Posterior Probabilities in Bayes Theorem
Posterior probabilities in Bayes Theorem are the revised probabilities of an event occurring based on new available information. These probabilities are updated using the prior probabilities and the likelihood of the event given the new information. The Bayes Theorem formula is expressed as:

P(A|B) = (P(B|A) * P(A)) / P(B)

Where:
- P(A|B) is the posterior probability of event A given event B
- P(B|A) is the likelihood of event B given event A
- P(A) is the prior probability of event A
- P(B) is the prior probability of event B

Explanation of the Correct Answer
The correct answer is option D, posterior probabilities. Here's why:

1. Independent probabilities:
- Independent probabilities refer to events that do not affect or depend on each other.
- In Bayes Theorem, the prior and posterior probabilities are not independent as the posterior probabilities are updated based on new information.

2. Dependent probabilities:
- Dependent probabilities refer to events that are influenced by or affect each other.
- In Bayes Theorem, the prior and posterior probabilities are dependent as the posterior probabilities are derived from the prior probabilities and the likelihood.

3. Interior probabilities:
- There is no concept of "interior probabilities" in Bayes Theorem.
- This option is incorrect and not relevant to the theorem.

4. Posterior probabilities:
- Posterior probabilities in Bayes Theorem are the revised probabilities of an event occurring based on new information.
- These probabilities are calculated by updating the prior probabilities with the likelihood of the event given the new information.
- The posterior probabilities reflect the updated belief or confidence in the event occurring.

Therefore, the correct answer is option D, posterior probabilities, as they are the probabilities that are changed with the new available information in Bayes Theorem.