README.md (4702B)
1 <!-- 2 3 @license Apache-2.0 4 5 Copyright (c) 2018 The Stdlib Authors. 6 7 Licensed under the Apache License, Version 2.0 (the "License"); 8 you may not use this file except in compliance with the License. 9 You may obtain a copy of the License at 10 11 http://www.apache.org/licenses/LICENSE-2.0 12 13 Unless required by applicable law or agreed to in writing, software 14 distributed under the License is distributed on an "AS IS" BASIS, 15 WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 16 See the License for the specific language governing permissions and 17 limitations under the License. 18 19 --> 20 21 # Logarithm of Probability Density Function 22 23 > [Beta][beta-distribution] distribution logarithm of probability density function (PDF). 24 25 <section class="intro"> 26 27 The [probability density function][pdf] (PDF) for a [beta][beta-distribution] random variable is 28 29 <!-- <equation class="equation" label="eq:beta_pdf" align="center" raw="f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) + \Gamma(\beta)}{x^{\alpha-1}(1-x)^{\beta-1}} & \text{ for } x \in (0,1) \\ 0 & \text{ otherwise } \end{cases}" alt="Probability density function (PDF) for a beta distribution."> --> 30 31 <div class="equation" align="center" data-raw-text="f(x;\alpha,\beta)= \begin{cases} \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) + \Gamma(\beta)}{x^{\alpha-1}(1-x)^{\beta-1}} & \text{ for } x \in (0,1) \\ 0 & \text{ otherwise } \end{cases}" data-equation="eq:beta_pdf"> 32 <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@51534079fef45e990850102147e8945fb023d1d0/lib/node_modules/@stdlib/stats/base/dists/beta/logpdf/docs/img/equation_beta_pdf.svg" alt="Probability density function (PDF) for a beta distribution."> 33 <br> 34 </div> 35 36 <!-- </equation> --> 37 38 where `alpha > 0` is the first shape parameter and `beta > 0` is the second shape parameter. 39 40 </section> 41 42 <!-- /.intro --> 43 44 <section class="usage"> 45 46 ## Usage 47 48 ```javascript 49 var logpdf = require( '@stdlib/stats/base/dists/beta/logpdf' ); 50 ``` 51 52 #### logpdf( x, alpha, beta ) 53 54 Evaluates the natural logarithm of the [probability density function][pdf] (PDF) for a [beta][beta-distribution] distribution with parameters `alpha` (first shape parameter) and `beta` (second shape parameter). 55 56 ```javascript 57 var y = logpdf( 0.5, 0.5, 1.0 ); 58 // returns ~-0.347 59 60 y = logpdf( 0.1, 1.0, 1.0 ); 61 // returns 0.0 62 63 y = logpdf( 0.8, 4.0, 2.0 ); 64 // returns ~0.717 65 ``` 66 67 If provided an input value `x` outside the support `[0,1]`, the function returns `-Infinity`. 68 69 ```javascript 70 var y = logpdf( -0.1, 1.0, 1.0 ); 71 // returns -Infinity 72 73 y = logpdf( 1.1, 1.0, 1.0 ); 74 // returns -Infinity 75 ``` 76 77 If provided `NaN` as any argument, the function returns `NaN`. 78 79 ```javascript 80 var y = logpdf( NaN, 1.0, 1.0 ); 81 // returns NaN 82 83 y = logpdf( 0.0, NaN, 1.0 ); 84 // returns NaN 85 86 y = logpdf( 0.0, 1.0, NaN ); 87 // returns NaN 88 ``` 89 90 If provided `alpha <= 0`, the function returns `NaN`. 91 92 ```javascript 93 var y = logpdf( 0.5, 0.0, 1.0 ); 94 // returns NaN 95 96 y = logpdf( 0.5, -1.0, 1.0 ); 97 // returns NaN 98 ``` 99 100 If provided `beta <= 0`, the function returns `NaN`. 101 102 ```javascript 103 var y = logpdf( 0.5, 1.0, 0.0 ); 104 // returns NaN 105 106 y = logpdf( 0.5, 1.0, -1.0 ); 107 // returns NaN 108 ``` 109 110 #### logpdf.factory( alpha, beta ) 111 112 Returns a `function` for evaluating the natural logarithm of the [PDF][pdf] for a [beta][beta-distribution] distribution with parameters `alpha` (first shape parameter) and `beta` (second shape parameter). 113 114 ```javascript 115 var mylogPDF = logpdf.factory( 0.5, 0.5 ); 116 117 var y = mylogPDF( 0.8 ); 118 // returns ~-0.228 119 120 y = mylogPDF( 0.3 ); 121 // returns ~-0.364 122 ``` 123 124 </section> 125 126 <!-- /.usage --> 127 128 <section class="notes"> 129 130 ## Notes 131 132 - 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. 133 134 </section> 135 136 <!-- /.notes --> 137 138 <section class="examples"> 139 140 ## Examples 141 142 <!-- eslint no-undef: "error" --> 143 144 ```javascript 145 var randu = require( '@stdlib/random/base/randu' ); 146 var EPS = require( '@stdlib/constants/float64/eps' ); 147 var logpdf = require( '@stdlib/stats/base/dists/beta/logpdf' ); 148 149 var alpha; 150 var beta; 151 var x; 152 var y; 153 var i; 154 155 for ( i = 0; i < 10; i++ ) { 156 x = randu(); 157 alpha = ( randu()*5.0 ) + EPS; 158 beta = ( randu()*5.0 ) + EPS; 159 y = logpdf( x, alpha, beta ); 160 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 ) ); 161 } 162 ``` 163 164 </section> 165 166 <!-- /.examples --> 167 168 <section class="links"> 169 170 [beta-distribution]: https://en.wikipedia.org/wiki/Beta_distribution 171 172 [pdf]: https://en.wikipedia.org/wiki/Probability_density_function 173 174 </section> 175 176 <!-- /.links -->