*exp(-1))/((1 + g *theta1* ct^(g*theta1) * FW^lambda)))
DSplam <- -fp0*theta1*ct^(g*theta1)*FW^lambda*log(FW)
DSpa1 <- -fp0*theta1*ct^(g*theta1)*lambda*FW^(lambda-1)*(DFa1)
DSpa2 <- -fp0*theta1*ct^(g*theta1)*lambda*FW^(lambda-1)*(DFa2)
DSpb0 <- -fp0*ct^(g*theta1)*FW^(lambda)*(1+g*theta1*exp(-1))*theta1
DSpb1 <- -fp0*ct^(g*theta1)*FW^(lambda)*(1+g*theta1*exp(-1))*theta1*x1
DSpb2 <- -fp0*ct^(g*theta1)*FW^(lambda)*(1+g*theta1*exp(-1))*theta1*x2
parg1c<-fp[D1]
parg2c<-p00[D0]
parg3c<-Sp[D0]
parg11<-Dfpg[D1]
parg21<-Dp0g[D0]
parg31<-DSpg[D0]
parg12<-Dfplam[D1]
parg22<-0
parg32<-DSplam[D0]
parg13<-Dfpa1[D1]
parg23<-0
parg33<-DSpa1[D0]
parg14<-Dfpa2[D1]
parg24<-0
parg34<-DSpa2[D0]
parg15<-Dfpb0[D1]
parg25<-Dp0b0[D0]
parg35<-DSpb0[D0]
parg16<-Dfpb1[D1]
parg26<-Dp0b1[D0]
parg36<-DSpb1[D0]
parg17<-Dfpb2[D1]
parg27<-Dp0b2[D0]
parg37<-DSpb2[D0]
com11<-sum(parg11/parg1c)
com21<-sum(parg21/parg2c)
com31<-sum(I_sim[D0]*((parg31*parg2c-parg3c*parg21)/(parg2c*(parg3c-parg2c))))
com12<-sum(parg12/parg1c)
com22<-sum(parg22/parg2c)
com32<-sum(I_sim[D0]*((parg32*parg2c-parg3c*parg22)/(parg2c*(parg3c-parg2c))))
com13<-sum(parg13/parg1c)
com23<-sum(parg23/parg2c)
com33<-sum(I_sim[D0]*((parg33*parg2c-parg3c*parg23)/(parg2c*(parg3c-parg2c))))
com14<-sum(parg14/parg1c)
com24<-sum(parg24/parg2c)
com34<-sum(I_sim[D0]*((parg34*parg2c-parg3c*parg24)/(parg2c*(parg3c-parg2c))))
com15<-sum(parg15/parg1c)
com25<-sum(parg25/parg2c)
com35<-sum(I_sim[D0]*((parg35*parg2c-parg3c*parg25)/(parg2c*(parg3c-parg2c))))
com16<-sum(parg16/parg1c)
com26<-sum(parg26/parg2c)
com36<-sum(I_sim[D0]*((parg36*parg2c-parg3c*parg26)/(parg2c*(parg3c-parg2c))))
com17<-sum(parg17/parg1c)
com27<-sum(parg27/parg2c)
com37<-sum(I_sim[D0]*((parg37*parg2c-parg3c*parg27)/(parg2c*(parg3c-parg2c))))
return(c(com11+com21+com31,com12+com22+com32,com13+com23+com33,com14+com24+com34,com15+com25+com35,com16+com26+com36,com17+com27+com37))}
#################################################
#########################test####################
log_prior_gradient <- function(g, lambda, a1, a2, b0, b1, b2 ,mu_g, s2_g, a_l, b_l, a_1, b_1, a_2, b_2, mu_b, Sigma){
log_gradient <- numeric(7)
log_gradient[1] <- if(g < 0){
(mu_g - 1)/g + s2_g
}else{
(mu_g - 1)/g - s2_g
}
log_gradient[2] <- -(a_l+1)/lambda + b_l/(lambda^2)
log_gradient[3] <- -(a_1+1)/a1 + b_1/(a1^2)
log_gradient[4] <- -(a_2+1)/a2 + b_2/(a2^2)
log_gradient[5:7] <- (mu_b - c(b0,b1,b2))/diag(Sigma)
return(log_gradient)
}
cure_rate_metropolis_hastings_cpp <- function( myData, m, alpha = 1,
mu_g = 0.2, s2_g = 0.1, a_l = 2.001, b_l = 1,
a_1 = 2.001, b_1 = 1, a_2 = 2.001, b_2 = 1,
mu_b = rep(0,3), Sigma = 100*diag(3),
g_prop_sd = 0.045,
lambda_prop_scale = 0.03,
a1_prop_scale = 0.2,
a2_prop_scale = 0.03,
b_prop_sd = rep(0.022, 3),
initialValues = NULL,
plot = TRUE,
verbose = TRUE,
tau_mala = 0.000015, mala = 0.33
){
# input data: myData
# this should be a matrix with the following columns:
# "Y", "Censoring_status", "Covariate1", "Covariate2"
# mala: probability of performing a mala move
# otherwise, usual MH update is performed
# m: denotes the total number of MCMC iterations (arbitrary for now)
# prior parameters
# gamma ~ 0.5G(a_g, b_g) + 0.5NG(a_g, b_g)
# NOTE mu_g = a_g and s2_g = b_g
# lambda ~ IG(a_l, b_l)
# a1 ~ IG(a_1, b_1)
# a2 ~ IG(a_2, b_2)
# b = (b0, b1, b2) ~ N(mu_b, Sigma)
# proposal parameters
# g_prop_sd <- 0.4
# lambda_prop_scale <- 0.1
# a1_prop_scale <- 0.2
# a2_prop_scale <- 0.1
# b_prop_sd = rep(0.05, 3)
# initialValues: should be a list with entries named as g_mcmc, b0_mcmc, b1_mcmc, b2_mcmc, lambda_mcmc, a1_mcmc, a2_mcmc
# and if it is NULL then random starting values are generated
n <- dim(myData)[1]
X1 <- myData[,3]
X2 <- myData[,4]
ct = exp(exp(-1))
D1 <- which(myData[,"Censoring_status"] == 1) # fixed
D0 <- which(myData[,"Censoring_status"] == 0) # fixed
g_mcmc <- b0_mcmc <- b1_mcmc <- b2_mcmc <- lambda_mcmc <- a1_mcmc <- a2_mcmc <- numeric(m)
cllValues <- numeric(m)
lpd <- numeric(m)
I_sim_values_D0 <- matrix(data = NA, ncol = length(D0), nrow = m)
