differential_entropy_uniform.h

// Description: Differential entropy of a uniform distribution

#ifndef TIM_DIFFERENTIAL_ENTROPY_UNIFORM_H
#define TIM_DIFFERENTIAL_ENTROPY_UNIFORM_H

#include "tim/core/mytypes.h"

#include <pastel/sys/real/real_concept.h>
#include <cmath>

namespace Tim
{

    //! Differential entropy of a uniform distribution.
    /*!
   Preconditions:
   supportVolume > 0

   supportVolume:
   The measure m of the support of the probability density
   function p: m({x in R^n: p(x) != 0})
   */
    template <typename Real>
    Real differentialEntropyUniform(
        const NoDeduction<Real>& supportVolume)
    {
        PENSURE_OP(supportVolume, >, 0);

        // Let X be a random variable in R^n 
        // with a probability density function p.
        // Let S subset R^n be the support set of p.
        // The measure of S is m(S). Then because
        // of uniformity p(x) = 1 / m(S).
        // Now
        // H(X) = -int_S p(x) ln(p(x)) dx
        // = -int_S (1 / m(s)) ln(1 / m(S)) dx
        // = -int_S (1 / m(s)) (ln(1) - ln(m(S))) dx
        // = -int_S (1 / m(s)) (-ln(m(S))) dx
        // = (ln(m(S)) / m(S)) int_S 1 dx
        // = (ln(m(S)) / m(S)) m(S)
        // = ln(m(S))

        return std::log(supportVolume);
    }

}

#endif