# include <cppad/cppad.hpp>
# include <cppad/mixed/cppad_mixed.hpp>
# include <cppad/mixed/manage_gsl_rng.hpp>
# include <Eigen/Dense>

namespace {
     using CppAD::vector;
     using CppAD::log;
     using CppAD::AD;
     //
     using CppAD::mixed::d_sparse_rcv;
     using CppAD::mixed::d_vector;

     class mixed_derived : public cppad_mixed {
     private:
          const size_t          n_fixed_;
          const size_t          n_random_;
          const d_vector&       y_;
     // ----------------------------------------------------------------------
     public:
          // constructor
          mixed_derived(
               size_t                 n_fixed       ,
               size_t                 n_random      ,
               bool                   quasi_fixed   ,
               bool                   bool_sparsity ,
               const d_vector&       y              ) :
               cppad_mixed(
                    n_fixed, n_random, quasi_fixed, bool_sparsity
               )                     ,
               n_fixed_(n_fixed)     ,
               n_random_(n_random)   ,
               y_(y)
          {    assert( n_fixed == 2);
               assert( y_.size() == n_random_ );
          }
     // ----------------------------------------------------------------------
          // implementation of ran_likelihood
          a1_vector ran_likelihood(
               const a1_vector&         theta  ,
               const a1_vector&         u      ) override
          {
               assert( theta.size() == n_fixed_ );
               assert( u.size() == y_.size() );
               a1_vector vec(1);

               // initialize part of log-density that is always smooth
               vec[0] = 0.0;

               // sqrt_2pi = CppAD::sqrt(8.0 * CppAD::atan(1.0) );

               for(size_t i = 0; i < n_random_; i++)
               {    a1_double mu     = u[i] + theta[0];
                    a1_double sigma  = theta[1];
                    a1_double res    = (y_[i] - mu) / sigma;

                    // p(y_i | u, theta)
                    vec[0] += log(sigma) + res * res / 2.0;
                    // following term does not depend on fixed or random effects
                    // vec[0] += log(sqrt_2pi);

                    // p(u_i | theta)
                    a1_double sq = u[i] * u[i];
                    if( i == 0 )
                         sq = (u[0] + u[1]) * (u[0] + u[1]);
                    vec[0] += sq / 2.0;
                    // following term does not depend on fixed or random effects
                    // vec[0] += log(sqrt_2pi);
               }
               return vec;
          }
     };
}

bool sample_random_xam(void)
{
     bool   ok = true;
     double inf = std::numeric_limits<double>::infinity();
     //
     // initialize gsl random number generator
     size_t random_seed = CppAD::mixed::new_gsl_rng(0);
     //
     size_t n_data   = 10;
     size_t n_fixed  = 2;
     size_t n_random = n_data;
     //
     d_vector data(n_data), random_in(n_random);
     for(size_t i = 0; i < n_data; i++)
     {    data[i]       = double(i + 1);
          random_in[i]    = 0.0;
     }
     d_vector fixed_vec(n_fixed);
     for(size_t i = 0; i < n_fixed; i++)
          fixed_vec[i] = double(i + 1);

     // object that is derived from cppad_mixed
     bool quasi_fixed   = true;
     bool bool_sparsity = true;
     mixed_derived mixed_object(
          n_fixed, n_random, quasi_fixed, bool_sparsity, data
     );
     mixed_object.initialize(fixed_vec, random_in);

     std::string random_ipopt_options =
          "Integer print_level     0\n"
          "String  sb              yes\n"
          "String  derivative_test second-order\n"
          "Numeric tol             1e-8\n"
     ;
     d_vector random_lower(n_random), random_upper(n_random);
     for(size_t i = 0; i < n_random; i++)
     {    random_lower[i] = -inf;
          random_upper[i] = +inf;
     }
     //
     // compute the optimal random effects
     d_vector random_opt = mixed_object.optimize_random(
          random_ipopt_options, fixed_vec, random_lower, random_upper, random_in
     );
     //
     // sample from the posterior for random effects given fixed effects
     // and compute the  sample covariance matrix
     size_t n_sample = 10000;
     d_vector sample(n_sample * n_random);
     std::string error_msg = mixed_object.sample_random(
          sample,
          random_ipopt_options,
          fixed_vec,
          random_lower,
          random_upper,
          random_in
     );
     ok &= error_msg == "";
     d_vector sample_cov(n_random * n_random);
     for(size_t i = 0; i < n_random; i++)
          for(size_t j = 0; j < n_random; j++)
               sample_cov[i * n_random + j] = 0;
     for(size_t i_sample = 0; i_sample < n_sample; i_sample++)
     {    d_vector diff(n_random);
          for(size_t j = 0; j < n_random; j++)
               diff[j] = sample[i_sample * n_random + j] - random_opt[j];
          for(size_t i = 0; i < n_random; i++)
               for(size_t j = 0; j < n_random; j++)
                    sample_cov[i * n_random + j] +=
                         diff[i] * diff[j] / double(n_sample);
     }
     //
     // For i > 1, the terms involving u[i] is
     //     0.5 * ( (y[i] - theta[0] - u[i])^2 / theta[1]^2 + u[i]^2  )
     // The terms involving u[0] and u[1] are
     //     0.5 * ( (y[1] - theta[0] - u[1])^2 / theta[1]^2 + u[1]^2  )
     //   + 0.5 * ( (y[0] - theta[0] - u[0])^2 / theta[1]^2 + (u[0]+u[1])^2  )
     double diag = 1.0 / (fixed_vec[1] * fixed_vec[1]) + 1.0;
     typedef Eigen::Matrix< double, Eigen::Dynamic, Eigen::Dynamic > double_mat;
     double_mat info_mat = double_mat::Zero(n_random, n_random);
     for(size_t i = 0; i < n_random; i++)
          info_mat(i, i) = diag;
     info_mat(0, 1)  = 1.0;
     info_mat(1, 0)  = 1.0;
     info_mat(1, 1) += 1.0;
     //
     double_mat cov_mat = info_mat.inverse();
     double max_err = 0.0;
     for(size_t i = 0; i < n_random; i++)
     {    for(size_t j = 0; j < n_random; j++)
          {    double value = sample_cov[i * n_random + j];
               double check = cov_mat(i, j);
               max_err = std::max(max_err, fabs(value - check) / diag);
          }
     }
     //
     if( ! ok )
          std::cout << "\nrandom_seed = " << random_seed << "\n";
     //
     CppAD::mixed::free_gsl_rng();
     return ok;
}