#ifndef STAN_MATH_PRIM_MAT_PROB_INV_WISHART_LPDF_HPP
#define STAN_MATH_PRIM_MAT_PROB_INV_WISHART_LPDF_HPP

#include <stan/math/prim/mat/err/check_ldlt_factor.hpp>
#include <stan/math/prim/scal/err/check_greater.hpp>
#include <stan/math/prim/scal/err/check_size_match.hpp>
#include <stan/math/prim/mat/err/check_square.hpp>
#include <stan/math/prim/mat/meta/index_type.hpp>
#include <stan/math/prim/mat/fun/log_determinant_ldlt.hpp>
#include <stan/math/prim/mat/fun/mdivide_left_ldlt.hpp>
#include <stan/math/prim/scal/fun/constants.hpp>
#include <stan/math/prim/scal/meta/include_summand.hpp>
#include <stan/math/prim/scal/fun/lmgamma.hpp>
#include <stan/math/prim/mat/fun/trace.hpp>

namespace stan {
  namespace math {
    /**
     * The log of the Inverse-Wishart density for the given W, degrees
     * of freedom, and scale matrix.
     *
     * The scale matrix, S, must be k x k, symmetric, and semi-positive
     * definite.
     *
     * \f{eqnarray*}{
     W &\sim& \mbox{\sf{Inv-Wishart}}_{\nu} (S) \\
     \log (p (W \, |\, \nu, S) ) &=& \log \left( \left(2^{\nu k/2} \pi^{k (k-1) /4} \prod_{i=1}^k{\Gamma (\frac{\nu + 1 - i}{2})} \right)^{-1}
     \times \left| S \right|^{\nu/2} \left| W \right|^{-(\nu + k + 1) / 2}
     \times \exp (-\frac{1}{2} \mbox{tr} (S W^{-1})) \right) \\
     &=& -\frac{\nu k}{2}\log(2) - \frac{k (k-1)}{4} \log(\pi) - \sum_{i=1}^{k}{\log (\Gamma (\frac{\nu+1-i}{2}))}
     +\frac{\nu}{2} \log(\det(S)) - \frac{\nu+k+1}{2}\log (\det(W)) - \frac{1}{2} \mbox{tr}(S W^{-1})
     \f}
     *
     * @param W A scalar matrix
     * @param nu Degrees of freedom
     * @param S The scale matrix
     * @return The log of the Inverse-Wishart density at W given nu and S.
     * @throw std::domain_error if nu is not greater than k-1
     * @throw std::domain_error if S is not square, not symmetric, or not
     * semi-positive definite.
     * @tparam T_y Type of scalar.
     * @tparam T_dof Type of degrees of freedom.
     * @tparam T_scale Type of scale.
     */
    template <bool propto,
              typename T_y, typename T_dof, typename T_scale>
    typename boost::math::tools::promote_args<T_y, T_dof, T_scale>::type
    inv_wishart_lpdf(const
                     Eigen::Matrix<T_y, Eigen::Dynamic, Eigen::Dynamic>& W,
                     const T_dof& nu,
                     const Eigen::Matrix
                     <T_scale, Eigen::Dynamic, Eigen::Dynamic>& S) {
      static const char* function("inv_wishart_lpdf");

      using boost::math::tools::promote_args;
      using Eigen::Dynamic;
      using Eigen::Matrix;

      typename index_type<Matrix<T_scale, Dynamic, Dynamic> >::type k
        = S.rows();
      typename promote_args<T_y, T_dof, T_scale>::type lp(0.0);

      check_greater(function, "Degrees of freedom parameter", nu, k-1);
      check_square(function, "random variable", W);
      check_square(function, "scale parameter", S);
      check_size_match(function,
                       "Rows of random variable", W.rows(),
                       "columns of scale parameter", S.rows());

      LDLT_factor<T_y, Eigen::Dynamic, Eigen::Dynamic> ldlt_W(W);
      check_ldlt_factor(function, "LDLT_Factor of random variable", ldlt_W);
      LDLT_factor<T_scale, Eigen::Dynamic, Eigen::Dynamic> ldlt_S(S);
      check_ldlt_factor(function, "LDLT_Factor of scale parameter", ldlt_S);

      if (include_summand<propto, T_dof>::value)
        lp -= lmgamma(k, 0.5 * nu);
      if (include_summand<propto, T_dof, T_scale>::value) {
        lp += 0.5 * nu * log_determinant_ldlt(ldlt_S);
      }
      if (include_summand<propto, T_y, T_dof, T_scale>::value) {
        lp -= 0.5 * (nu + k + 1.0) * log_determinant_ldlt(ldlt_W);
      }
      if (include_summand<propto, T_y, T_scale>::value) {
        //    L = crossprod(mdivide_left_tri_low(L));
        //    Eigen::Matrix<T_y, Eigen::Dynamic, 1> W_inv_vec = Eigen::Map<
        //      const Eigen::Matrix<T_y, Eigen::Dynamic, Eigen::Dynamic> >(
        //      &L(0), L.size(), 1);
        //    Eigen::Matrix<T_scale, Eigen::Dynamic, 1> S_vec = Eigen::Map<
        //      const Eigen::Matrix<T_scale, Eigen::Dynamic, Eigen::Dynamic> >(
        //      &S(0), S.size(), 1);
        //    lp -= 0.5 * dot_product(S_vec, W_inv_vec); // trace(S * W^-1)
        Eigen::Matrix<typename promote_args<T_y, T_scale>::type,
                      Eigen::Dynamic, Eigen::Dynamic>
          Winv_S(mdivide_left_ldlt
                 (ldlt_W,
                  static_cast<Eigen::Matrix
                  <T_scale, Eigen::Dynamic, Eigen::Dynamic> >
                  (S.template selfadjointView<Eigen::Lower>())));
        lp -= 0.5*trace(Winv_S);
      }
      if (include_summand<propto, T_dof, T_scale>::value)
        lp += nu * k * NEG_LOG_TWO_OVER_TWO;
      return lp;
    }

    template <typename T_y, typename T_dof, typename T_scale>
    inline
    typename boost::math::tools::promote_args<T_y, T_dof, T_scale>::type
    inv_wishart_lpdf(const
                     Eigen::Matrix<T_y, Eigen::Dynamic, Eigen::Dynamic>& W,
                     const T_dof& nu,
                     const Eigen::Matrix
                     <T_scale, Eigen::Dynamic, Eigen::Dynamic>& S) {
      return inv_wishart_lpdf<false>(W, nu, S);
    }

  }
}
#endif
