Implement Bayesian inference using PHP: Part 1, IBM developerWorks, March 16, 2004.
Conditional probability -- the probability of observing one event given that you have observed another event -- is a potentially important factor in designing intelligent Web applications. Paul Meagher introduces Bayesian inference by discussing the basic mathematical concepts involved and demonstrating how to implement the underlying conditional probability calculations using PHP. In this article, the author discusses how Bayesian inference can be used to build an online PHP-based wizard that guides a user through the process making a medical diagnosis. This three-part series features interesting applications designed to help you appreciate the power and potential of Bayesian inference concepts.
Last updated: April 14, 2004 - 2:12 PM AST
Demos:
<?php
/**
* Bayes.php
*
* @author Paul Meagher <paul@datavore.com>
* @license PHP License v3.0
*
* Calculates posterior probabilities for m hypotheses and n evidence
* alternatives. The code was inspired by a procedural TrueBasic version
* (Bayes.tru) bundled with Grimstead and Snell's excellent online
* textbook "Introduction to Probability".
*
* @see http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html
*/
class Bayes {
/**
* Number of evidence alternatives (i.e., number of rows).
*/
var $m;
/**
* Number of hypothesis alternatives (i.e., number of columns).
*/
var $n;
/**
* Output labels for evidence alternatives.
*/
var $row_labels = array();
/**
* Output labels for hypothesis alternatives.
*/
var $column_labels = array();
/**
* Vector container for prior probabilities.
*/
var $priors = array();
/**
* Matrix container for likelihood of evidence e given hypothesis h.
*/
var $likelihoods = array();
/**
* Matrix container for posterior probabilties.
*/
var $posterior = array();
/**
* Vector container for evidence probabilties.
*/
var $evidence = array();
/**
* Initialize the Bayes algorithm by setting the priors, likelihoods and
* dimensions of the likelihood and posterior matrices.
*/
function Bayes($priors, $likelihoods) {
$this->priors = $priors;
$this->likelihoods = $likelihoods;
$this->m = count($this->likelihoods); // num rows
$this->n = count($this->likelihoods[0]); // num cols
return true;
}
/**
* Output method for setting row labels prior to display.
*/
function setRowLabels($row_labels) {
$this->row_labels = $row_labels;
return true;
}
/**
* Output method for setting column labels prior to display.
*/
function setColumnLabels($column_labels) {
$this->column_labels = $column_labels;
return true;
}
/**
* Compute the posterior probability matrix given the priors and likelihoods.
*
* The first set of loops computes the denominator of the canonical Bayes
* equation. The probability appearing in the denominator serves a normalizing
* role in the computation - it ensures that posterior probabilities sum to 1.
*
* The second set of loops:
*
* 1. multiplies the prior[$h] by the likelihood[$h][$e]
* 2. divides the result by the evidence[$e]
* 3. assigns the result to the posterior[$e][$h] probability matrix
*/
function getPosterior() {
// Find probability of evidence e
for($e=0; $e < $this->n; $e++) {
for ($h=0; $h < $this->m; $h++) {
$this->evidence[$e] += $this->priors[$h] * $this->likelihoods[$h][$e];
}
}
// Find probability of hypothesis given evidence
for($e=0; $e < $this->n; $e++) {
for ($h=0; $h < $this->m; $h++) {
$this->posterior[$e][$h] = $this->priors[$h] * $this->likelihoods[$h][$e] / $this->evidence[$e];
}
}
return true;
}
/**
* Output method for displaying posterior probability matrix
*/
function toHTML($number_format="%01.3f") {
?>
<table border='1' cellpadding='5' cellspacing='0'>
<tr>
<td> </td>
<?php
for ($h=0; $h < $this->m; $h++) {
?>
<td align='center'><b><?php echo $this->column_labels[$h] ?></b></td>
<?php
}
?>
</tr>
<?php
for($e=0; $e < $this->n; $e++) {
?>
<tr>
<td><b><?php echo $this->row_labels[$e] ?></b></td>
<?php
for ($h=0; $h < $this->m; $h++) {
?>
<td align='right'><?php printf($number_format, $this->posterior[$e][$h]) ?></td>
<?php
}
?>
</tr>
<?php
}
?>
</table>
<?php
}
}
?>