inv_gamma.hpp

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#ifndef __STAN__PROB__DISTRIBUTIONS__UNIVARIATE__CONTINUOUS__INV_GAMMA_HPP__
#define __STAN__PROB__DISTRIBUTIONS__UNIVARIATE__CONTINUOUS__INV_GAMMA_HPP__

#include <boost/random/gamma_distribution.hpp>
#include <boost/random/variate_generator.hpp>

#include <stan/agrad.hpp>
#include <stan/math/error_handling.hpp>
#include <stan/math/functions/value_of.hpp>
#include <stan/meta/traits.hpp>
#include <stan/prob/constants.hpp>
#include <stan/prob/traits.hpp>
#include <stan/prob/internal_math.hpp>

namespace stan {

  namespace prob {

    template <bool propto,
              typename T_y, typename T_shape, typename T_scale, 
              class Policy>
    typename return_type<T_y,T_shape,T_scale>::type
    inv_gamma_log(const T_y& y, const T_shape& alpha, const T_scale& beta, 
                  const Policy&) {
      static const char* function = "stan::prob::inv_gamma_log(%1%)";

      using stan::is_constant_struct;
      using stan::math::check_not_nan;
      using stan::math::check_positive;
      using stan::math::check_finite;
      using boost::math::tools::promote_args;
      using stan::math::check_consistent_sizes;
      using stan::math::value_of;

      // check if any vectors are zero length
      if (!(stan::length(y) 
            && stan::length(alpha) 
            && stan::length(beta)))
        return 0.0;

      // set up return value accumulator
      double logp(0.0);

      if (!check_not_nan(function, y, "Random variable", &logp, Policy()))
        return logp;
      if (!check_finite(function, alpha, "Shape parameter", 
                        &logp, Policy())) 
        return logp;
      if (!check_positive(function, alpha, "Shape parameter",
                          &logp, Policy())) 
        return logp;
      if (!check_finite(function, beta, "Scale parameter",
                        &logp, Policy())) 
        return logp;
      if (!check_positive(function, beta, "Scale parameter", 
                          &logp, Policy())) 
        return logp;
      if (!(check_consistent_sizes(function,
                                   y,alpha,beta,
                                   "Random variable","Shape parameter",
                                   "Scale parameter",
                                   &logp, Policy())))
        return logp;

      // check if no variables are involved and prop-to
      if (!include_summand<propto,T_y,T_shape,T_scale>::value)
        return 0.0;

      // set up template expressions wrapping scalars into vector views
      VectorView<const T_y> y_vec(y);
      VectorView<const T_shape> alpha_vec(alpha);
      VectorView<const T_scale> beta_vec(beta);

      for (size_t n = 0; n < length(y); n++) {
        const double y_dbl = value_of(y_vec[n]);
        if (y_dbl <= 0)
          return LOG_ZERO;
      }

      size_t N = max_size(y, alpha, beta);
      agrad::OperandsAndPartials<T_y, T_shape, T_scale> 
        operands_and_partials(y, alpha, beta);
      
      using boost::math::lgamma;
      using stan::math::multiply_log;
      using boost::math::digamma;

      DoubleVectorView<include_summand<propto,T_y,T_shape>::value,
                       is_vector<T_y>::value>
        log_y(length(y));
      DoubleVectorView<include_summand<propto,T_y,T_scale>::value,
                       is_vector<T_y>::value>
        inv_y(length(y));
      for(size_t n = 0; n < length(y); n++) {
        if (include_summand<propto,T_y,T_shape>::value)
          if (value_of(y_vec[n]) > 0)
            log_y[n] = log(value_of(y_vec[n]));
        if (include_summand<propto,T_y,T_scale>::value)
          inv_y[n] = 1.0 / value_of(y_vec[n]);
      }

      DoubleVectorView<include_summand<propto,T_shape>::value,
                       is_vector<T_shape>::value>
        lgamma_alpha(length(alpha));
      DoubleVectorView<!is_constant_struct<T_shape>::value,
                       is_vector<T_shape>::value>
        digamma_alpha(length(alpha));
      for (size_t n = 0; n < length(alpha); n++) {
        if (include_summand<propto,T_shape>::value)
          lgamma_alpha[n] = lgamma(value_of(alpha_vec[n]));
        if (!is_constant_struct<T_shape>::value)
          digamma_alpha[n] = digamma(value_of(alpha_vec[n]));
      }

