Bayesian Reasoning(Naive Bayes)

Bayesian reasoning(Naive Bayes Algorithm)
This algorithm is one of the important algorithm in Machine Learning. Many people make mistake in this.
Bayesian Reasoning can be very important for any field which includes Statistics. Lets try to grab the concept.
Bayesian reasoning is logical reasoning.  Even doctors make this mistake. That can be very dangerous.
Question:
Imagine a women in her forties and she wants to know if she has breast cancer. If she does, statistics show that there is 75% chance that a mammogram will detect it but if she doesn’t there is a 10% chance that the mammogram will give her a false positive, that is, it will say she has cancer and she doesn’t. It is a fact that 1.4% of all women in their forties  have breast cancer. Now imagine a hypothetical Women gets a mammogram and it comes back positive. What are the odds that the test is right?
Answer:
First of all change the percent to per thousand.
1.4 % means out of 1000 people 14 women has breast cancer. 986 don’t. 75% of 14 will test positive in mammogram which is (10.5). 10% of 986 will receive a positive test result 98.6 which is false positive. Total number of positives is 109.1 . So 10.5/109.1 which is 9.6% chance that the women who has positive test result will actually have cancer.
Now why does this happen? It’s because we think 75% is more than 10% but it’s not. 75% of 10 is very less than 10% of 986.

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