# initial values (random)
if(is.null(initialValues)){
g = rnorm(1, sd = 2)
b0 = rnorm(1, sd = 2)
b1 = rnorm(1, sd = 2)
b2 = rnorm(1, sd = 2)
lambda = rgamma(1, shape = 1, rate = 1)
a1 = rgamma(1, shape = 1, rate = 1)
a2 = rgamma(1, shape = 1, rate = 1)
}else{
g = initialValues[['g_mcmc']]
b0 = initialValues[['b0_mcmc']]
b1 = initialValues[['b1_mcmc']]
b2 = initialValues[['b2_mcmc']]
lambda = initialValues[['lambda_mcmc']]
a1 = initialValues[['a1_mcmc']]
a2 = initialValues[['a2_mcmc']]
}
b <- c(b0,b1,b2)
g_mcmc[1] <- g
b0_mcmc[1] <- b0
b1_mcmc[1] <- b1
b2_mcmc[1] <- b2
a1_mcmc[1] <- a1
a2_mcmc[1] <- a2
lambda_mcmc[1] <- lambda
# latent binary variables (I = 1 => susceptible, I = 0 => cured)
I_sim <- rep(1, n)
# for i in D1: I_i = 1
if(is.null(initialValues)){
susceptible_prob <- runif(n)
I_sim[D0] <- rbinom(n = length(D0), size = 1, prob = susceptible_prob[D0])
}else{
I_sim[D0] <- initialValues[["I_sim_D0"]]
}
I_sim_values_D0[1, ] <- I_sim[D0]
cll <- complete_log_likelihood(myData = myData, g = g, lambda = lambda, a1 = a1, a2 = a2, b0 = b0, b1 = b1, b2 = b2, I_sim = I_sim, alpha = alpha)
cllValues[1] <- cll$cll
# print("0")
# print(cllValues[1])
# print(c(g, lambda, a1, a2, b0, b1, b2))
# a: shape, b: scale
log_inv_gamma_kernel <- function(x, a, b){
if(min(c(x,a,b)) < 0){
stop("input should be positive")
}
return(-b/x - (a+1) * log(x))
}
# print(c(g, lambda, a1, a2, b0, b1, b2))
# log_0.5 <- log(0.5)
log_gamma_mix_kernel <- function(g, a, b){
return((a-1)*log(abs(g)) - b*abs(g))
# if(g < 0){
# return((a-1)*log(-g) + b*g)
# }else{
# return((a-1)*log(g) - b*g)
# }
}
log_prior_density <- -alpha*log_gamma_mix_kernel(g, mu_g, s2_g) +
alpha*log_inv_gamma_kernel(lambda, a_l, b_l) +
alpha*log_inv_gamma_kernel(a1, a_1, b_1) +
alpha*log_inv_gamma_kernel(a2, a_2, b_2) -
alpha*0.5 * mahalanobis(b, mu_b, Sigma)^2
lpd[1] <- log_prior_density
mala_accept_rate <- 0
g_accept_rate <- lambda_accept_rate <- a1_accept_rate <- a2_accept_rate <- b_accept_rate <- 0
if(plot){
dev.new(width=18, height=8, unit="in")
}
######################################
# MCMC sampler
######################################
if(m > 1){
mh_iter <- 0
mala_iter <- 0
for(iter in 2:m){
joe <- runif(1)
if(joe < 1 - mala){
#------------------------------------------------
# Metropolis-Hastings proposal for gamma: normal proposal distribution
#------------------------------------------------
g_prop <- rnorm(1, mean = g, sd = g_prop_sd)
cll_prop <- complete_log_likelihood(myData = myData, g = g_prop, lambda = lambda, a1 = a1,
a2 = a2, b0 = b0, b1 = b1, b2 = b2, I_sim = I_sim, alpha = alpha)
cll_diff <- cll_prop$cll - cll$cll
# print("1")
# print(cll_diff)
log_prior_diff <- alpha*diff(log_gamma_mix_kernel(c(g, g_prop), mu_g, s2_g))
log_proposal_ratio <- 0
mh_ar <- cll_diff + log_prior_diff + log_proposal_ratio
# print("gamma")
# print(cll_prop)
if(is.na(mh_ar)==F){
if(log(runif(1)) < mh_ar){
g = g_prop
cll <- cll_prop
g_accept_rate <- g_accept_rate + 1
}
}
g_mcmc[iter] <- g
#------------------------------------------------
# Metropolis-Hastings proposal for lambda: lognormal proposal distribution
#------------------------------------------------
lambda_prop = rlnorm(1, meanlog = log(lambda), sdlog = lambda_prop_scale)
cll_prop <- complete_log_likelihood(myData = myData, g = g, lambda = lambda_prop, a1 = a1, a2 = a2,
b0 = b0, b1 = b1, b2 = b2, I_sim = I_sim, alpha = alpha)
cll_diff <- cll_prop$cll - cll$cll
# print("2")
# print(cll_diff)
log_prior_diff <- alpha*diff(log_inv_gamma_kernel(c(lambda, lambda_prop), a_l, b_l))
log_proposal_ratio <- log(lambda_prop) - log(lambda)
mh_ar <- cll_diff + log_prior_diff + log_proposal_ratio
# print("lambda")
# print(cll_prop)
if(is.na(mh_ar)==F){
if(log(runif(1)) < mh_ar){
lambda = lambda_prop
cll <- cll_prop
lambda_accept_rate <- lambda_accept_rate + 1
}
}
lambda_mcmc[iter] <- lambda
#------------------------------------------------
# Metropolis-Hastings proposal for a1: lognormal proposal distribution
#------------------------------------------------
