Fundamental Loopholes in Bayesian Probability Theory

There has to be careful when Bayes Theorwm can be applied and when not to estimate probability in Causal terms ! 

1st Problem  (Taken from renowned Daniel Kahneman Book Thinking Fast and Slow )

Consider the following scenario and note your intuitive answer to the question. A cab was involved in a hit-and-run accident at night. Two cab companies, the Green and the Blue, operate in the city.

You are given the following data:

• 85% of the cabs in the city are Green and 15% are Blue.

• A witness identified the cab as Blue. The court tested the reliability of the witness under the circumstances that existed on the night of the accident and concluded that the witness correctly identified each one of the two colours 80% of the time and failed 20% of the time.

What is the probability that the cab involved in the accident was Blue rather than Green? This is a standard problem of Bayesian inference. There are two items of information: a base rate and the imperfectly reliable testimony of a witness. In the absence of a witness, the probability of the guilty cab being Blue is 15%, which is the base rate of that outcome. If the two cab companies had been equally large, the base rate would be uninformative and you would consider only the reliability of the witness, concluding that the probability is 80%. The two sources of information can be combined by Bayes’s rule.

 The correct answer is 41%. However, you can probably guess what people do when faced with this problem: they ignore the base rate and go with the witness. The most common answer is 80%.

Explanation: Here the two approaches of base rate is ignored because that dimension of green and blue and witness’s probability are independent to each other and not derived one within the other.

There could be other witness too for example having their own probability or other independent opinions too, but that can’t be included in the Bayesian formula, the reason being different approaches are independent not one inside the other.

So the correct answer won’t be 41% as the Bayesian formula won’t be conceptually applied.

2nd Problem

You are given the following data: • The two companies operate the same number of cabs, but Green cabs are involved in 85% of accidents.

 • The information about the witness is as in the previous version. The two versions of the problem are mathematically indistinguishable, but they are psychologically quite different. People who read the first version do not know how to use the base rate and often ignore it. In contrast, people who see the second version give considerable weight to the base rate, and their average judgment is not too far from the Bayesian solution. Why?

Explanation: Here the Green and Cab % are like causal factors affecting the likelihood of the event and hence base rate would be taken.

Example of Bayesian’s Formula :

The key issue and Reality Check of Bayesian Statistics

What are the hidden Bayes’ theorem can be applied ?

The Bayesian formula can be applied when the different sources of information is lying one as the subset of the other in hierarchy not when different sources of information are parallel and independent to each other.

For example if N witnesses have their own independent opinions of probability, then can’t combine them in the Bayes equation.

One has to understand the basic difference! If one goes into the proof of Bayes theorem, one can sense the feeling of one information with the other information as the subset not parallel to each other.

In Example A

To be contd...