Question

H. Jeffreys (1961) in his work Theory of Probability published by Oxford Univ. Press, suggested a rule to generate a prior distribution of a parameter $\theta$ associated with the sampling distribution $p(y | \theta)$. The Jeffreys prior distribution is of the form: $p_J(\theta) \propto \sqrt{I(\theta)}$, Where: $I(\theta) = -E_{y|\theta} \left(\frac{d^2}{d\theta^2} \log p(y|\theta)\right)$ is the expected Fisher information. As defined above, you are asked to determine the following, a) Consider a single observation from the sampling distribution, $x_i | \theta, \sigma^2 \stackrel{iid}{\sim} N(\theta, \sigma^2)$, donde $\theta > 0$. As defined at the beginning, determine Jeffrey's prior, and represent the distribution graphically and analyze. b) The Jeffreys prior for the beta binomial model is known to be, $p(\theta) \propto \theta^{-\frac{1}{2}}(1-\theta)^{-\frac{1}{2}}$ Reparameterize the Binomial distribution with $\phi = logit(\theta)$, and obtain the Jeffreys prior distribution for this sampling distribution. To check the above result, apply the change of variables formula to obtain the prior density of $\phi$. Does the invariance of the Jeffreys prior hold under reparameterizations?

          H. Jeffreys (1961) in his work Theory of Probability published by Oxford Univ. Press, suggested
a rule to generate a prior distribution of a parameter $\theta$ associated with the sampling
distribution $p(y | \theta)$. The Jeffreys prior distribution is of the form:

$p_J(\theta) \propto \sqrt{I(\theta)}$,

Where:

$I(\theta) = -E_{y|\theta} \left(\frac{d^2}{d\theta^2} \log p(y|\theta)\right)$

is the expected Fisher information.
As defined above, you are asked to determine the following,
a) Consider a single observation from the sampling distribution,
$x_i | \theta, \sigma^2 \stackrel{iid}{\sim} N(\theta, \sigma^2)$, donde $\theta > 0$.
As defined at the beginning, determine Jeffrey's prior, and represent the distribution
graphically and analyze.
b) The Jeffreys prior for the beta binomial model is known to be,
$p(\theta) \propto \theta^{-\frac{1}{2}}(1-\theta)^{-\frac{1}{2}}$
Reparameterize the Binomial distribution with $\phi = logit(\theta)$, and obtain the
Jeffreys prior distribution for this sampling distribution.
To check the above result, apply the change of variables formula
to obtain the prior density of $\phi$. Does the invariance of the Jeffreys prior hold
under reparameterizations?

H. Jeffreys (1961) in his work Theory of Probability published by Oxford Univ. Press, suggested
a rule to generate a prior distribution of a parameter θ associated with the sampling
distribution p(y | θ). The Jeffreys prior distribution is of the form:

pJ(θ) ∝√(I(θ)),

Where:

I(θ) = -Ey|θ((d^2)/(dθ^2)log p(y|θ))

is the expected Fisher information.
As defined above, you are asked to determine the following,
a) Consider a single observation from the sampling distribution,
xi | θ, σ^2 iid∼ N(θ, σ^2), donde θ > 0.
As defined at the beginning, determine Jeffrey's prior, and represent the distribution
graphically and analyze.
b) The Jeffreys prior for the beta binomial model is known to be,
p(θ) ∝θ^-(1)/(2)(1-θ)^-(1)/(2)
Reparameterize the Binomial distribution with ϕ = logit(θ), and obtain the
Jeffreys prior distribution for this sampling distribution.
To check the above result, apply the change of variables formula
to obtain the prior density of ϕ. Does the invariance of the Jeffreys prior hold
under reparameterizations?

Added by Darren B.

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