You have a prior belief is that the gambler's coin is biased (remember that the bias can be either way: either towards Heads or Tails). One reasonable criterion for the prior to reflect the belief stated above is for \pi ( heta ) to attain its minimum at 1/2, be strictly decreasing from [\epsilon , \frac{1}{2}], and lastly strictly increasing from [\frac{1}{2}, 1-\epsilon ]. Which of the following priors satisfies this given criterion? (Choose all that apply.)

Question

You have a prior belief is that the gambler's coin is biased (remember that the bias can be either way: either towards Heads or Tails). One reasonable criterion for the prior to reflect the belief stated above is for \pi (	heta ) to attain its minimum at 1/2, be strictly decreasing from [\epsilon , \frac{1}{2}], and lastly strictly increasing from [\frac{1}{2}, 1-\epsilon ]. Which of the following priors satisfies this given criterion? (Choose all that apply.)

(Note: for each answer choice, the denominator Z is chosen to make sure that the integral of the density over its support is equal to 1. Moreover, we restrict ourself to the bias values in the interval [\epsilon , 1-\epsilon ], rather than the actual interval, (0,1). Here we assume that \epsilon is a very small number, e.g. \epsilon = 10^{-5}. You may want to use software to graph the given functions.)

\pi (	heta )\sim {m Unif}[\epsilon ,1-\epsilon ]

\pi (	heta )= \frac{	heta ^2+(1-	heta )^2}{Z}, 	heta \in [\epsilon ,1-\epsilon ]

\pi (	heta )= \frac{	heta (1-	heta )}{Z}, 	heta \in [\epsilon ,1-\epsilon ]

\pi (	heta )= \frac{1/	heta +1/(1-	heta )}{Z}, 	heta \in [\epsilon ,1-\epsilon ]

GPT 3.5 · Answer

To determine which priors satisfy the given criterion, let's analyze each option:

1. \(\pi (\theta )\sim {\rm Unif}[\epsilon ,1-\epsilon ]\):
This prior is a uniform distribution over the interval \([\epsilon ,1-\epsilon ]\). It does not meet the criterion of being strictly decreasing from \([\epsilon , \frac{1}{2}]\) and then strictly increasing from \([\frac{1}{2}, 1-\epsilon ]\). Therefore, this option does not satisfy the given criterion.

2. \(\pi (\theta )= \frac{\theta ^2+(1-\theta )^2}{Z}\), \(\theta \in [\epsilon ,1-\epsilon ]\):
This prior is a symmetric distribution that attains its minimum at \(\frac{1}{2}\). It is strictly decreasing from \([\epsilon , \frac{1}{2}]\) and strictly increasing from \([\frac{1}{2}, 1-\epsilon ]\). Therefore, this option satisfies the given criterion.

3. \(\pi (\theta )= \frac{\theta (1-\theta )}{Z}\), \(\theta \in [\epsilon ,1-\epsilon ]\):
This prior is a Beta distribution with parameters \(\alpha = 1\) and \(\beta = 1\). It attains its minimum at \(\frac{1}{2}\), is strictly decreasing from \([\epsilon , \frac{1}{2}]\), and strictly increasing from \([\frac{1}{2}, 1-\epsilon ]\). Therefore, this option satisfies the given criterion as well.

4. \(\pi (\theta )= \frac{1/\theta +1/(1-\theta )}{Z}\), \(\theta \in [\epsilon ,1-\epsilon ]\):
This prior is not defined at \(\theta = 0\) and \(\theta = 1\), which means it does not satisfy the criterion of being defined over \([\epsilon ,1-\epsilon ]\). Therefore, this option does not satisfy the given criterion.

In summary, options 2 and 3 satisfy the given criterion.