Question

Consider the case of a binary parameter \theta \in \{ 0, 1\}. We have the prior distribution \pi (\theta ) that satisfies \pi (0)=p and \pi (1)=1-p, for some 0 \leq p \leq 1. Then, we observe X_1, \ldots, X_ n with corresponding conditional likelihood function L(X_1, \ldots , X_ n | \theta ) that is positive for both \theta =0 and \theta =1. Which of the following statements about the posterior distribution \pi (\theta |X_1, \ldots , X_ n) is true? (Choose all that apply.)



If p=0, then the posterior distribution will be identical to the prior distribution.

If p=1, then the posterior distribution will be identical to the prior distribution.

If p < \frac{1}{2}, then in the posterior distribution, \pi (\theta =0|X_1, \ldots , X_ n) \leq \pi (\theta =1|X_1, \ldots , X_ n) will necessarily be true.

If p > \frac{1}{2}, then in the posterior distribution, \pi (\theta =0|X_1, \ldots , X_ n) \geq \pi (\theta =1|X_1, \ldots , X_ n) will necessarily be true.

Answer 1

The posterior distribution $\pi(\theta|X_1,\ldots,X_n)$ is proportional to the product of the prior distribution $\pi(\theta)$ and the likelihood function $L(X_1,\ldots,X_n|\theta)$. Let's consider each statement separately:

1) If $p=0$, then the prior distribution is $\pi(0)=0$ and $\pi(1)=1$. In this case, the likelihood function $L(X_1,\ldots,X_n|\theta)$ will not change the posterior, as any non-zero probability multiplied by 0 will still be 0. Therefore, the posterior distribution will be identical to the prior distribution.

2) If $p=1$, then the prior distribution is $\pi(0)=1$ and $\pi(1)=0$. Again, the likelihood function will not change the posterior, as any non-zero probability multiplied by 0 will still be 0. So the posterior distribution will be identical to the prior distribution.

3) If $p<\frac{1}{2}$, then $\pi(\theta=0|X_1,\ldots,X_n) \leq \pi(\theta=1|X_1,\ldots,X_n)$ is not necessarily true. The posterior distribution depends on both the prior and the likelihood function, so it is possible for the likelihood to favor $\theta=0$ over $\theta=1$ even if $p<\frac{1}{2}$.

4) If $p>\frac{1}{2}$, then $\pi(\theta=0|X_1,\ldots,X_n) \geq \pi(\theta=1|X_1,\ldots,X_n)$ is not necessarily true. The posterior distribution depends on both the prior and the likelihood function, so it is possible for the likelihood to favor $\theta=1$ over $\theta=0$ even if $p>\frac{1}{2}$.

Therefore, the correct statements are:

- If $p=0$, then the posterior distribution will be identical to the prior distribution.
- If $p=1$, then the posterior distribution will be identical to the prior distribution.