      DoubleVectorView<include_summand<propto,T_shape,T_scale>::value,
                       is_vector<T_scale>::value>
        log_beta(length(beta));
      if (include_summand<propto,T_shape,T_scale>::value)
        for (size_t n = 0; n < length(beta); n++)
          log_beta[n] = log(value_of(beta_vec[n]));

      for (size_t n = 0; n < N; n++) {
        // pull out values of arguments
        const double alpha_dbl = value_of(alpha_vec[n]);
        const double beta_dbl = value_of(beta_vec[n]);

        if (include_summand<propto,T_shape>::value)
          logp -= lgamma_alpha[n];
        if (include_summand<propto,T_shape,T_scale>::value)
          logp += alpha_dbl * log_beta[n];
        if (include_summand<propto,T_y,T_shape>::value)
          logp -= (alpha_dbl+1.0) * log_y[n];
        if (include_summand<propto,T_y,T_scale>::value)
          logp -= beta_dbl * inv_y[n];
  
        // gradients
        if (!is_constant<typename is_vector<T_y>::type>::value)
          operands_and_partials.d_x1[n] 
            += -(alpha_dbl+1) * inv_y[n] + beta_dbl * inv_y[n] * inv_y[n];
        if (!is_constant<typename is_vector<T_shape>::type>::value)
          operands_and_partials.d_x2[n] 
            += -digamma_alpha[n] + log_beta[n] - log_y[n];
        if (!is_constant<typename is_vector<T_scale>::type>::value)
          operands_and_partials.d_x3[n] += alpha_dbl / beta_dbl - inv_y[n];
      }
      return operands_and_partials.to_var(logp);
    }

    template <bool propto,
              typename T_y, typename T_shape, typename T_scale>
    inline
    typename return_type<T_y,T_shape,T_scale>::type
    inv_gamma_log(const T_y& y, const T_shape& alpha, const T_scale& beta) {
      return inv_gamma_log<propto>(y,alpha,beta,stan::math::default_policy());
    }

    template <typename T_y, typename T_shape, typename T_scale, 
              class Policy>
    inline
    typename return_type<T_y,T_shape,T_scale>::type
    inv_gamma_log(const T_y& y, const T_shape& alpha, const T_scale& beta, 
                  const Policy&) {
      return inv_gamma_log<false>(y,alpha,beta,Policy());
    }

    template <typename T_y, typename T_shape, typename T_scale>
    inline
    typename return_type<T_y,T_shape,T_scale>::type
    inv_gamma_log(const T_y& y, const T_shape& alpha, const T_scale& beta) {
      return inv_gamma_log<false>(y,alpha,beta,stan::math::default_policy());
    }

    template <typename T_y, typename T_shape, typename T_scale, class Policy>
    typename return_type<T_y,T_shape,T_scale>::type
    inv_gamma_cdf(const T_y& y, const T_shape& alpha, const T_scale& beta, 
                  const Policy&) { 
          
      // Size checks
      if (!(stan::length(y) && stan::length(alpha) && stan::length(beta))) 
        return 1.0;
          
      // Error checks
      static const char* function = "stan::prob::inv_gamma_cdf(%1%)";
          
      using stan::math::check_finite;      
      using stan::math::check_positive;
      using stan::math::check_not_nan;
      using stan::math::check_consistent_sizes;
      using stan::math::check_greater_or_equal;
      using stan::math::check_less_or_equal;
      using stan::math::check_nonnegative;
      using stan::math::value_of;

      using boost::math::tools::promote_args;
          
      double P(1.0);
          
      if (!check_finite(function, alpha, "Shape parameter", &P, Policy())) 
        return P;
          
      if (!check_positive(function, alpha, "Shape parameter", &P, Policy())) 
        return P;
          
      if (!check_finite(function, beta, "Scale parameter", &P, Policy())) 
        return P;
          
      if (!check_positive(function, beta, "Scale parameter", &P, Policy())) 
        return P;
          
      if (!check_not_nan(function, y, "Random variable", &P, Policy()))
        return P;
          
      if (!check_nonnegative(function, y, "Random variable", &P, Policy())) 
        return P;
          
      if (!(check_consistent_sizes(function, y, alpha, beta,
                                   "Random variable", "Shape parameter", 
                                   "Scale Parameter",
                                   &P, Policy())))
        return P;
          
