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In Bayesian statistics, A and A′ correspond to different hypotheses H1 and H2​, and D corresponds to the observed data (X).

An equation for hypothesis H1 can be given as:$$P(H_1|D) = \frac{P(H_1) \times P(X|H_1)}{P(X)}$$​
​Given below are statements related to the above equation:

A. The equation represents the conditional probability of hypothesis H1H_1H1​, given the data.
B. The equation represents the probability of the data, given the hypothesis.
C. P(X∣H1) is called a prior probability, which is assigned to the hypothesis before the data is observed or analyzed.
D. P(X∣H1) represents the likelihood under the hypothesis H1H_1H1​.

Select the option that represents all correct statements with respect to the equation above.

Correct option is B

Explanation:

The given equation is Bayes' Theorem, which describes the probability of a hypothesis given observed data. It is written as:
​$$P(H_1|D) = \frac{P(H_1) \times P(X|H_1)}{P(X)}$$​​

where:

·       P(H1∣D) → Posterior probability (Probability of H1 after observing data X).

·       P(H1) → Prior probability (Initial belief about H1 before seeing data).

·       P(X∣H1) → Likelihood (Probability of data X given H1​).

·       P(X) → Evidence (Marginal likelihood) (Overall probability of observing X across all hypotheses).

Evaluating the statements:

Statement A is correct:

• P(H1∣D) is the posterior probability, which represents the probability of hypothesis H1​ given the observed data D.

Statement B is incorrect:

• P(X∣H1) represents the likelihood of the data given the hypothesis, not the probability of the data itself.
• The probability of the data is given by P(X) , also known as marginal likelihood or evidence.

Statement C is incorrect:

• P(X∣H1) is not the prior probability. Instead, P(H1) is the prior probability.
• The prior probability is our belief about the hypothesis before seeing data.

Statement D is correct:

• P(X∣H1) is the likelihood, which measures how well hypothesis H1 explains the observed data X.