As an example, if we flip a fair coin three times independently, the probability of three heads (HHH) appearing is 1/8. After the first flip comes up heads, however, the probability changes to reflect the new information. We write

\[Pr[ HHH | H] = 1/4\] where the vertical bar means "given that".

Sometimes we don't know the conditional probabilities, however, and have to use a worst case/best case approach to get a range of possible probabilities. Probability dilation is when the range gets

*larger*when we

*add*information. That is, knowing more information leads to greater uncertainty. I read about dilation in this paper (pdf). Here is a paraphrased version of the example in the paper. (I also posted this on reddit.)

Apparently this sort of worst case/best case analysis is used in machine learning algorithms, making this a practical problem.Doctor: Your screening test for Krankheit came back positive. That means you have a 50% chance of having the condition. We want to do a more accurate test that will tell us for sure.

Patient: 50% Wow. I guess I need to know. Let's do the test.

------ 30 min later ----------

Doctor: Well, I'm sorry to say I have bad news and more bad news.

Patient: Oh no. The test was positive?

Doctor: Yes, it came back positive. But the other bad news is that they gave you the wrong blood test: one for an unrelated condition. We have no idea what the conditional probabilities are for having tested positive on this test and actually having Krankheit.

Patient: What! Why is that BAD news?

Doctor: Well, we used to be sure that Pr[Krankheit] = .5. Now we're dealing with Pr[Krankheit | Test+], which could be anywhere from zero to 100%. We know LESS than we did before.

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