differential_entropy_generalized_normal.h

// Description: Differential entropy of a generalized normal distribution

#ifndef TIM_DIFFERENTIAL_ENTROPY_GENERALIZED_NORMAL_H
#define TIM_DIFFERENTIAL_ENTROPY_GENERALIZED_NORMAL_H

#include "tim/core/mytypes.h"

namespace Tim
{

    //! Differential entropy of a generalized normal distribution.
    /*!
   Preconditions:
   dimension > 0
   shape > 0
   scale > 0

   dimension:
   The dimension of the distribution.

   shape:
   The shape parameter of the distribution.

   scale:
   The scale parameter of the distribution.
   */
    template <typename Real>
    Real differentialEntropyGeneralizedNormal(
        integer dimension, 
        const NoDeduction<Real>& shape, 
        const NoDeduction<Real>& scale)
    {
        PENSURE_OP(dimension, >, 0);
        PENSURE_OP(shape, >, 0);
        PENSURE_OP(scale, >, 0);

        // Let
        // a = scale
        // b = shape
        //
        // Then
        //
        // differential entropy 
        // = (1 / b) - log(b / (2a gamma(1 / b)))
        // = (1 / b) - (log(b / (2a)) - log(gamma(1 / b)))
        // = (1 / b) - log(b / (2a)) + log(gamma(1 / b))
        // = (1 / b) + log((2a) / b) + log(gamma(1 / b))
        // = (1 / b) + log((2a) / b) + log(gamma(1 / b))
        //
        // The point of this derivation is to evaluate lnGamma 
        // instead of gamma for better numerical behaviour.

        Real invShape = inverse(shape);

        return dimension * (
            inverse(shape) + 
            std::log(2 * scale * invShape) +
            lnGamma<dreal>(invShape));
    }

}

#endif