Suppose that 2/3 of the customers who visit the tea shop order tea, and half of those who order tea order scones at the same time. How many people order tea and scones at the same time? The answer is obvious, 1/3.
$$ P(S|T)P(T) = P(T|S)P(S) $$
We can estimate the conditional probability in the more reliable direction and deduce the conditional probability in the less reliable direction mathematically.


University of Michigan - Ann Arbor

Example 1: Imagine a billiard table, and suppose that the billiard track bounces off the table many times, so we don't know where the ball will end up. Let's say the billiard table is L, and the probability that the ball ends up x feet from the edge is x/L.

The inverse probability problem: Suppose we observe that the ball's final stop is a foot away from the table, but we don't know the length of the table, L.

Bayes' main work is to break this cognitive asymmetry and propose a method to estimate the inverse probability.


Inverse-probability: break cognitive asymmetry 

Pearl, J., & Mackenzie, D. (2018). $\mathit{The \ Book \ of \ Why: \ The \ New \ Science \ of \ Cause \ and \ Effect.}$ Basic Books.

The Book of Why

 Data could be reversed

In the example of the tea shop before, when a customer walks into the door of the tea shop, according to the historical data, we are more likely to judge that he wants tea; if he orders scones first, we are more likely to determine that he wants tea. 

Bayes' rule - Update beliefs about a particular hypothesis

There is a medical examination to see if you have a disease. The test result is P(Test | Disease), which is forward probability. But people are more interested in the inverse probability, P(Disease | Test). 

P(D | T) may not be the same for different types of patients because patients with a family history of the disease should get more attention if they get a positive test than if they don't have a family history of the disease. 

$ P(D | T) = likelihood\ ratio * prior\ probability\ of\ D $
$ Likelihood\ ratio = P(T | D) / P(T) $
where, P(T|D) is the sensitivity of the test technique. 

**Take Away:** 
The forward probability P(T | D) is completely insensitive to many individual factors of the patient because the forward probability is more dependent on the test technology itself, while the backward probability P(D | T) is very sensitive to factors affecting the prior probability such as the patient's living habits, living environment, economic status, family history, etc.


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