Possible error in user guide? "statistically equivalent"

Modeling
1

In the linear regression portion of the Stan user guide, it states that the following are “statistically equivalent”. Could someone please clarify why this is the case?

model {
  y ~ normal(x * beta + alpha, sigma);  // likelihood
}

vs.

model {
  for (n in 1:N) {
    y[n] ~ normal(x[n] * beta, sigma);
  }
}

How does the alpha get incorporated into beta and x[n] in the second model code block?

Thanks!

2

This could certainly be clearer. The ambiguity is due to the preceding sentence:

 It would be possible to include a column of ones in the data matrix x to remove the alpha parameter.

Does that make sense?

3

Yes, that makes sense. Sorry, I missed that statement about redefining x and beta. I should have paid closer attention.