What is a common conjugate prior for the Binomial likelihood?

What is a common conjugate prior for the Binomial likelihood?

• A. Normal prior for p
• B. Gamma prior for p
• C. Beta prior for p
• D. Uniform prior for p

Answer: (C)

The idea being tested is conjugacy for a Binomial likelihood. When you observe k successes out of n trials, the likelihood as a function of p is proportional to p^k (1−p)^(n−k) for p in [0,1]. A Beta(a, b) prior has density proportional to p^(a−1) (1−p)^(b−1) on [0,1]. When you multiply the likelihood by this prior, the posterior is proportional to p^(a+k−1) (1−p)^(b+n−k−1), which is another Beta distribution with updated parameters a' = a + k and b' = b + n − k. This keeps you in the same family after updating—exact conjugacy. That’s why the Beta prior is the standard choice for Binomial data. A Normal prior isn’t suitable here because it assigns positive probability outside [0,1], which doesn’t make sense for a probability parameter. A Gamma prior lives on (0, ∞) and doesn’t fit p ∈ [0,1]. A Uniform prior is a special case of the Beta family (Beta(1,1)), but the Beta family generalizes this to reflect different beliefs about p.

The idea being tested is conjugacy for a Binomial likelihood. When you observe k successes out of n trials, the likelihood as a function of p is proportional to p^k (1−p)^(n−k) for p in [0,1]. A Beta(a, b) prior has density proportional to p^(a−1) (1−p)^(b−1) on [0,1]. When you multiply the likelihood by this prior, the posterior is proportional to p^(a+k−1) (1−p)^(b+n−k−1), which is another Beta distribution with updated parameters a' = a + k and b' = b + n − k. This keeps you in the same family after updating—exact conjugacy. That’s why the Beta prior is the standard choice for Binomial data.

A Normal prior isn’t suitable here because it assigns positive probability outside [0,1], which doesn’t make sense for a probability parameter. A Gamma prior lives on (0, ∞) and doesn’t fit p ∈ [0,1]. A Uniform prior is a special case of the Beta family (Beta(1,1)), but the Beta family generalizes this to reflect different beliefs about p.