a1_prop = rlnorm(1, meanlog = log(a1), sdlog = a1_prop_scale)
cll_prop <- complete_log_likelihood(myData = myData, g = g, lambda = lambda, a1 = a1_prop, a2 = a2,
b0 = b0, b1 = b1, b2 = b2, I_sim = I_sim, alpha = alpha)
cll_diff <- cll_prop$cll - cll$cll
# print("3")
# print(cll_diff)
log_prior_diff <- alpha*diff(log_inv_gamma_kernel(c(a1, a1_prop), a_1, b_1))
log_proposal_ratio <- log(a1_prop) - log(a1)
mh_ar <- cll_diff + log_prior_diff + log_proposal_ratio
# print("a1")
# print(cll_prop)
if(is.na(mh_ar)==F){
if(log(runif(1)) < mh_ar){
a1 = a1_prop
cll <- cll_prop
a1_accept_rate <- a1_accept_rate + 1
}
}
a1_mcmc[iter] <- a1
#------------------------------------------------
# Metropolis-Hastings proposal for a2: lognormal proposal distribution
#------------------------------------------------
a2_prop = rlnorm(1, meanlog = log(a2), sdlog = a2_prop_scale)
cll_prop <- complete_log_likelihood(myData = myData, g = g, lambda = lambda, a1 = a1, a2 = a2_prop,
b0 = b0, b1 = b1, b2 = b2, I_sim = I_sim, alpha = alpha)
cll_diff <- cll_prop$cll - cll$cll
# print("4")
# print(cll_diff)
log_prior_diff <- alpha*diff(log_inv_gamma_kernel(c(a2, a2_prop), a_2, b_2))
log_proposal_ratio <- log(a2_prop) - log(a2)
mh_ar <- cll_diff + log_prior_diff + log_proposal_ratio
# print("a2")
# print(cll_prop)
if(is.na(mh_ar)==F){
if(log(runif(1)) < mh_ar){
a2 = a2_prop
cll <- cll_prop
a2_accept_rate <- a2_accept_rate + 1
}
}
a2_mcmc[iter] <- a2
#------------------------------------------------
# Metropolis-Hastings proposal for b = (b0,b1,b2): multivariate normal proposal distribution
#------------------------------------------------
b_prop <- b + b_prop_sd*rnorm(3)
cll_prop <- complete_log_likelihood(myData = myData, g = g, lambda = lambda, a1 = a1, a2 = a2,
b0 = b_prop[1], b1 = b_prop[2], b2 = b_prop[3], I_sim = I_sim, alpha = alpha)
cll_diff <- cll_prop$cll - cll$cll
# print("5")
# print(cll_diff)
log_prior_diff <- -alpha*0.5 * mahalanobis(b_prop, mu_b, Sigma)^2 + alpha*0.5 * mahalanobis(b, mu_b, Sigma)^2
log_proposal_ratio <- 0
mh_ar <- cll_diff + log_prior_diff + log_proposal_ratio
# print("b")
# print(cll_prop)
if(is.na(mh_ar)==F){
if(log(runif(1)) < mh_ar){
b = b_prop
b0 <- b[1]
b1 <- b[2]
b2 <- b[3]
cll <- cll_prop
b_accept_rate <- b_accept_rate + 1
}
}
b0_mcmc[iter] <- b[1]
b1_mcmc[iter] <- b[2]
b2_mcmc[iter] <- b[3]
mh_iter <- mh_iter + 1
}else{
theta_previous <- c(g, lambda, a1, a2, b0, b1, b2)
gradient_vec <- derv_complete_log_likelihood_fotis(
theta_previous[1],theta_previous[2],theta_previous[3],
theta_previous[4],theta_previous[5],theta_previous[6],
theta_previous[7],I_sim)
#c(derv_complete_log_likelihood_fotis(theta_previous[1],theta_previous[2],theta_previous[3],
# theta_previous[4],theta_previous[5],theta_previous[6],theta_previous[7],I_sim,1),
# derv_complete_log_likelihood_fotis(theta_previous[1],theta_previous[2],theta_previous[3],
# theta_previous[4],theta_previous[5],theta_previous[6],theta_previous[7],I_sim,2),
# derv_complete_log_likelihood_fotis(theta_previous[1],theta_previous[2],theta_previous[3],
# theta_previous[4],theta_previous[5],theta_previous[6],theta_previous[7],I_sim,3),
# derv_complete_log_likelihood_fotis(theta_previous[1],theta_previous[2],theta_previous[3],
# theta_previous[4],theta_previous[5],theta_previous[6],theta_previous[7],I_sim,4),
# derv_complete_log_likelihood_fotis(theta_previous[1],theta_previous[2],theta_previous[3],
# theta_previous[4],theta_previous[5],theta_previous[6],theta_previous[7],I_sim,5),
# derv_complete_log_likelihood_fotis(theta_previous[1],theta_previous[2],theta_previous[3],
# theta_previous[4],theta_previous[5],theta_previous[6],theta_previous[7],I_sim,6),
# derv_complete_log_likelihood_fotis(theta_previous[1],theta_previous[2],theta_previous[3],
# theta_previous[4],theta_previous[5],theta_previous[6],theta_previous[7],I_sim,7)
# )
lpg <- log_prior_gradient(g, lambda, a1, a2, b0, b1, b2 , mu_g, s2_g, a_l, b_l, a_1, b_1, a_2, b_2, mu_b, Sigma)
gradient_vec <- alpha*gradient_vec + alpha*lpg
