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Conditional Probability of Startup Success [Bayesian Inference
Posted on April 8, 2013 @ 08:39:00 AM by Paul Meagher

In this blog post, I'll be going over the concept of Conditional Probability (i.e., P(H|E). I'll be reusing some of my earlier writings on bayesian inference using a medical example and substituting in an angel investing example. The concept of conditional probability is central to Bayesian inference. A bayesian angel investor is always computing the probability of some hypothesis given some pattern of evidence P(H|E). There are many mathematical techniques you can use to compute a conditional probability P(H|E), but the simplest way involves set enumeration and it is what clergyman Thomas Bayes had in mind when he proposed his new method of inference. So hopefully you will learn one important method for computing a conditional probability from reading this blog post.

Imagine that H refers to "Company is Successful" and E refers to "Quality Business Plan". P(H | E) would then read as the "probability that a company is successful (H) given that they have a quality business plan (E)." If H tends to occur when E occurs, then knowing that E has occurred allows you to assign a higher probability to H's occurrence than in a situation in which you did not know that E occurred.

More generally, if H and E systematically co-vary in some way, then P(H | E) will not be equal to P(H). Conversely, if H and E are independent events, then P(H | E) would be expected to equal P(H).

The need to compute a conditional probability thus arises any time you think the occurence of some event has a bearing on the probability of another event's occurring.

The most basic and intuitive method for computing P(H | E) is the set enumeration method. Using this method, P(H | E) can be computed by counting the number of times H and E occur together {H & E} and dividing by the number of times E occurs {E}:

P(H | E) = {H & E} / {E}

If you gave your ok to 12 business plans to date, and observed that 10 of those companies were successful, then P(H | E) would be estimated at 10/12 or 0.833. In other words, the probability of a company being successful given that they have a quality business plan can be estimated at 83 percent by using a method that involves enumerating the relative frequencies of H and E events from the data gathered to date.

Computing a conditional probability becomes a form of inference when we take into account that the prior probability P(H) that a startup would be successful was probably lower than 83 percent. So conditionalizing our hypothesis (company will succeed) on other information (business plan quality) helped to increase our estimate of the probability that a startup would be successful. We can make decisions to proceed further based upon this improved knowledge.

You can compute a conditional probability using the set enumeration method with the PHP code below.

<?php

/**
* @script conditional_probability.php
* @author: paul@datavore.com
*
* @purpose: Illustrates how to compute a conditional probability 
* using set enumeration
*/

/**
* Returns conditional probability of $A given $B and $Data.
* $Data is an indexed array.  Each element of the $Data array 
* consists of an A measurement and B measurment on a sample 
* item.
*/
function getConditionalProbabilty($A$B$Data) {
  
$NumAB   0;
  
$NumB    0;
  
$NumData count($Data);
  for (
$i=0$i $NumData$i++) {
    if (
in_array($B$Data[$i])) {
      
$NumB++;
      if (
in_array($A$Data[$i])) {
        
$NumAB++;
      }
    }
  }
  return 
$NumAB $NumB;
}

/**
* The elements of the $Data array use this coding convention:
*
* +success - company is successful
* -success - company is not successful
* +bizplan - bizplan passed quality test
* -buzplan - bizplan failed quality test
*/

$Data[0]  = array("+success""+bizplan");
$Data[1]  = array("+success""+bizplan");
$Data[2]  = array("+success""+bizplan");
$Data[3]  = array("+success""+bizplan");
$Data[4]  = array("+success""+bizplan");
$Data[5]  = array("+success""+bizplan");
$Data[6]  = array("+success""+bizplan");
$Data[7]  = array("+success""+bizplan");
$Data[8]  = array("+success""+bizplan");
$Data[9]  = array("+success""+bizplan");
$Data[10] = array("-success""+bizplan");
$Data[11] = array("-success""+bizplan");

// specify query variable $A and conditioning variable $B
$A "+success";
$B "+bizplan";

// compute the conditional probability of having cancer given 1) 
// a positive test and 2) a sample of covariation data
$probability getConditionalProbabilty($A$B$Data);

echo 
"P($A|$B) = $probability";

// P(+success|+bizplan) = 0.83333333333333

?>

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