Entropy

Gamma distribution differential entropy.

The differential entropy (in nats) for a gamma random variable with shape parameter α > 0 and rate parameter β > 0 is

h left-parenthesis upper X right-parenthesis equals alpha minus ln beta plus ln left-bracket normal upper Gamma left-parenthesis alpha right-parenthesis right-bracket plus left-parenthesis 1 minus alpha right-parenthesis psi left-parenthesis alpha right-parenthesis

where Γ and Ψ denote the gamma and digamma function, respectively.

Usage

var entropy = require( '@stdlib/math/base/dists/gamma/entropy' );

entropy( alpha, beta )

Returns the differential entropy of a gamma distribution with shape parameter alpha and rate parameter beta (in nats).

var v = entropy( 1.0, 1.0 );
// returns 1.0

v = entropy( 4.0, 12.0 );
// returns ~-0.462

v = entropy( 8.0, 2.0 );
// returns ~1.723

If provided NaN as any argument, the function returns NaN.

var v = entropy( NaN, 2.0 );
// returns NaN

v = entropy( 2.0, NaN );
// returns NaN

If provided alpha <= 0, the function returns NaN.

var v = entropy( 0.0, 1.0 );
// returns NaN

v = entropy( -1.0, 1.0 );
// returns NaN

If provided beta <= 0, the function returns NaN.

var v = entropy( 1.0, 0.0 );
// returns NaN

v = entropy( 1.0, -1.0 );
// returns NaN

Examples

var randu = require( '@stdlib/random/base/randu' );
var EPS = require( '@stdlib/constants/math/float64-eps' );
var entropy = require( '@stdlib/math/base/dists/gamma/entropy' );

var alpha;
var beta;
var v;
var i;

for ( i = 0; i < 10; i++ ) {
    alpha = ( randu()*10.0 ) + EPS;
    beta = ( randu()*10.0 ) + EPS;
    v = entropy( alpha, beta );
    console.log( 'α: %d, β: %d, h(X;α,β): %d', alpha.toFixed( 4 ), beta.toFixed( 4 ), v.toFixed( 4 ) );
}