mean_prop <- theta_previous + tau_mala*gradient_vec
sd_prop <- sqrt(2*tau_mala)
theta_prop <- rnorm(7, mean_prop, sd_prop)
rejectedMove = FALSE
if(sum(is.na(theta_prop))){rejectedMove = TRUE}else{
if(min(theta_prop[2:4]) < 0){rejectedMove = TRUE}}
if(rejectedMove == FALSE){
log_prop_density <- sum(dnorm(theta_prop, mean_prop, sd = sd_prop, log = TRUE))
gradient_vec_prop <- derv_complete_log_likelihood_fotis(
theta_prop[1],theta_prop[2],theta_prop[3],
theta_prop[4],theta_prop[5],theta_prop[6],
theta_prop[7],I_sim)
# c(derv_complete_log_likelihood_fotis(theta_prop[1],theta_prop[2],theta_prop[3],
# theta_prop[4],theta_prop[5],theta_prop[6],theta_prop[7],I_sim,1),
# derv_complete_log_likelihood_fotis(theta_prop[1],theta_prop[2],theta_prop[3],
# theta_prop[4],theta_prop[5],theta_prop[6],theta_prop[7],I_sim,2),
# derv_complete_log_likelihood_fotis(theta_prop[1],theta_prop[2],theta_prop[3],
# theta_prop[4],theta_prop[5],theta_prop[6],theta_prop[7],I_sim,3),
# derv_complete_log_likelihood_fotis(theta_prop[1],theta_prop[2],theta_prop[3],
# theta_prop[4],theta_prop[5],theta_prop[6],theta_prop[7],I_sim,4),
# derv_complete_log_likelihood_fotis(theta_prop[1],theta_prop[2],theta_prop[3],
# theta_prop[4],theta_prop[5],theta_prop[6],theta_prop[7],I_sim,5),
# derv_complete_log_likelihood_fotis(theta_prop[1],theta_prop[2],theta_prop[3],
# theta_prop[4],theta_prop[5],theta_prop[6],theta_prop[7],I_sim,6),
# derv_complete_log_likelihood_fotis(theta_prop[1],theta_prop[2],theta_prop[3],
# theta_prop[4],theta_prop[5],theta_prop[6],theta_prop[7],I_sim,7)
# )
lpg <- log_prior_gradient(theta_prop[1],theta_prop[2],theta_prop[3],
theta_prop[4],theta_prop[5],theta_prop[6],theta_prop[7] ,
mu_g, s2_g, a_l, b_l, a_1, b_1, a_2, b_2, mu_b, Sigma)
gradient_vec_prop <- alpha*gradient_vec_prop + alpha*lpg
mean_prop_rev <- theta_prop + tau_mala * gradient_vec_prop
log_prop_density_reverse <- sum(dnorm(theta_previous, mean_prop_rev, sd = sd_prop, log = TRUE))
cll_prop <- complete_log_likelihood(myData = myData, g = theta_prop[1], lambda = theta_prop[2],
a1 = theta_prop[3], a2 = theta_prop[4], b0 = theta_prop[5], b1 = theta_prop[6],
b2 = theta_prop[7], I_sim = I_sim, alpha = alpha)
cll_diff <- cll_prop$cll - cll$cll
log_proposal_ratio <- log_prop_density_reverse - log_prop_density
log_prior_diff <- diff(dnorm(c(g, theta_prop[1]), mean = mu_g, sd = sqrt(s2_g), log = TRUE))
log_prior_diff <- log_prior_diff + diff(log_inv_gamma_kernel(c(lambda, theta_prop[2]), a_l, b_l))
log_prior_diff <- log_prior_diff + diff(log_inv_gamma_kernel(c(a1, theta_prop[3]), a_1, b_1))
log_prior_diff <- log_prior_diff + diff(log_inv_gamma_kernel(c(a2, theta_prop[4]), a_2, b_2))
b_prop <- theta_prop[-(1:4)]
log_prior_diff <- log_prior_diff -0.5 * mahalanobis(b_prop, mu_b, Sigma)^2 + 0.5 * mahalanobis(b, mu_b, Sigma)^2
mh_ar <- cll_diff + alpha*log_prior_diff + log_proposal_ratio
if(is.na(mh_ar)==F){
if(log(runif(1)) < mh_ar){
g = theta_prop[1]
lambda = theta_prop[2]
a1 = theta_prop[3]
a2 = theta_prop[4]
b = b_prop
b0 <- b[1]
b1 <- b[2]
b2 <- b[3]
cll <- cll_prop
mala_accept_rate <- mala_accept_rate + 1
}
}
}
g_mcmc[iter] <- g
lambda_mcmc[iter] <- lambda
a1_mcmc[iter] <- a1
a2_mcmc[iter] <- a2
b0_mcmc[iter] <- b[1]
b1_mcmc[iter] <- b[2]
b2_mcmc[iter] <- b[3]
mala_iter <- mala_iter + 1
}
#--------------------------------------------
# Gibbs step for latent variables
#--------------------------------------------
logS <- cll$logS
logP0 <- cll$logP0
# print(cll)
susceptible_prob <- 1/(1 + (exp(logS - logP0) - 1)^{-alpha}) #
na_ind1 <- which(is.na(susceptible_prob))
na_ind2 <- which(logS - logP0 <= 0)
na_ind <- union(na_ind1, na_ind2)
if(length(na_ind) > 0){
# print(length(na_ind))
# print(logS[na_ind]-logP0[na_ind])
susceptible_prob[na_ind] <- 1/(1 + (1e-6)^{-alpha})
}
# latent binary variables (I = 1 => susceptible, I = 0 => cured)
I_sim <- rep(1, n)
# for i in D1: I_i = 1
I_sim[D0] <- rbinom(n = length(D0), size = 1, prob = susceptible_prob[D0])
I_sim_values_D0[iter, ] <- I_sim[D0]
