Computing the Posterior Probability Using Bayes' Theorem

To determine

P(J∣F,I)

the probability Jill Stein spoke the words 'freedom' and 'immigration', we'll apply Bayes' Theorem:

P(J∣F,I)
=P(J)×P(F∣J)×P(I∣J) / P(F,I)

Where:

• P(J) is the prior probability (the overall likelihood of Jill Stein giving a speech). In our case, P(J)=0.5P(J)=0.5.
• P(F∣J) and P(I∣J) are the likelihoods. These represent the probabilities of Jill Stein saying the words 'freedom' and 'immigration' respectively. Both are given as 0.1.
• P(F,I) is the evidence or marginal likelihood. This is the overall probability of the words 'freedom' and 'immigration' being said, regardless of the speaker.
  It can be computed as:
  

P(F,I)
=P(J)×P(F∣J)×P(I∣J) + P(G)×P(F∣G)×P(I∣G)

By plugging in the numbers, we can compute P(J∣F,I):

P(J∣F,I)
=0.5×0.1×0.1 / P(F,I)

After calculating P(F,I) from the formula above, insert its value into the equation to get P(J∣F,I).

This approach, rooted in Bayes' Theorem, provides the foundation for the Naive Bayes classifier in the realm of Natural Language Processing.