Bayes' Theorem states that the conditional probability of an event, based on the occurrence of another event, is equal to the likelihood of the second event given the first event multiplied by the probability of the first event.
Blaa Blaa Blaa - I find definitions to be strange, only after understanding the concept I do understand the definition.
If a random variable is independent then the probability of the event is irrespective of the outcomes of other random variables. In simple words, it's like looking at the probability of something occurring without taking into account any other factors.
The probability of 2 or more simultaneous events happening together. Eg Probability of watching TV and Eating.
Probability of one (or more) event given the occurrence of another event. Eg the probability of your father having dessert given that tomorrow he is having a diabetes test is very low. If you notice carefully if there is no diabetes test tomorrow then the probability would have been almost 100%.
Note : P(A,B) = P(B,A) (Symmetrical)
Note : P(A|B) != P(B|A) (Not Symmetrical)
The numerator P(B|A) * P(A) is the joint probability equation given in (1)
P(A|B) ===> Posterior Probability
P(B|A) ===> Likelihood
P(A) ===> Prior Probability
P(B) ===> Evidence
Question: What is the probability that there is fire given that there is smoke?
P(Fire|Smoke) ===> Posterior Probability
P(Smoke|Fire) ===> Likelihood
P(Fire) ===> Prior Probability
P(Smoke) ===> Evidence
The probability of fire given that there is smoke is equal to the likelihood multiplied by the probability of fire divided by the probability of smoke. And this is Bayes' theorem to understand its use-case better read further.
One very common space where you can find the theorem applied is in the evaluation of medical diagnostic tests.
Let us consider a diagnostic test that determines whether a person has a lesion that is malignant or not.
From observation, it is given that
The above statement means that the probability of diagnostic test results being Positive given that he/she has a malignant tumour is 85%.
What will be a normal person's understanding of the above probability???
If a person takes this diagnostic test and the result turns out to be Positive since the above statement shows that for someone with a Malignant tumour, the test detects 85% per cent correctly there is a good chance that the person assumes that he/she might have a malignant tumour and that's scary, right?
Now let's look at what Bayes' got to say about it
P(Malignant=True|Test=Positive), this is what we are going to analyse using Bayes' theorem
There are a few assumptions that we have to make
This assumption means that on average only 1 in 5000 will have malignant tumours and the probability of the test is positive regardless of whether the person has a malignant tumour or not is 0.05016.
Plugging the values that we have
Wait what... this is a terrible diagnostic test because the the above probability shows that if this diagnostic test for a malignant tumour turns out to be True the probability of it being correct is only 0.33 per cent.
Note: This result was obtained on a few assumptions and if those assumptions are verified and updated the result could change.
Bayes' theorem is a significant contribution to the field of statistics and is widely used in machine learning. Bayes' Theorem provides a systematic way to update prior probabilities with new information or evidence. In other words, it helps us adjust our beliefs about the likelihood of an event occurring based on the data we observe.
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