cll <- complete_log_likelihood(myData = myData, g = g, lambda = lambda, a1 = a1, a2 = a2, b0 = b0, b1 = b1, b2 = b2, I_sim = I_sim, alpha = alpha)
# print("6")
# print(cll$cll)
log_prior_density <- alpha*log_gamma_mix_kernel(g,mu_g, s2_g) +
alpha*log_inv_gamma_kernel(lambda, a_l, b_l) +
alpha*log_inv_gamma_kernel(a1, a_1, b_1) +
alpha*log_inv_gamma_kernel(a2, a_2, b_2) -
alpha*0.5 * mahalanobis(b, mu_b, Sigma)^2
cllValues[iter] <- cll$cll
lpd[iter] <- log_prior_density
if(iter %% 10000 == 0 & verbose){
cat(paste0("* iteration = ", iter, ". Current ergodic means (after discarding 30% of iterations):"), "\n")
burn <- floor(0.3*iter)
ergMean <- c(mean(g_mcmc[(burn+1):iter]), mean(lambda_mcmc[(burn+1):iter]),
mean(a1_mcmc[(burn+1):iter]), mean(a2_mcmc[(burn+1):iter]),
mean(b0_mcmc[(burn+1):iter]), mean(b1_mcmc[(burn+1):iter]), mean(b2_mcmc[(burn+1):iter]))
names(ergMean) <- c("gamma", "lambda", "a1", "a2", "b0", "b1", "b2")
print(round(ergMean,2))
cat(paste0(" g_accept_rate = ", round(100*g_accept_rate/mh_iter, 2), "%. "), "\n")
cat(paste0(" lambda_accept_rate = ", round(100*lambda_accept_rate/mh_iter, 2), "%. "), "\n")
cat(paste0(" a1_accept_rate = ", round(100*a1_accept_rate/mh_iter, 2), "%. "), "\n")
cat(paste0(" a2_accept_rate = ", round(100*a2_accept_rate/mh_iter, 2), "%. "), "\n")
cat(paste0(" b_accept_rate = ", round(100*b_accept_rate/mh_iter, 2), "%. "), "\n")
cat(paste0(" mala_accept_rate = ", round(100*mala_accept_rate/mala_iter, 2), "%. "), "\n")
cat(paste0("\n"))
if(plot){
par(mfrow = c(2,3))
plot(g_mcmc[1:iter], type = "l", xlab = "MCMC iteration", ylab = bquote(gamma))
plot(lambda_mcmc[1:iter], type = "l", xlab = "MCMC iteration", ylab = bquote(lambda))
plot(a1_mcmc[1:iter], type = "l", xlab = "MCMC iteration", ylab = bquote(alpha[1]))
plot(a2_mcmc[1:iter], type = "l", xlab = "MCMC iteration", ylab = bquote(alpha[2]))
matplot(cbind(b0_mcmc[1:iter], b1_mcmc[1:iter], b2_mcmc[1:iter]), type = "l", xlab = "MCMC iteration", ylab = 'regression coefficients')
lText <- c("b0", "b1", "b2")
legend("topright", lText, col = 1:3, lty = 1:3)
plot(cllValues[1:iter], type = "l", xlab = "MCMC iteration", ylab = "complete log-likelihood")
}
}
}
result <- vector("list", length = 5)
mcmc_sample <- as.mcmc(cbind(g_mcmc, lambda_mcmc, a1_mcmc, a2_mcmc, b0_mcmc, b1_mcmc, b2_mcmc))
latent_status_D0 <- I_sim_values_D0
colnames(latent_status_D0) <- D0
complete_log_likelihood <- cllValues
result[[1]] <- mcmc_sample
result[[2]] <- complete_log_likelihood
result[[3]] <- c(g_accept_rate/mh_iter, lambda_accept_rate/mh_iter,
a1_accept_rate/mh_iter, a2_accept_rate/mh_iter, b_accept_rate/mh_iter, mala_accept_rate/mala_iter)
result[[4]] <- latent_status_D0
result[[5]] <- lpd
names(result[[3]]) <- c("gamma", "lambda", "a1", "a2", "betas","mala")
names(result) <- c("mcmc_sample", "complete_log_likelihood",
"acceptance_rates", "latent_status_censored", "log_prior_density")
}else{
result <- vector("list", length = 5)
complete_log_likelihood <- cllValues
result[[2]] <- complete_log_likelihood
result[[5]] <- lpd
names(result) <- c("mcmc_sample", "complete_log_likelihood",
"acceptance_rates", "latent_status_censored", "log_prior_density")
}
return(result)
}
cure_rate_MC3 <- function( myData, nChains = 16,
mcmc_cycles = 15000,
alpha = NULL,
nCores = 8,
sweep = 10,
mu_g = 0.2, s2_g = 0.1, a_l = 2.001, b_l = 1,
a_1 = 2.001, b_1 = 1, a_2 = 2.001, b_2 = 1,
mu_b = rep(0,3), Sigma = 100*diag(3),
g_prop_sd = 0.045,
lambda_prop_scale = 0.03,
a1_prop_scale = 0.2,
a2_prop_scale = 0.03,
b_prop_sd = rep(0.022, 3),
initialValues = NULL,
plot = TRUE,
adjust_scales = TRUE,
adjust_alpha = FALSE,
verbose = TRUE, tau_mala = 0.000015, mala = 0.33
){
if(nChains < 2){stop("at least two chains should be considered.")}
if(is.null(alpha)){
alpha <- 1/1.001^{(1:nChains)^2.5 - 1} # temperature per chain (the target chain always corresponds to chain 1)
cat("Default (inverse) temperatures: ", "\n")
print(alpha)
}else{
if(length(alpha)!= nChains){
stop("number of temperatures does not match number of chains.")
