@stdlib/stats/base/dists/invgamma/entropy/README.md

Standard library statistical functions.

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# Entropy

> [Inverse gamma][invgamma-distribution] distribution [differential entropy][entropy].

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The [differential entropy][entropy] (in [nats][nats]) for an [inverse gamma][invgamma-distribution] random variable is

<!-- <equation class="equation" label="eq:invgamma_entropy" align="center" raw="h\left( X \right) = \alpha \!+\!\ln(\beta \, \Gamma (\alpha ))\!-\!(1\!+\!\alpha )\Psi (\alpha )" alt="Differential entropy for an inverse gamma distribution."> -->

<div class="equation" align="center" data-raw-text="h\left( X \right) = \alpha \!+\!\ln(\beta \, \Gamma (\alpha ))\!-\!(1\!+\!\alpha )\Psi (\alpha )" data-equation="eq:invgamma_entropy">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/invgamma/entropy/docs/img/equation_invgamma_entropy.svg" alt="Differential entropy for an inverse gamma distribution.">
    <br>
</div>

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where `α > 0` is the shape parameter, `β > 0` is the rate parameter, `Γ` and denotes the [gamma][gamma-function] and `Ψ` the [digamma][digamma] function.

</section>

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<section class="usage">

## Usage

```javascript
var entropy = require( '@stdlib/stats/base/dists/invgamma/entropy' );
```

#### entropy( alpha, beta )

Returns the [differential entropy][entropy] of an [inverse gamma][invgamma-distribution] distribution with shape parameter `alpha` and rate parameter `beta` (in [nats][nats]).

```javascript
var v = entropy( 1.0, 1.0 );
// returns ~2.154

v = entropy( 4.0, 12.0 );
// returns ~1.996

v = entropy( 8.0, 2.0 );
// returns ~-0.922
```

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

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

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

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

```javascript
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`.

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

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

</section>

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</section>

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## Examples

<!-- eslint no-undef: "error" -->

```javascript
var randu = require( '@stdlib/random/base/randu' );
var EPS = require( '@stdlib/constants/float64/eps' );
var entropy = require( '@stdlib/stats/base/dists/invgamma/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 ) );
}
```

</section>

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</section>

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</section>

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<section class="links">

[digamma]: https://en.wikipedia.org/wiki/Digamma_function

[gamma-function]: https://en.wikipedia.org/wiki/Gamma_function

[invgamma-distribution]: https://en.wikipedia.org/wiki/Inverse-gamma_distribution

[entropy]: https://en.wikipedia.org/wiki/Entropy_%28information_theory%29

[nats]: https://en.wikipedia.org/wiki/Nat_%28unit%29

</section>

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