For a likelihood $p(y | \theta)$ and pdf $f(y)$:

Suppose that a likelihood is location invariant i.e.

$p(y | \theta) = f(y - \theta)$

Show that Jeffrey's prior is of the form $p(θ) ∝ 1$.

I understand that we have to use the fisher information to solve this, but am confused about the idea of location invariance.

Fisher = $\sqrt{(-E[d^2/(d\theta)^2 log (f(y - \theta))]}$

which has to be proportional to $1$, but how do I use $f(y - \theta)$ in this equation for Fishers information?

For a likelihood $p(y | \theta)$ and pdf $f(y)$:

Suppose that a likelihood is location invariant i.e.

$p(y | \theta) = f(y - \theta)$

Show that the Jeffreys prior is of the form $p(θ) ∝ 1$.

I understand that we have to use the Fisher information to solve this, but am confused about the idea of location invariance.

Fisher = $\sqrt{-E[d^2/(d\theta)^2 \log (f(y - \theta))]}$

which has to be proportional to $1$, but how do I use $f(y - \theta)$ in this equation for Fishers information?

For a likelihood $p(y | \theta)$ and pdf $f(y)$:

Suppose that a likelihood is location invariant i.e.

$$p(y | \theta) = f(y - \theta)$$

Show that Jeffrey's prior is of the form $p(θ) ∝ 1$.

I understand that we have to use the Fisher information to solve this, but am confused about the idea of location invariance.

$$I^\text{Fisher} = \sqrt{-\mathbb E[d^2/(d\theta)^2 \log (f(y - \theta))]}$$

which has to be proportional to 1, but how do I use $f(y - \theta)$ in this equation for the Fisher information?

For a likelihood $p(y | \theta)$ and pdf $f(y)$:

Suppose that a likelihood is location invariant i.e.

$p(y | \theta) = f(y - \theta)$

Show that Jeffrey's prior is of the form $p(θ) ∝ 1$.

I understand that we have to use the fisher information to solve this, but am confused about the idea of location invariance.

Fisher = $\sqrt{(-E[d^2/(d\theta)^2 log (f(y - \theta))]}$

which has to be proportional to $1$, but how do I use $f(y - \theta)$ in this equation for Fishers information?

For a likelihood p(y | theta) and pdf f(y):

Suppose that a likelihood is location invariant i.e.

p(y | theta) = f(y - theta)

Show that Jeffrey's prior is of the form p(θ) ∝ 1.

I understand that we have to use the fisher information to solve this, but am confused about the idea of location invariance.

Fisher = sqrt(-E[d^2/(dtheta)^2 log (f(y - theta))]

which has to be proportional to 1, but how do I use f(y - theta) in this equation for Fishers information?

For a likelihood $p(y | \theta)$ and pdf $f(y)$:

Suppose that a likelihood is location invariant i.e.

$$p(y | \theta) = f(y - \theta)$$

Show that Jeffrey's prior is of the form $p(θ) ∝ 1$.

I understand that we have to use the Fisher information to solve this, but am confused about the idea of location invariance.

$$I^\text{Fisher} = \sqrt{-\mathbb E[d^2/(d\theta)^2 \log (f(y - \theta))]}$$

which has to be proportional to 1, but how do I use $f(y - \theta)$ in this equation for the Fisher information?