In Part I we posed the project situation of 'A' and 'B', where 'A' is a probabilistic event--in our example 'A' is the weather--and 'B' is another probabilistic event, the results of tests. We hypothesized that 'B' had a dependency on 'A', but not the other way 'round.
The Figure below is a Bayes' Grid for this situation. 'A+' is good weather, and 'B+' is a good test result. 'A' is independent of 'B', but 'B' has dependencies on 'A'. The notation, 'B+ | A' means a good test result given any conditions of the weather, whereas 'B+ | A+' [shown in another figure] means a good test result given the condition of good weather. 'B+ and A+' means a good test result when at the same time the weather is good. Note the former is a dependency and the latter is a intersection of two conditions; they are not the same.
There are a few basic math rules that govern Bayes' Grid.
- The dark blue space [4 cells] is every condition of 'A' and 'B', so the numbers in this 'space' must sum 1.0, representing the total 'A' and 'B' union
- The light blue row just under the 'A' is every condition of 'A', so this row must sum to 1.0
- The light blue column just adjacent to 'B' is every condition of 'B' so this column must sum to 1.0
- The dark blue columns or rows must sum to their light blue counter parts
First, let's say the empirical observations of the weather are that 60% of the time it is good and 40% of the time it is bad. Going forward, using the empirical observations, we can say that our 'confidence' of good weather is 60%-or-less. We can begin to fill in the grid, as shown below.
In spite of the intersections of A and B shown on the grid, it's very rare for the project to observe them. More commonly, observations are made of conditional results. Suppose we observe that given good weather, 90% of the test results are good. This is a conditional statement of the form P(B+ | A+) which is read: "probability of B+ given the condition of A+". Now, the situation of 'B+ | A+' per se is not shown on the grid. What is shown is 'B+ and A+'. However, our friend Bayes gave us this equation:
Take note: B+ is not 90%; in fact, we don't know yet what B+ is. However, we know the value of 'B+ and A+' is 0.54 because of Bayes' equation given above.
Now, since the grid has to add in every direction, we also know that the second number in the A+ column is 0.06, P(B- and A+).
However, we can go no farther until we obtain another independent emprical observation.
To be continued