Implementing the calculation with Bayes.php
The Bayes.php class implements the Bayes theorem calculation. The getPosterior method is where most of the mathematically interesting code resides.
Listing 5. Implementing the calculation with the Bayes.php class
<?php
/**
* Bayes
*
* 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".
*/
class Bayes {
/**
* Number of evidence alternatives (that is, number of rows).
*/
var $m;
/**
* Number of hypothesis alternatives (that is, 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 denominator
* 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
}
}
?>
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Sensitivity analysis
An important aspect of Bayesian inference involves examining the effect of
small changes to your prior and likelihood distributions. If the prior
probability values you are using are viewed as best guesses, then you
might want to see what happens when you adjust the prior probabilities
of each hypothesis slightly. You may notice that this significantly
changes the posterior distribution values or it might have little
effect. It is good to know how sensitive your results are to the exact
prior values (or likelihood values) used.
The final screen of the Bayes diagnosis wizard gives you the options to
- Start again
- Re-enter labels
- Re-enter your priors
- Re-enter your likelihoods
If you decide to re-enter your priors, the wizard remembers your
previously entered likelihoods. After you re-enter your priors, you can
click forward to Step 5 without having to re-enter your likelihood
values (or you can modify the likelihoods as well). In other words, the
design of the Bayes wizard encourages you to engage in sensitivity
analysis prior to drawing any final conclusions.
