Bayes' theorem, Purpose of Bayes theorem in natural language processing, statistical nlp and bayes theorem, definition of Bayes theorem, example application of Bayes' theorem
Bayes Theorem
Bayes’
theorem is a way for finding conditional probability. Conditional probability
is a probability to be found when we know about certain other probabilities
(usually related).
In
other words, a conditional probability could be stated as follows;
Conditional
probability is about finding the probability of hypothesis H given an evidence
E. That is, how often the hypothesis is true if the evidence is true?
The
following tasks are some of the examples for conditional probabilities;
Task
|
What is to be found?
(Hypothesis)
|
What is known/given?
(Evidence)
|
What is the
probability of raining if it is cloudy?
|
Probability of
rain
|
It is cloudy
|
What is the
probability of fire if there is smoke?
|
Probability of
fire
|
There is smoke
|
What is the
probability of the flight arriving at right time if it is a bad weather?
|
Probability of
arrival of flight at right time
|
It is bad weather
|
Bayes’
theorem is a formula used to calculate conditional probability. If A and B are
two events, Bayes’ theorem finds the conditional probability P(A|B) by relating
another conditional probability P(B|A) with prior probabilities P(A) and P(B)
as follows;
The
above given Bayes’ formula tells us the following;
Bayes’ theorem finds
|
how frequently A
can happen if B happened, written as P(A|B) which is Posterior
probability
|
If we know
|
·
how frequently B can happen if A happened, written as
P(B|A) which is Likelihood
·
how frequently A can happen on its own, written as
P(A) which is Prior probability
·
how frequently B can happen on its own, written as
P(B) which is Prior probability
|
Example:
Given
the following;
·
It may be hot and dry during the month
April in India. Hence, the possibility for rain is less, say 3%. [P(Rain)
= 0.03]
·
Sometimes, it may be cloudy during this
month (say 12%) due to moisture. [P(Cloudy) = 0.12]
·
60% of times cloudy climate causes rain during
April. [P(Cloudy|Rain) = 0.6]
Find
the probability of rain during the month of April if it is cloudy;
P(Rain
| Cloudy) = [P(Rain) * P(Cloudy|Rain)] / P(Cloudy)
= [0.03 * 0.6] /
0.12
= 0.15 = 15%
Conclusion:
15%
is the probability of rain if it is cloudy there.
***********
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