I’m reading the Stan manual section on Matt trick (Ch 27.6 Reparameterization). The example using Neal’s funnel makes sense, but it isn’t clear how the example corresponds the claims made about the Matt trick.
I’d like to better understand these claims:

“[Matt trick] is a general transform from a centered to a noncentered parameterization”. What’s centered and noncentered in the Neal’s funnel example?

“This reparameterization is helpful when there is not much data, because it separates the hierarchical parameters and lowerlevel parameters in the prior.”
 Does this refer to a hierarchical model rather than Neal’s funnel?
 (Assuming that this is about a hierarchical model) Isn’t the Matt trick about separating hierarchical parameters and lowerlevel parameters in the sampling statement rather than in the prior?
For reference, below is the Matt trick, presented in the manual.
Before transformation
model {
y ~ normal(0, 3);
x ~ normal(0, exp(y/2));
}
After transformation
transformed parameters {
...
y = 3.0 * y_raw;
x = exp(y/2) * x_raw;
}
model {
y_raw ~ normal(0, 1); // implies y ~ normal(0, 3)
x_raw ~ normal(0, 1); // implies x ~ normal(0, exp(y/2))
}