Calculating Probability

Moose Gimmick

Listing 2 uses the CPAN Moose module, which saves me from having to code the Distrib constructor in Perl. However, the code instead needs to use has (line 3) to declare and initialize the class attributes. This is just a neat gimmick in the present case; for more attributes, however, the code would be much leaner than manually creating classes.

The set() method expects the name of a hypothesis (e.g., BB) and its a priori probability and stores this in the object's internal values array. To multiply a hypothesis value by a constant value, both are passed to mult(); it finds the previously stored value in values and multiplies it with the value passed in for $prob.

The normalize() method iterates over all values previously inserted into the hash, adds them up in $sum, and then divides all the values by the sum. Thus, the new sum of all probability values after a multiplication is again 1. Each value can thus be interpreted as a probability between 0 and 1. At the bottom of the module, prob() finds the value for the searched-for hypothesis by retrieving the hash value using the values() method. The latter has been automatically generated by Moose and returns the value stored under the key of the desired hypothesis.

More Abstraction

If you perform several such tests for various problems, you will identify a pattern: After establishing the hypotheses, the probabilities of all defined hypotheses are always multiplied by the likelihood of newly incoming data. It makes sense (as shown in Think Bayes [3]) to define a class derived from Distrib by the name of HypoTest, much as in Listing 3. The class uses an update() method to update all the values for all hypotheses, based on the probabilities of incoming data.

Listing 3

HypoTest.pm

 

HypoTest also relies on classes derived from it (e.g., CardHypoTest in Listing 4) to overload the abstract likelihood() method and return the value for P(D|H) based on the probability of the additionally available data D under the assumption that hypothesis H is true.

Listing 4

hypotest

 

The HypoTest framework in Listing 3 calls the likelihood() method repeatedly to obtain the data probabilities under the assumption of individual hypotheses before storing them in the Distrib distribution. The framework further provides a print() method, which is used to output the values of all the updated probabilities for each hypothesis.

Testing Hypotheses

In Listing 4, likelihood() accepts the letter R from the main program as $data – to record a drawn card with red front as an additional condition. Then, based on the hypothesis also passed in (RR, RB, BB), it computes how likely it is that the test candidate will look at a red surface: 1 (i.e., 100 percent) for RR, 0.5 for RB, and 0 for BB.

For this calculation, the function uses a regular expression that counts the number of Rs in the hypothesis and divides the result by 2 as a floating-point value, so that Perl does not perform integer division and dump the remainder.

Finally, hypotest outputs the probabilities for all hypotheses in the distribution using the print() method from the HypoTest module and correctly reports that the red-red card will appear in 2/3 of all cases:

$ ./hypotest
RC 0.333333333333333
RR: 0.666666666666667
BB 0

In other words, the test candidate, who has just drawn a card with a red front and now turns over this card, will see a red back with probability of 2/3.

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