Preliminaries

A decision tree with no choice nodes is sometimes called an "event tree." It can be a useful way to layout a sequence of contingencies to help us identify the possible combinations of outcomes and their probabilities.

Consider a simple example: Suppose that if I order a dish in a restaurant I may like it (0.6) or not (0.4) and if I order dessert after the meal I may like it (0.75) or not (0.25). What are the possible combinations of outcomes and what are their probabilities?

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Here's what this looks like generically:

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Tree Flipping

What, though, if the information about the test was in a slightly different form. Suppose we knew that if the generator was going to work, then the test is 90% accurate, but when the generator is not going to work it is only 60% accurate.

STOP AND THINK. Do you see the difference?

What we have:

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What we need:

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Since the upper branches represent all the cases where the generator tests OK, we have the upper branch of the left-most node as 75% and the lower branch must be 25%. We can then work out c, d, e, and f as shown:

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Important but Subtle Point

There is a difference between knowing how good a test is GIVEN that you already know the truth and knowing the likelihood of the truth being one way or another GIVEN that you know the test result.

The latter is how we like to think, the former is usually the form the information comes in.

For example, suppose we have a new medical test. We give it to some people we know have a disease and we give it to others we know do not have the disease. We record how often the test comes up positive/negative in the presence of the disease and negative/positive in the absence. Then we have to "flip the tree" to come up with numbers on how likely a positive test is to be correct, etc.
The Value of Imperfect Information

Once we have flipped the information into the right order, we can do the tree analysis we did above.

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The take-away here is that a test that has a false positive rate of 10% and a false negative rate of 40% would still be worth $0.25 million in this case.

HOW TO

  1. We start with information about the test — GIVEN a particular reality, how likely is the test to reflect that reality vs. how likely is it to give a false result.
  2. Arrange this information in an EVENT TREE (a tree with just chance nodes) that has "reality" as left-most node.
  3. Calculate the joint probabilities for each branch all the way from the left to the right.
  4. Redraw the tree with the test as the left-most node.
  5. Insert all known probabilities.
  6. Calculate missing probabilities.