multi logit normal density

[bob-carpenter]

added multi logic normal density example as a function.

I wrote some code to test, but didn't know where to put it if anywhere:

functions {
  /**
   * Return the multivariate logistic normal density for the specified log simplex.
   *
   * See: https://en.wikipedia.org/wiki/Logit-normal_distribution#Multivariate_generalization
   * 
   * @param theta a simplex (N rows)
   * @param mu location of normal (N-1 rows)
   * @param L_Sigma Cholesky factor of covariance (N-1 rows, N-1 cols)
   */
  real multi_logit_normal_cholesky_lpdf(vector theta, vector mu, matrix L_Sigma) {
    int N = rows(theta);
    return sum(-log(theta))
      + multi_normal_cholesky_lpdf(log(theta[1:N - 1] / theta[N]) | mu, L_Sigma);
  }
}
data {
  int<lower=1> N;
  vector[N - 1] mu;
  matrix[N - 1, N - 1] L_Sigma;
}
parameters {
  simplex[N] theta;
}
model {
  theta ~ multi_logit_normal_cholesky(mu, L_Sigma);
}

When I fit that for N = 10, I get

>>> fit.summary()
               Mean      MCSE    StdDev         5%       50%       95%    N_Eff   N_Eff/s     R_hat
lp__      -9.707250  0.118998  2.287330 -13.981800 -9.352490 -6.594010  369.468   9722.84  1.002890
theta[1]   0.101758  0.003245  0.080631   0.017135  0.078687  0.279115  617.543  16251.10  0.999185
theta[2]   0.093105  0.003114  0.065519   0.016626  0.078280  0.222328  442.704  11650.10  0.999218
theta[3]   0.091287  0.002735  0.061166   0.019894  0.078057  0.207519  500.311  13166.10  0.999022
theta[4]   0.086258  0.002170  0.053007   0.023276  0.074910  0.187406  596.702  15702.70  0.999006
theta[5]   0.086327  0.001873  0.052215   0.023367  0.073449  0.183068  777.025  20448.00  1.000590
theta[6]   0.086985  0.001846  0.052987   0.022538  0.075478  0.190856  823.772  21678.20  1.001630
theta[7]   0.087918  0.002201  0.057098   0.021416  0.076411  0.203125  673.113  17713.50  0.999299
theta[8]   0.089039  0.002516  0.059121   0.023527  0.075369  0.205456  552.153  14530.30  0.999014
theta[9]   0.092057  0.003195  0.065958   0.020491  0.075871  0.221874  426.106  11213.30  0.999156
theta[10]  0.095794  0.003665  0.078532   0.017671  0.073461  0.261481  459.097  12081.50  0.999713
theta[11]  0.089472  0.002335  0.061425   0.021900  0.074570  0.211407  691.799  18205.20  1.000490

I'm not 100% sure how to test here without doing simulation-based calibration, but the correlations are along the lines I'd expect:

>>> theta_draws = fit.stan_variable('theta')

>>> np.shape(theta_draws)
(1000, 11)

>>> np.corrcoef(theta_draws[:, 1], theta_draws[:, 2])
array([[1.      , 0.451464],
       [0.451464, 1.      ]])

>>> np.corrcoef(theta_draws[:, 1], theta_draws[:, 3])
array([[1.        , 0.11187134],
       [0.11187134, 1.        ]])

>>> np.corrcoef(theta_draws[:, 1], theta_draws[:, 4])
array([[ 1.       , -0.1824799],
       [-0.1824799,  1.       ]])

>>> np.corrcoef(theta_draws[:, 1], theta_draws[:, 10])
array([[ 1.        , -0.05364937],
       [-0.05364937,  1.        ]])

>>> np.corrcoef(theta_draws[:, 2], theta_draws[:, 3])
array([[1.        , 0.45617112],
       [0.45617112, 1.        ]])

The entry at [0, 1] is the correlation here---sorry for not making this neater, but I closed the session.