Is "soft" censoring a thing?

soft-censor.R

# Testing a soft-left-censored Stan model:
# Values above some minimum are detected normally, 
# values less than minimum go detected with some probability > 0.

set.seed(2345767)
library(rstan)
rstan_options(auto_write = TRUE)
options(mc.cores = 7)

sim_mu = 3
sim_sigma = 1.5
detection_limit = 4

# Simulated data: lognormal, 1/3 of values below detection limit are missing
lcens_log = rlnorm(1000, sim_mu, sim_sigma)
lcens_ptrunc_log = lcens_log
lcens_ptrunc_log[lcens_ptrunc_log < detection_limit] = sapply(
	lcens_ptrunc_log[lcens_ptrunc_log < detection_limit],
	function(x)sample(c(x,x,NA),size=1))
print(summary(lcens_log))
print(summary(lcens_ptrunc_log))

stanstr = "
data{
	real<lower=0> L;
	int<lower=0> N_obs;
	int<lower=0> N_cens;
	real<lower=0> y_obs[N_obs];
}
parameters{
	real mu;
	real<lower=0> sigma;
}
model{
	sigma ~ normal(0,100); // Weak prior on sigma to keep sampler away from Inf
	y_obs ~ lognormal(mu, sigma);
	increment_log_prob(N_cens * lognormal_cdf_log(L, mu, sigma));
}"

fit = stan(
	model_code=stanstr,
	data=list(
		L = detection_limit,
		N_obs=length(which(!is.na(lcens_ptrunc_log))),
		N_cens=length(which(is.na(lcens_ptrunc_log))),
		y_obs=lcens_ptrunc_log[!is.na(lcens_ptrunc_log)]),
	chains=7,
	iter=10000,
	warmup=1000)

print(fit)
plot(fit, pars=c("mu", "sigma"))