# Is correlation between a parameter and the log posterior inherently problematic?

In fitting quasi-Poisson models using RStan, I often see a negative correlation between the variance of the overdispersion term and the overall log posterior (lp__). Other diagnostics such as Rhat, effective sample size, and trace plots look good. Is a correlation between a model parameter and lp__ inherently problematic (i.e. indicate a lack of convergence or other problem with the model)?

Example:

Quasi-Poisson RNG function to generate overdispersed-Poisson counts (with variance = mean * theta):

rqpois <- function(n, mu, theta) {
rnbinom(n = n, mu = mu, size = mu/(theta-1))
}


Generate counts y as log-link function of continuous variable x:

set.seed(4536)
N <- 200
x <- runif(N, 0, 10)
alpha <- 1
beta <- 0.3
y_pred <- exp(alpha + beta * x)
y <- rqpois(n = N, mu = y_pred, theta = 2)


Stan model:

qpois_stan <- "
data {
int<lower=0> N;
real<lower=0> x[N];
int<lower=0> y[N];
}

parameters {
real alpha;
real beta;
vector[N] overdisp;
real<lower=0> sigma_overdisp;
}

transformed parameters {
vector[N] mu;

for(i in 1:N) {
mu[i] <- alpha + beta * x[i] + overdisp[i];
}
}

model {
alpha ~ normal(0, 10);
beta ~ normal(0, 10);
overdisp ~ normal(0, sigma_overdisp);
sigma_overdisp ~ cauchy(0, 10);

y ~ poisson_log(mu);
}"


Fit model using rstan:

mod <- stan(
model_code = qpois_stan,
data = list(N = N, x = x, y = y),
warmup = 2000,
iter = 4000,
chains = 2
)


Negative correlation between lp__ and sigma_overdisp:

You should be able to see this if you plot the log density vs. sigma_overdisp with the other parameters fixed to some reasonable value.