Interpretation Mean Parameter Random Intercept Model

Modeling
1

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))
2

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.

3

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.

4

In a sense, yes, but it’s a bit more complicated than that because these parameters are directly related.