To determine which of the options will result in a posterior distribution that is a Beta distribution, regardless of the observations, we need to consider the conjugate prior property for the Bernoulli likelihood function.
The Beta distribution is a conjugate prior for the Bernoulli likelihood function, meaning that if we use a Beta prior distribution and update it with Bernoulli observations, the resulting posterior distribution will also be a Beta distribution.
Among the given options, the ones that are Beta distributions are:
1. \textsf{Unif}(0, 1)
The uniform distribution on the interval [0, 1] is equivalent to a Beta(1, 1) distribution, which is a valid choice for a prior when using a Bernoulli likelihood function.
2. \textsf{Ber}(0.3)
This option specifies a point mass at 0.3, which is equivalent to a Beta(0, 0) distribution. While this technically does not fit the standard definition of a Beta distribution, it can be considered as a limiting case in the Beta family when the shape parameters approach zero. Therefore, it can be considered a valid choice for a prior distribution.
The option that does not result in a Beta posterior distribution is:
3. \textsf{N}(0.5, 0.25)
This option specifies a normal distribution, which is not a conjugate prior for the Bernoulli likelihood function. Updating a normal prior with Bernoulli observations will not result in a posterior distribution that is a Beta distribution.
The option \pi (p) \propto \sqrt{\frac{1}{p(1-p)}} (0 < p < 1) is missing the prior distribution specification. The given expression is proportional to the square root of the reciprocal of p(1-p). Without a specific form for the prior distribution, it is impossible to determine whether the posterior distribution will be a Beta distribution.
Therefore, the options that will result in a posterior distribution that is a Beta distribution, regardless of the observations, are:
- \textsf{Unif}(0, 1)
- \textsf{Ber}(0.3)
Consider the case where we still have observations X_1, X_2, \cdots X_ n, \stackrel{\text {i.i.d}}{\sim } \textsf{Ber}(p). Suppose we are to change our prior distribution from \textsf{Beta}(a, b) to one of the options listed below. Which of these will result into the posterior distribution being a Beta distribution, regardless of the observations? Assume that the likelihood function is only defined for 0 \leq p \leq 1.
(Choose all that apply.)
\textsf{Unif}(0, 1)
\textsf{Unif}(0.3, 0.8)
\textsf{Ber}(0.3)
\textsf{N}(0.5, 0.25)
\pi (p) \propto \sqrt{\frac{1}{p(1-p)}} (0 < p < 1)
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