Test: Soft vs Hard sum-to-zero constrain + choosing the right prior for soft constrain

Modeling
27

Cool. I didn’t know covariance matrices were flexible enough for that kind of constraint, but I guess I should learn how to think about matrices as bundles of equations.

Your implementation does give me what I need, the scaling. When I apply that to the constrained implementation in Stan,

transformed data {
  int<lower = 2> N = 4;
}
parameters {
  vector[N - 1] mu_prefix;
}
transformed parameters {
  vector[N] mu = append_row(mu_prefix, -sum(mu_prefix));
}
model {
  mu ~ normal(0, inv(sqrt(1 - inv(N))));
}

I get the correct marginals:

             mean se_mean   sd
mu[1]        -0.04    0.02 1.03
mu[2]         0.02    0.02 1.02
mu[3]         0.04    0.02 1.01
mu[4]        -0.02    0.02 1.01

With more iterations the mean and sd are correct to more decimal places, as expected.

The direct constrained version might be more efficient, because it doesn’t require the matrix multiply A_qr * x_raw, which is a quadratic operation with lots of autodiff to do for the N x N Jacobian. On the other hand, it’s less of a normal sampling and reconstruct approach, which should mix very well.

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