Tuesday, August 17, 2010

Our friend Bayes -- Part IV

This is the last of our series on Bayes Theorem of conditional probabilities. If you missed Part I, II, or III, click to go directly to them.

We ended Part III with this grid:

Graphically, the situation is represented in the figure below. Note how the light blue area is 54.8% of the total area; it is the test success area. Note there is a sliver of light blue, representing 2% B+ | A-, in the 'bad weather column'.  The ratio of the good/bad weather is 60/40, per the example in this series.

To fully understand the figure and the example we have been discussing, it's important to realize that the dependent event, testing, occurs in an opportunity space that is defined by the independent event (or condition), which in this case is the weather.  So, good results are forecasted to be attained in 54.8% of the opportunity space.

In this part, we address a few Tricks and Traps:

  • It's often confusing to properly identify 'A' and 'B' and the cause-and-effect between them.  After all, in our example of test results and the weather, we hypothesize a dependency, but is there really an effect from the alledged cause? 
  • The single largest confusion is misunderstanding the difference between 'B+ | A+' and 'B+ and A+'.  One is a conditioned probability and the other is the chance of an intersection between conditions. 
  • Validating the independence and dependence of the events or conditions in the Bayes' Grid is sometimes no easy task. 
  • Validation of the grid is the single most important analytical thing to do.  The Grid will not numerically add properly if the events are not defined correctly because the observation data won't fit properly into the Grid.
  • The grid has four unknowns, A+, A-, B+ and B- so four equations are needed.  Usually three independent relationships, and the data observations to go with them are required. In our example, we observed the weather, the 90% test success during the 60% chance of good weather, and the 2% test success during bad weather.  We were able to compute other relationships without observations.
  • You can enter Bayes' Grid with with different sets of observations.  The ones in our example are typical but other observation sets are possible.
Bayes' Grid is not my invention; it's been written about for years and can be found in many statistics books.  Another web link that explains it pretty well is under this click.

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