Logarithm of Probability Density Function

Beta prime distribution logarithm of probability density function (PDF).

The probability density function (PDF) for a beta prime random variable is

f left-parenthesis x semicolon alpha comma beta right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column StartFraction normal upper Gamma left-parenthesis alpha plus beta right-parenthesis Over normal upper Gamma left-parenthesis alpha right-parenthesis plus normal upper Gamma left-parenthesis beta right-parenthesis EndFraction x Superscript alpha minus 1 Baseline left-parenthesis 1 plus x right-parenthesis Superscript negative alpha minus beta Baseline 2nd Column for x greater-than 0 2nd Row 1st Column 0 2nd Column otherwise EndLayout

where α > 0 is the first shape parameter and β > 0 is the second shape parameter.

Usage

var logpdf = require( '@stdlib/stats/base/dists/betaprime/logpdf' );

logpdf( x, alpha, beta )

Evaluates the natural logarithm of the probability density function (PDF) for a beta prime distribution with parameters alpha (first shape parameter) and beta (second shape parameter).

var y = logpdf( 0.5, 0.5, 1.0 );
// returns ~-0.955

y = logpdf( 0.1, 1.0, 1.0 );
// returns ~-0.191

y = logpdf( 0.8, 4.0, 2.0 );
// returns ~-1.2

If provided an input value x outside smaller or equal to zero, the function returns -Infinity.

var y = logpdf( -0.1, 1.0, 1.0 );
// returns -Infinity

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

var y = logpdf( NaN, 1.0, 1.0 );
// returns NaN

y = logpdf( 0.0, NaN, 1.0 );
// returns NaN

y = logpdf( 0.0, 1.0, NaN );
// returns NaN

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

var y = logpdf( 0.5, 0.0, 1.0 );
// returns NaN

y = logpdf( 0.5, -1.0, 1.0 );
// returns NaN

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

var y = logpdf( 0.5, 1.0, 0.0 );
// returns NaN

y = logpdf( 0.5, 1.0, -1.0 );
// returns NaN

logpdf.factory( alpha, beta )

Returns a function for evaluating the natural logarithm of the PDF for a beta prime distribution with parameters alpha (first shape parameter) and beta (second shape parameter).

var mylogPDF = logpdf.factory( 0.5, 0.5 );

var y = mylogPDF( 0.8 );
// returns ~-1.62

y = mylogPDF( 0.3 );
// returns ~-0.805

Notes

  • In virtually all cases, using the logpdf or logcdf functions is preferable to manually computing the logarithm of the pdf or cdf, respectively, since the latter is prone to overflow and underflow.

Examples

var randu = require( '@stdlib/random/base/randu' );
var EPS = require( '@stdlib/constants/float64/eps' );
var logpdf = require( '@stdlib/stats/base/dists/betaprime/logpdf' );

var alpha;
var beta;
var x;
var y;
var i;

for ( i = 0; i < 10; i++ ) {
    x = randu();
    alpha = ( randu()*5.0 ) + EPS;
    beta = ( randu()*5.0 ) + EPS;
    y = logpdf( x, alpha, beta );
    console.log( 'x: %d, α: %d, β: %d, ln(f(x;α,β)): %d', x.toFixed( 4 ), alpha.toFixed( 4 ), beta.toFixed( 4 ), y.toFixed( 4 ) );
}
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