# Divergence in fitting a mixture of censored and truncated data

Hello,

I’ve got some trouble to estimate my parameters from a convoluted signal stemming from a mix of truncated and censored distributions.
To diagnose my model I generated some simulated data.

I can recover the parameters quite well from the convoluted signal and the traceplots look ok, but I’m constantly getting a lot of diverging transitions. I feel like I’m missing something and have tried different priors or increasing adapt_delta but nothing really helped to get rid of the divergences.

Generate data:

``````K <- 2
N <- 1000
TRUNC <- c(3, 0)
CENS <- 21
MIXING_PROBABILITIES <- c(0.7, 0.3)
DISTS <- list(
z1 = list(mu = 10, sd = 4),
z2 = list(mu = 2, sd = 2)
)

y <- matrix(-Inf, nrow = N, ncol = 1)
for (n in 1:N) {
# under which stressor does n die?
z <- rcategorical(1, p = MIXING_PROBABILITIES)
while (y[n, 1] < TRUNC[z]) {
y[n, 1] <- rnorm(1, DISTS[[z]]\$mu, DISTS[[z]]\$sd)
}
}
y <- ifelse(y > CENS, CENS, y)

hist(y[, 1], breaks = 20)

PRIOR_DELAY <- matrix(
c(7, 4,
2, 4),
byrow = TRUE, ncol = 2, nrow = 2)

standata <- list(
# dataset dimensions
N = N,
K = K,

# dataset
y = as.vector(y),
WEIGHTS = MIXING_PROBABILITIES,
CENS = CENS,
TRUNC = TRUNC,
PRIOR_DELAY = PRIOR_DELAY
)
``````

This is the corresponding stan code. Generated quantities section is responsible to reconstruct the censored-truncated signal

``````functions {
real normal_rcensed_ltrunc( real x, real alpha, real beta, real ul, real lt) {
real mylp;
if(x < lt){
mylp = negative_infinity();               // probability is infinity if x falls below lower truncation bound
} else {
if(x < ul){
mylp = normal_lpdf (x  | alpha, beta);   // probability of x falling on normal gamma
mylp += -normal_lccdf(lt | alpha, beta); // rescale due to truncation (divide over integral)
} else {
mylp = normal_lccdf(ul | alpha, beta);
mylp += -normal_lccdf(lt | alpha, beta);
}
}
return mylp;
}
}
data {
int<lower=1> N;                // number of data points
int<lower=1> K;
real y[N];                     // observations (death counts at day)
vector[K] WEIGHTS;
real CENS;
real TRUNC[K];
matrix[K, 2] PRIOR_DELAY;
}
parameters {
// here parameters are drawn
real<lower=0> delay[K];       // locations of mixture components
real<lower=0> sigma[K];      // scales of mixture components
}
model {
// there are only ever evaluations on how likely a parameter is given the prior
for (k in 1:K){
delay ~ normal(PRIOR_DELAY[k,1], PRIOR_DELAY[k,2]);
sigma ~ cauchy(0, 5);
}

for (n in 1:N) {                                       // add log probabilities to log posterior probability vector
vector[K] lps = log(WEIGHTS);
// calculate the center of the distribution as estimated time until effect
// plus time lag until the stressor was applied
for (k in 1:K) {
lps[k] += normal_rcensed_ltrunc(y[n], delay[k], sigma[k], CENS, TRUNC[k]);
}
target += log_sum_exp(lps);
}
}
generated quantities {
matrix[N,K] yk_predict;
vector[N] y_predict;
int choice[N];
// posterior predictions
for (n in 1:N){
// add truncating -- sample again until the sample falls inside the truncated distribution
// one iteration is ensured if LAG = 0
for(k in 1:K){
yk_predict[n,k] = negative_infinity();
// add truncating -- sample again until the sample falls inside the truncated distribution
// one iteration is ensured if LAG = 0
while(yk_predict[n,k] < TRUNC[k]){
yk_predict[n,k] = normal_rng(delay[k], sigma[k]);
}
}
choice[n] = categorical_rng(WEIGHTS);
y_predict[n] = yk_predict[n,choice[n]];
// set to limit if
if(y_predict[n] > CENS) y_predict[n] = CENS;
}
}
``````

I would be extremely happy to get some leads or ideas of ways I could explore this problem better and get rid of the divergences.