}
if(alpha[1] != 1){
stop("alpha[1] should be equal to 1.")
}
}
nPars <- 7
D0 <- which(myData[,"Censoring_status"] == 0)
nCensored <- length(D0)
parallel_mcmc_samples <- array(data = NA, dim = c(1, nPars + nCensored, nChains))
target_mcmc <- array(data = NA, dim = c(mcmc_cycles, nPars + nCensored))
dimnames(parallel_mcmc_samples) = list(paste0("current_value"),
c("g_mcmc", "lambda_mcmc", "a1_mcmc", "a2_mcmc", "b0_mcmc", "b1_mcmc","b2_mcmc",paste0("status_",D0)),
paste0("chain_",1:nChains))
dimnames(parallel_mcmc_samples)[[3]][1] <- "chain_1_(target_chain)"
dimnames(target_mcmc) = list(paste0("cycle_",1:mcmc_cycles),
c("g_mcmc", "lambda_mcmc", "a1_mcmc", "a2_mcmc", "b0_mcmc", "b1_mcmc","b2_mcmc",paste0("status_",D0)))
cllValues <- numeric(mcmc_cycles)
g_prop_sd <- rep(g_prop_sd, nChains)
lambda_prop_scale <- rep(lambda_prop_scale, nChains)
a1_prop_scale <- rep(a1_prop_scale, nChains)
a2_prop_scale <- rep(a2_prop_scale, nChains)
b_prop_sd <- array(data = b_prop_sd, dim = c(nChains, 3))
if(adjust_scales == TRUE){
cat("Making a warm-up run in order to adjust the proposal scale per chain...","\n")
chain_iter <- 1
# g_prop_sd[chain_iter] = g_prop_sd
# lambda_prop_scale[chain_iter] = lambda_prop_scale
# a1_prop_scale[chain_iter] = a1_prop_scale
# a2_prop_scale[chain_iter] = a2_prop_scale
# b_prop_sd[chain_iter,] = b_prop_sd
criterion <- TRUE
adjFactor <- 0.75
iter <- 0
while(criterion & (iter < 20)){
run <- cure_rate_metropolis_hastings_cpp( myData = myData,
m = 1000,
mu_g = mu_g, s2_g = s2_g, a_l = a_l, b_l = b_l,
a_1 = a_1, b_1 = b_1, a_2 = a_2, b_2 = b_2,
mu_b = mu_b, Sigma = Sigma,
alpha = alpha[chain_iter],
g_prop_sd = g_prop_sd[chain_iter],
lambda_prop_scale = lambda_prop_scale[chain_iter],
a1_prop_scale = a1_prop_scale[chain_iter],
a2_prop_scale = a2_prop_scale[chain_iter],
b_prop_sd = b_prop_sd[chain_iter,], plot = FALSE, verbose = FALSE, tau_mala = tau_mala, mala = mala)
if(mala>0){
indUp <- which((run$acceptance_rates > 0.6) == TRUE)
indDown <- which((run$acceptance_rates < 0.4) == TRUE)
}else{
indUp <- which((run$acceptance_rates[-6] > 0.3) == TRUE)
indDown <- which((run$acceptance_rates[-6] < 0.15) == TRUE)
}
if(length(indUp) > 0){
criterion = TRUE;
if(1 %in% indUp){g_prop_sd[chain_iter] <- g_prop_sd[chain_iter]/adjFactor}
if(2 %in% indUp){lambda_prop_scale[chain_iter] <- lambda_prop_scale[chain_iter]/adjFactor}
if(3 %in% indUp){a1_prop_scale[chain_iter] <- a1_prop_scale[chain_iter]/adjFactor}
if(4 %in% indUp){a2_prop_scale[chain_iter] <- a2_prop_scale[chain_iter]/adjFactor}
if(5 %in% indUp){b_prop_sd[chain_iter,] <- b_prop_sd[chain_iter,]/adjFactor}
if(mala>0){
if(6 %in% indUp){tau_mala <- tau_mala/adjFactor}
}
criterion1 = TRUE
}else{criterion1 = FALSE}
if(length(indDown) > 0){
criterion = TRUE
if(1 %in% indDown){g_prop_sd[chain_iter] <- g_prop_sd[chain_iter] * adjFactor}
if(2 %in% indDown){lambda_prop_scale[chain_iter] <- lambda_prop_scale [chain_iter]* adjFactor}
if(3 %in% indDown){a1_prop_scale[chain_iter] <- a1_prop_scale[chain_iter] * adjFactor}
if(4 %in% indDown){a2_prop_scale[chain_iter] <- a2_prop_scale[chain_iter] * adjFactor}
if(5 %in% indDown){b_prop_sd[chain_iter,] <- b_prop_sd[chain_iter,] * adjFactor}
if(mala>0){
if(6 %in% indDown){tau_mala <- tau_mala*adjFactor}
}
criterion2 = TRUE
}else{criterion2 = FALSE}
if((criterion1 == FALSE) & (criterion2 == FALSE)){criterion = FALSE}
iter <- iter + 1
#print(iter)
}
cat(paste0(" proposal scale for chain: ",chain_iter),"\n")
print(c(g_prop_sd[chain_iter], lambda_prop_scale[chain_iter], a1_prop_scale[chain_iter], a2_prop_scale[chain_iter], b_prop_sd[chain_iter,], tau_mala), digits = 4)
for(chain_iter in 2:nChains){
# initial metropolis-hastings proposal parameters (they will be adjusted)
g_prop_sd[chain_iter] = g_prop_sd[chain_iter - 1]
lambda_prop_scale[chain_iter] = lambda_prop_scale[chain_iter-1]
a1_prop_scale[chain_iter] = a1_prop_scale[chain_iter-1]
a2_prop_scale[chain_iter] = a2_prop_scale[chain_iter-1]
b_prop_sd[chain_iter,] = b_prop_sd[chain_iter-1,]
