For each of the statements below about Jeffreys prior, determine whether it is true or false. Select all the true statements.
It allows us to reflect our prior belief about the possible hypotheses. In other words, Jeffreys prior is not obtained from the statistical model alone.
Jeffreys prior is always proper.
For a Bernoulli statistical model, the Jeffreys prior \pi (\theta _1), computed from using \theta _1=p^2 as the parameter (i.e. the model is \textsf{Ber}(\theta _1)=\textsf{Ber}(p^2)), and the Jeffreys prior \tilde{\pi }(\theta _2), computed from using \theta _2=p^3 as the parameter (i.e. the model is \textsf{Ber}(\theta _2)=\textsf{Ber}(p^3)), satisfy \mathbf{P}_{\pi (\theta _1)}(a^2<\theta _1<b^2)=\mathbf{P}_{\tilde{\pi }(\theta _2)}(a^3<\theta _2<b^3) for any 0<a<b<1. That is the probability of \theta _1 being between a^2 and b^2 under the distribution \pi (\theta _1) is equal to the probability of \theta _2 being between a^3 and b^3 under the distribution \tilde{\pi }(\theta _2), for any pair a<b within (0,1).
The true statements are:
1. It allows us to reflect our prior belief about the possible hypotheses. In other words, Jeffreys prior is not obtained from the statistical model alone.
2. For a Bernoulli statistical model, the Jeffreys prior π(θ₁), computed from using θ₁ = p² as the parameter (i.e. the model is Ber(θ₁) = Ber(p²)), and the Jeffreys prior π̃(θ₂), computed from using θ₂ = p³ as the parameter (i.e. the model is Ber(θ₂) = Ber(p³)), satisfy Pπ(θ₁)(a² < θ₁ < b²) = Pπ̃(θ₂)(a³ < θ₂ < b³) for any 0 < a < b < 1. That is, the probability of θ₁ being between a² and b² under the distribution π(θ₁) is equal to the probability of θ₂ being between a³ and b³ under the distribution π̃(θ₂), for any pair a < b within (0,1).
The false statement is:
1. Jeffreys prior is always proper.
Please note that the answer assumes that the information provided is accurate.
The true statements are:
- It allows us to reflect our prior belief about the possible hypotheses. In other words, Jeffreys prior is not obtained from the statistical model alone.
- For a Bernoulli statistical model, the Jeffreys prior pi(theta_1), computed from using theta_1=p^2 as the parameter, and the Jeffreys prior tilde(pi)(theta_2), computed from using theta_2=p^3 as the parameter, satisfy P_pi(theta_1)(a^2<theta_1<b^2)=P_tilde(pi)(theta_2)(a^3<theta_2<b^3) for any 0<a<b<1. That is the probability of theta_1 being between a^2 and b^2 under the distribution pi(theta_1) is equal to the probability of theta_2 being between a^3 and b^3 under the distribution tilde(pi)(theta_2), for any pair a<b within (0,1).