Friday, August 13, 2010

Our friend Bayes -- Part III

In this Part III [of IV] of our series on Bayes we complete the Bayes Grid. If you missed Part I or Part II, you can click to catch up.

Bayes' Grid
Here's where we left off in Part II. 'A' is an independent probabilistic event, in this case the weather, and we have empirical observations that give us the probability of good weather, 'A+', as 0.6 .  We are seeking information about the project test results, B, for which we have one project observation: the conditional situation of  'B+' when 'A+' is present, 90% probable.  And again, this is not an intersection of two events--good weather and good results happening in the same timeframe--it's a dependency: good results because of good weather.

Now, as we said in Part II, without another independent observation, we can go no farther. 

New observation
Let's assume that in the course of testing, the test manager observes that given bad weather conditions, 'B+ | A-', the B+ success rate is 2%, thus showing that even given the condition of "the weather is not favorable", there are positive test results. 

Take note: the 2% success of 'B+ | A-' may be erroroneous results.  In other words, the test may be designed to fail if the weather is not good [ie, test results are dependent on weather which is our theme for this example].  Or, there may be a misunderstanding of cause and effect.  In any event, again we return to Bayes' equation:
P(B+ | A-) = P(B+ and A-) * P(A-) = 0.02
Solving for the intersection, we find
P(B+ and A-) = 0.008

Using the result from above and the grid math to compute 'B- and A-' = 0.392, we now have this grid:

Grid results

The computed figures in the light blue column adjacent to the test results arise from the grid math that requires all columns and rows to add.

We also can validate this grid: the dark blue cells sum to 1.0.  They sum to their counterparts in the light blue by column and row, and the light blue columns and rows sum to 1.0.  All space has been accounted for.

We now have a result for the elusive B+: We see that 54.8% of the time the test results will be good, and 54% of the time good results will coincide--that is, intersect--with good weather.  In fact, if the weather is good, as it is 60% of the time, then we forecast 90% test success.

To be continued
In the final Part, we'll address a few tricks and traps in this method, and provide some insight to the what the grid is telling the risk manager.

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