      // Wrap arguments in vectors
      VectorView<const T_y> y_vec(y);
      VectorView<const T_shape> alpha_vec(alpha);
      VectorView<const T_scale> beta_vec(beta);
      size_t N = max_size(y, alpha, beta);
          
      agrad::OperandsAndPartials<T_y, T_shape, T_scale> 
        operands_and_partials(y, alpha, beta);
          
      std::fill(operands_and_partials.all_partials,
                operands_and_partials.all_partials 
                + operands_and_partials.nvaris, 0.0);
          
      // Explicit return for extreme values
      // The gradients are technically ill-defined, but treated as zero
          
      for (size_t i = 0; i < stan::length(y); i++) {
        if (value_of(y_vec[i]) == 0) 
          return operands_and_partials.to_var(0.0);
      }
          
      // Compute CDF and its gradients
      using boost::math::gamma_p_derivative;
      using boost::math::gamma_q;
      using boost::math::digamma;
      using boost::math::tgamma;
          
      // Cache a few expensive function calls if nu is a parameter
      DoubleVectorView<!is_constant_struct<T_shape>::value,
                       is_vector<T_shape>::value> 
        gamma_vec(stan::length(alpha));
      DoubleVectorView<!is_constant_struct<T_shape>::value,
                       is_vector<T_shape>::value> 
        digamma_vec(stan::length(alpha));
          
      if (!is_constant_struct<T_shape>::value) {
        for (size_t i = 0; i < stan::length(alpha); i++) {
          const double alpha_dbl = value_of(alpha_vec[i]);
          gamma_vec[i] = tgamma(alpha_dbl);
          digamma_vec[i] = digamma(alpha_dbl);
        }
      }
          
      // Compute vectorized CDF and gradient
      for (size_t n = 0; n < N; n++) {
        // Explicit results for extreme values
        // The gradients are technically ill-defined, but treated as zero
        if (value_of(y_vec[n]) == std::numeric_limits<double>::infinity())
          continue;
              
        // Pull out values
        const double y_dbl = value_of(y_vec[n]);
        const double y_inv_dbl = 1.0 / y_dbl;
        const double alpha_dbl = value_of(alpha_vec[n]);
        const double beta_dbl = value_of(beta_vec[n]);
              
        // Compute
        const double Pn = gamma_q(alpha_dbl, beta_dbl * y_inv_dbl);
              
        P *= Pn;
              
        if (!is_constant_struct<T_y>::value)
          operands_and_partials.d_x1[n] 
            += beta_dbl * y_inv_dbl * y_inv_dbl 
            * gamma_p_derivative(alpha_dbl, beta_dbl * y_inv_dbl) 
            / Pn;
              
        if (!is_constant_struct<T_shape>::value)
          operands_and_partials.d_x2[n] 
            += stan::math::gradRegIncGamma(alpha_dbl, beta_dbl
                                           * y_inv_dbl, gamma_vec[n],
                                           digamma_vec[n]) / Pn;
              
        if (!is_constant_struct<T_scale>::value)
          operands_and_partials.d_x3[n] 
            += - y_inv_dbl * gamma_p_derivative(alpha_dbl, 
                                                beta_dbl * y_inv_dbl) / Pn;
              
      }
          
      if (!is_constant_struct<T_y>::value)
        for (size_t n = 0; n < stan::length(y); ++n) 
          operands_and_partials.d_x1[n] *= P;
          
      if (!is_constant_struct<T_shape>::value)
        for (size_t n = 0; n < stan::length(alpha); ++n) 
          operands_and_partials.d_x2[n] *= P;
          
      if (!is_constant_struct<T_scale>::value)
        for (size_t n = 0; n < stan::length(beta); ++n) 
          operands_and_partials.d_x3[n] *= P;
          
      return operands_and_partials.to_var(P);
    }
      
    template <typename T_y, typename T_shape, typename T_scale>
    inline typename return_type<T_y, T_shape, T_scale>::type
    inv_gamma_cdf(const T_y& y, const T_shape& alpha, const T_scale& beta) {
      return inv_gamma_cdf(y, alpha, beta, stan::math::default_policy());
    }

    template <class RNG>
    inline double
    inv_gamma_rng(const double alpha,
                  const double beta,
                  RNG& rng) {
      using boost::variate_generator;
      using boost::random::gamma_distribution;
      variate_generator<RNG&, gamma_distribution<> >
        gamma_rng(rng, gamma_distribution<>(alpha, 1 / beta));
      return 1 / gamma_rng();
    }          
  }
}

#endif