criterion <- TRUE
adjFactor <- 0.75
iter <- 0
while(criterion & (iter < 5)){
run <- cure_rate_metropolis_hastings_cpp( myData = myData,
m = 500,
mu_g = mu_g, s2_g = s2_g, a_l = a_l, b_l = b_l,
a_1 = a_1, b_1 = b_1, a_2 = a_2, b_2 = b_2,
mu_b = mu_b, Sigma = Sigma,
alpha = alpha[chain_iter],
g_prop_sd = g_prop_sd[chain_iter],
lambda_prop_scale = lambda_prop_scale[chain_iter],
a1_prop_scale = a1_prop_scale[chain_iter],
a2_prop_scale = a2_prop_scale[chain_iter],
b_prop_sd = b_prop_sd[chain_iter,], plot = FALSE, verbose = FALSE, tau_mala = tau_mala, mala = mala)
if(mala>0){
indUp <- which((run$acceptance_rates > 0.6) == TRUE)
indDown <- which((run$acceptance_rates < 0.4) == TRUE)
}else{
indUp <- which((run$acceptance_rates[-6] > 0.3) == TRUE)
indDown <- which((run$acceptance_rates[-6] < 0.15) == TRUE)
}
if(length(indUp) > 0){
criterion = TRUE;
if(1 %in% indUp){g_prop_sd[chain_iter] <- g_prop_sd[chain_iter]/adjFactor}
if(2 %in% indUp){lambda_prop_scale[chain_iter] <- lambda_prop_scale[chain_iter]/adjFactor}
if(3 %in% indUp){a1_prop_scale[chain_iter] <- a1_prop_scale[chain_iter]/adjFactor}
if(4 %in% indUp){a2_prop_scale[chain_iter] <- a2_prop_scale[chain_iter]/adjFactor}
if(5 %in% indUp){b_prop_sd[chain_iter,] <- b_prop_sd[chain_iter,]/adjFactor}
if(mala>0){
if(6 %in% indUp){tau_mala <- tau_mala/adjFactor}
}
criterion1 = TRUE
}else{criterion1 = FALSE}
if(length(indDown) > 0){
criterion = TRUE
if(1 %in% indDown){g_prop_sd[chain_iter] <- g_prop_sd[chain_iter] * adjFactor}
if(2 %in% indDown){lambda_prop_scale[chain_iter] <- lambda_prop_scale [chain_iter]* adjFactor}
if(3 %in% indDown){a1_prop_scale[chain_iter] <- a1_prop_scale[chain_iter] * adjFactor}
if(4 %in% indDown){a2_prop_scale[chain_iter] <- a2_prop_scale[chain_iter] * adjFactor}
if(5 %in% indDown){b_prop_sd[chain_iter,] <- b_prop_sd[chain_iter,] * adjFactor}
if(mala>0){
if(6 %in% indDown){tau_mala <- tau_mala*adjFactor}
}
criterion2 = TRUE
}else{criterion2 = FALSE}
if((criterion1 == FALSE) & (criterion2 == FALSE)){criterion = FALSE}
iter <- iter + 1
#print(iter)
}
cat(paste0(" proposal scale for chain: ", chain_iter),"\n")
print(c(g_prop_sd[chain_iter], lambda_prop_scale[chain_iter], a1_prop_scale[chain_iter], a2_prop_scale[chain_iter], b_prop_sd[chain_iter,], tau_mala), digits = 4)
}
cat(" done Metropolis-Hastings proposal scale adjustements!","\n")
}
if(adjust_alpha){
alpha0 <- 1.05
alpha <- 1/alpha0^{1:nChains - 1}
cat(paste0(" Initial temperatures: "),"\n")
print(alpha)
mcmc_cyc <- 100
if(mcmc_cyc > 4){mcmc_cyc <- 2*mcmc_cyc}
if(mcmc_cyc > 8){mcmc_cyc <- 2*mcmc_cyc}
cat(paste0("Adjusting temperatures..."),"\n")
run <- cure_rate_MC3_cpp( myData = myData, nChains = nChains,
mcmc_cycles = mcmc_cyc,
alpha = alpha,
nCores = nCores,
sweep = sweep,
mu_g = mu_g, s2_g = s2_g, a_l = a_l, b_l = b_l,
a_1 = a_1, b_1 = b_1, a_2 = a_2, b_2 = b_2,
mu_b = mu_b, Sigma = Sigma,
lambda_prop_scale = lambda_prop_scale[chain_iter],
a1_prop_scale = a1_prop_scale[chain_iter],
a2_prop_scale = a2_prop_scale[chain_iter],
b_prop_sd = b_prop_sd[chain_iter,],
initialValues = NULL,
plot = FALSE,
adjust_scales = FALSE,
adjust_alpha = FALSE,
verbose = FALSE, tau_mala = tau_mala, mala = mala
)
criterion <- criterion_up <- criterion_down <- FALSE
if ( min(run[["swap_accept_rate"]]) < 0.35 ){
criterion <- TRUE
criterion_down <- TRUE
}
if ( min(run[["swap_accept_rate"]]) > 0.75 ){
criterion <- TRUE
criterion_up <- TRUE
}
cat(paste0("Swap accept rates:"), "\n")
print(run[["swap_accept_rate"]])
iter <- 0
while( criterion & (iter < 30)){
if(criterion_down){
alpha0 <- max(c(alpha0 - 0.01, 1.001))
}
if(criterion_up){
alpha0 <- alpha0 + 0.01
}
alpha <- 1/alpha0^{1:nChains - 1}
cat(paste0(" Adjusted temperatures: "),"\n")
print(alpha)
run <- cure_rate_MC3_cpp( myData = myData, nChains = nChains,
mcmc_cycles = mcmc_cyc,
alpha = alpha,
nCores = nCores,
sweep = sweep,
mu_g = mu_g, s2_g = s2_g, a_l = a_l, b_l = b_l,
a_1 = a_1, b_1 = b_1, a_2 = a_2, b_2 = b_2,
