# Interpretation Mean Parameter Random Intercept Model

I am getting reacquainted with Stan. I am currently trying to understand the following issue. Below, I fit a simple random intercept model. My intercepts are in the alpha vector. I have the following questions:

How do I interpret `mean_alpha`, and why do I get such a different estimate for it compared to `mu_alpha`? Specifically, both have a posterior mean of 0.41 but the posterior standard deviation for `mean_alpha` is essentially 0 but for `mu_alpha` it is 0.1

``````library(rstan)

# Set up the data
set.seed(123)
N <- 1000 # Number of observations
J <- 100  # Number of persons

person <- sample(1:J, N, replace = TRUE) # Group/person indices
group_intercepts <- rnorm(J, mean = 0.5, sd = 1) # Generating random intercepts for simplicity
Y <- group_intercepts[person] + rnorm(N, mean = 0, sd = 0.1) # Response variable

data_list <- list(N = N, J = J, person = person, Y = Y)

# Stan model code
stan_code <- '
data {
int<lower=0> N;
int<lower=0> J;
int<lower=1, upper=J> person[N];
vector[N] Y;
}

parameters {
vector[J] alpha;
real mu_alpha;
real<lower=0> sigma_alpha;
real<lower=0> sigma_Y;
}

model {
alpha ~ normal(mu_alpha, sigma_alpha);

for (n in 1:N) {
Y[n] ~ normal(alpha[person[n]], sigma_Y);
}
}

generated quantities {
real mean_alpha = mean(alpha);
}
'

# Compile and fit the model
fit <- stan(model_code = stan_code, data = data_list, iter = 2000, chains = 4, warmup = 1000, seed = 123)

# Print the fit summary
print(fit, pars = c("mu_alpha", "mean_alpha"))
stan_dens(fit, pars = c("mu_alpha", "mean_alpha"))

# Plot traceplots for diagnostics
library("bayesplot")
traceplot(fit, pars = c("sigma_Y", "mu_alpha"))

library(lme4)
lmeFit <- lmer(Y ~ 1 + (1|Person), data = data.frame(Person = person, Y = Y))
``````

Hello @karchjd, I believe this is expected behaviour. Your parameter `mu_alpha` represents the location of the distribution from which the individual `alpha` are drawn, whereas `mean_alpha` represents the simple mean of the individual values of `alpha` for this dataset. If the number of grouping levels are large enough then these should converge on the same value.

I expect that in the case where the individual `alpha` are strongly informed, then their mean will be more precise than the underlying population location, in that the data are more informative about themselves than the proposed generative model.

Thanks for the explanation. So, essentially mu_alpha would be the population mean but mean_alpha the sample mean, right? It also intuitively makes sense to me that the posterior of the population mean and sample have the same average but we have more uncertainty regarding the population mean.

In a sense, yes, but itâ€™s a bit more complicated than that because these parameters are directly related.