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)
To determine which option will result in the posterior distribution being a Beta distribution regardless of the observations, we need to consider the conjugate prior property for the Bernoulli likelihood.
In Bayesian inference, a prior distribution is called conjugate to a likelihood function if the posterior distribution belongs to the same family of distributions as the prior distribution. In the case of the Bernoulli likelihood, the conjugate prior is the Beta distribution.
Therefore, the options that will result in the posterior distribution being a Beta distribution are:
1. \textsf{Unif}(0, 1)
2. \textsf{Ber}(0.3)
The option \textsf{Unif}(0.3, 0.8) is not a Beta distribution, so it will not result in a Beta posterior distribution.
The option \textsf{N}(0.5, 0.25) is a normal distribution, not a Beta distribution, so it will not result in a Beta posterior distribution.
The option \(\pi (p) \propto \sqrt{\frac{1}{p(1-p)}}\) is a non-standard distribution known as the Jeffreys' prior for the Bernoulli likelihood. While it is a valid prior, it does not belong to the Beta distribution family, so it will not result in a Beta posterior distribution.
Therefore, the correct answers are:
1. \textsf{Unif}(0, 1)
2. \textsf{Ber}(0.3)
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)