mu_b = mu_b, Sigma = Sigma,
lambda_prop_scale = lambda_prop_scale[chain_iter],
a1_prop_scale = a1_prop_scale[chain_iter],
a2_prop_scale = a2_prop_scale[chain_iter],
b_prop_sd = b_prop_sd[chain_iter,],
initialValues = NULL,
plot = FALSE,
adjust_scales = FALSE,
adjust_alpha = FALSE,
verbose = FALSE, tau_mala = tau_mala, mala = mala
)
print(run[["swap_accept_rate"]])
criterion <- criterion_up <- criterion_down <- FALSE
if ( min(run[["swap_accept_rate"]]) < 0.25 ){
criterion <- TRUE
criterion_down <- TRUE
}
if ( min(run[["swap_accept_rate"]]) > 0.65 ){
criterion <- TRUE
criterion_up <- TRUE
}
iter <- iter + 1
}
cat(" done temperature adjustements!","\n")
}
if(verbose){
cat(paste0("Running MC^3 with ", nChains, " heated chains...", "\n"))
}
swap_accept_per_chain <- numeric(nChains - 1)
n_attempts_per_chain <- numeric(nChains - 1)
all_cll_values <- array(NA, dim = c(nChains, mcmc_cycles))
swap_accept <- 0
# initialization
cycle <- 1
registerDoParallel(nCores)
parLoop <- foreach(chain_iter = 1:nChains, .export = ls(envir = globalenv())) %dopar% {
# if(chain_iter == 1){perform_mala = mala}else{
# if(alpha[chain_iter] == 1){
# perform_mala = mala
# }else{perform_mala = 0}
# }
mcmc_chain <- cure_rate_metropolis_hastings_cpp( myData = myData,
m = sweep,
mu_g = mu_g, s2_g = s2_g, a_l = a_l, b_l = b_l,
a_1 = a_1, b_1 = b_1, a_2 = a_2, b_2 = b_2,
mu_b = mu_b, Sigma = Sigma,
alpha = alpha[chain_iter],
g_prop_sd = g_prop_sd[chain_iter],
lambda_prop_scale = lambda_prop_scale[chain_iter],
a1_prop_scale = a1_prop_scale[chain_iter],
a2_prop_scale = a2_prop_scale[chain_iter],
b_prop_sd = b_prop_sd[chain_iter,], plot = FALSE,
tau_mala = tau_mala, mala = mala)
}
stopImplicitCluster()
for(chain_iter in 1:nChains){
parallel_mcmc_samples[1, ,chain_iter] <- c(parLoop[[chain_iter]]$mcmc_sample[sweep,],
parLoop[[chain_iter]]$latent_status_censored[sweep,])
target_mcmc[cycle,] <- parallel_mcmc_samples[1,,1]
all_cll_values[chain_iter,cycle] <- parLoop[[chain_iter]]$complete_log_likelihood[sweep]
}
cllValues[1] <- parLoop[[1]]$complete_log_likelihood[sweep]
for(cycle in 2:mcmc_cycles){
#attempt chain-swap
# myPair <- sample(nChains,2,replace = FALSE)
# j1 <- myPair[1]
# j2 <- myPair[2]
j1 <- sample(nChains - 1 ,1,replace = FALSE)
j2 <- j1 + 1
n_attempts_per_chain[j1] <- n_attempts_per_chain[j1] + 1
log_posterior1 <- parLoop[[j1]]$complete_log_likelihood[sweep] #+ parLoop[[j1]]$log_prior_density[sweep]
log_posterior2 <- parLoop[[j2]]$complete_log_likelihood[sweep] #+ parLoop[[j2]]$log_prior_density[sweep]
denom <- log_posterior1 + log_posterior2
# print(denom)
state_j1 <- as.list(parLoop[[j1]]$mcmc_sample[sweep,])
state_j1[["I_sim_D0"]] <- parLoop[[j1]]$latent_status_censored[sweep,]
state_j2 <- as.list(parLoop[[j2]]$mcmc_sample[sweep,])
state_j2[["I_sim_D0"]] <- parLoop[[j2]]$latent_status_censored[sweep,]
nom1 <- cure_rate_metropolis_hastings_cpp( myData = myData,
m = 1, # only the log-posterior is computed at the initial values
mu_g = mu_g, s2_g = s2_g, a_l = a_l, b_l = b_l,
a_1 = a_1, b_1 = b_1, a_2 = a_2, b_2 = b_2,
mu_b = mu_b, Sigma = Sigma,
alpha = alpha[j1], plot = FALSE,
initialValues = state_j2, tau_mala = tau_mala, mala = mala)
nom2 <- cure_rate_metropolis_hastings_cpp( myData = myData,
m = 1, # only the log-posterior is computed at the initial values
alpha = alpha[j2], plot = FALSE,
mu_g = mu_g, s2_g = s2_g, a_l = a_l, b_l = b_l,
a_1 = a_1, b_1 = b_1, a_2 = a_2, b_2 = b_2,
mu_b = mu_b, Sigma = Sigma,
initialValues = state_j1, tau_mala = tau_mala, mala = mala)
nom <- nom1$complete_log_likelihood + #nom1$log_prior_density
+ nom2$complete_log_likelihood
#+ nom2$log_prior_density
# print(nom)
# print(nom-denom)
mmm <- log(runif(1)) < nom - denom
if (is.na(mmm) == FALSE){
if(mmm){
# chain switching accepted
tmp <- parallel_mcmc_samples[1,,j2]
parallel_mcmc_samples[1,,j2] <- parallel_mcmc_samples[1,,j1]
parallel_mcmc_samples[1,,j1] <- tmp
swap_accept <- swap_accept + 1
swap_accept_per_chain[j1] <- swap_accept_per_chain[j1] + 1