Consider the case of a binary parameter heta \in \{ 0, 1\}. We have the prior distribution \pi ( heta ) 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 | heta ) that is positive for both heta =0 and heta =1. Which of the following statements about the posterior distribution \pi ( heta |X_1, \ldots , X_ n) is true? (Choose all that apply.)

Question

Consider the case of a binary parameter 	heta \in \{ 0, 1\}. We have the prior distribution \pi (	heta ) 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 | 	heta ) that is positive for both 	heta =0 and 	heta =1. Which of the following statements about the posterior distribution \pi (	heta |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 (	heta =0|X_1, \ldots , X_ n) \leq \pi (	heta =1|X_1, \ldots , X_ n) will necessarily be true.

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

GPT 3.5 · Answer

The posterior distribution $\pi( heta|X_1,\ldots,X_n)$ is proportional to the product of the prior distribution $\pi( heta)$ and the likelihood function $L(X_1,\ldots,X_n| heta)$. 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 $L(X_1,\ldots,X_n| heta)$ 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 $\pi( heta=0|X_1,\ldots,X_n) \leq \pi( heta=1|X_1,\ldots,X_n)$ 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 $ heta=0$ over $ heta=1$ even if $p<\frac{1}{2}$. 4) If $p>\frac{1}{2}$, then $\pi( heta=0|X_1,\ldots,X_n) \geq \pi( heta=1|X_1,\ldots,X_n)$ 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 $ heta=1$ over $ heta=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.