Linear parametrization: `a + b(x - mu) `

Modeling
1

I came across this parametrization in Statistical Rethinking section 4.4.

stan_program <- '
data {
  int<lower=1> n;
  real xbar;
  vector[n] height;
  vector[n] weight;
}
parameters {
  real<lower=0,upper=50> sigma;
  real<lower=0> b;
  real a;
}
model {
  vector[n] mu;
  mu = a + b * (weight - xbar);
  height ~ normal(mu, sigma);
  a ~ normal(178, 20);
  b ~ lognormal(0, 1);
  sigma ~ uniform(0, 50);
}

I’m curious why the author would define a linear model such as \mu_{Y|X=x} = a + b(x - \mu_x). Consider when x = \mu_x, the equation would collapse to \mu_{Y|X=x}=a (just the intercept.)

As I understand it, this would have the effect of linking mu_Y to mu_X; in other words, if you supply the global average X, you get the global average Y in response. (I might be wrong here!)

Why would this design be preferable over the more common \mu_{Y|X=x} = a + b*x? Is this related to model interpretability or to HMC/NUTS sampling/geometry (or perhaps both)? Is it context dependent?

Thanks!

2

You are not wrong here but this is not the result of the parametrization, it is the result of the linear model. If the linear model is correct, y = \mu_y is expected to happen when x = \mu_x. The parametrization is mainly to ease the interpretation of a and sampling efficiency as you say.

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