1.9.2.2 Ignoring the Prior Probability
Another common mistake is ignoring the Prior Probability of the Hypothesis or in other words, ignoring the ratio of the Hypothesis. In our case, it would mean that while thinking about the probability that someone has Hypoillness, given that the person is experiencing some of its symptoms, we ignore the fact that the ratio of having Hypoillness is extremely low as it is a very rare illness, so the probability of a person having it lessens a lot right away. This is an extremely important aspect and ignoring it can give us drastically incorrect results.
Example 1.31. Walk or watch?
You want to go for a walk this morning, you look outside your window and see the weather. It's a bright sunny morning but as you are about to step out of the house, you get an alert from your weather app that there is a forecast for rain in the next few minutes, should you give up on your walk?
The morning looks too pleasant to pass up a walk, so you decide to do quick research to find out the accuracy of the app's forecasts. You go to the forecaster's website and find that the app has a rain forecast accuracy of 90% i.e., out of the 100 days on which it rained, the app predicted the rain on 90 of those days. This is pretty good! Along with this piece of information, you also find that out of the 100 dry days, the app predicted the dry days on 80 of those days. An accuracy of 80%, is not bad either!
Based on the accuracy, it looks like the app forecasts pretty reliably so you decide to give up on your walk and continue to watch Netflix. The whole day passed by but it did not rain and you missed your walk! Why? Because you did not use Bayes' Theorem!
Let's take a step back and ask the same questions that we asked in the example of Hypoillness
- How common is rain in your region, in other words, out of all the days (Say, in that season), how many days does it rain in that region? This is similar to the…