# In Bayesian statistics, what do mu, eta, and tau tend to represent?

In the eight schools example from Gelman, he sets his parameters as mu, eta, and tau.

// saved as 8schools.stan
data {
int<lower=0> J; // number of schools
real y[J]; // estimated treatment effects
real<lower=0> sigma[J]; // s.e. of effect estimates
}
parameters {
real mu;
real<lower=0> tau;
real eta[J];
}
transformed parameters {
real theta[J];
for (j in 1:J)
theta[j] = mu + tau * eta[j];
}
model {
target += normal_lpdf(eta | 0, 1);
target += normal_lpdf(y | theta, sigma);
}


I see these parameter names often used in Bayesian statistics. Can someone provide a general intuition for what they'll tend to mean?

Also why is mu modeled as unique to every school but the other two parameters are not?

• Usually $\mu$ is a mean and $\eta, \tau$ are variance but there's no reason to expect this to be true in general. – Sycorax Jun 24 '17 at 23:08
• $\mu$ and $\tau$ are often parameters for mean and scale (or variance or precision) respectively, but $\eta$ may often be any number of different things. Sometimes it's an "error" or "noise", sometimes it's a "y-like" variable (where by contrast $\xi$ is an "x-like" variable), sometimes its a location-type effect, sometimes a variance, sometimes a covariance type quantity, etc – Glen_b Jun 25 '17 at 1:03
• eta is unique to every school - it is a latent (random) effect. The other parameters are global hyperparameters in this hierarchical model. This should give you good keywords to investigate. – Björn Jun 25 '17 at 6:14
• There is no rationale in seeking a universal meaning for a given Greek symbol in mathematics: symbols can be used for any purpose without impacting the validity of the induced expression. – Xi'an Jun 27 '17 at 12:43