We demonstrate the property of reparametrization invariance with a simple example on a Bernoulli statistical model. We start with the model \textsf{Ber}(q), which has parameter q. What is its Jeffreys prior? Express your answer as an un-normalized pdf \pi (q) in proportionality notation such that \pi (0.5)=2.
\pi (q) \propto
\pi (q) \propto \sqrt{\frac{1}{q(1-q)}}
To find the Jeffreys prior on a Bernoulli statistical model, we need to compute the Fisher information. Suppose we have a Bernoulli distribution with parameter q.
The probability mass function (pmf) for this distribution is given by:
P(x|q) = q^x * (1-q)^(1-x)
To compute the Fisher information, we need to calculate the second derivative of the log-likelihood function with respect to q. The log-likelihood function is given by:
L(q) = log(P(X|q)) = x * log(q) + (1-x) * log(1-q)
Taking the second derivative of the log-likelihood function with respect to q, we have:
d^2(L(q))/dq^2 = -x/q^2 - (1-x)/(1-q)^2
Now, the Fisher information is defined as the expected value of the negative second derivative of the log-likelihood function:
I(q) = E[-d^2(L(q))/dq^2] = -E[x/q^2 + (1-x)/(1-q)^2]
For a Bernoulli distribution, the expected value of x is simply q. Therefore, we have:
I(q) = -q/q^2 - (1-q)/(1-q)^2 = -1/q - 1/(1-q)
Now, the Jeffreys prior is defined as the square root of the Fisher information:
pi(q) ∝ sqrt(I(q)) = sqrt(-1/q - 1/(1-q))
However, we are given that pi(0.5) = 2. Let's substitute q = 0.5 into the equation and solve for the constant of proportionality:
2 ∝ sqrt(-1/0.5 - 1/(1-0.5))
2 ∝ sqrt(-2 - 2) = sqrt(-4) = 2i
Since this is an imaginary number, it seems there is a mistake in the question or calculation. The Jeffreys prior for the Bernoulli statistical model should be a real-valued function. Please double-check the question or provide additional information for further assistance.