For a family of distribution \, \{ \textsf{Ber}(p)\} _{p\in (0,1)} \, , Jeffreys prior is proportional to:
\, \pi _ j(p) \propto \,
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Therefore, the Jeffreys prior is:
Proper
Improper
The Jeffreys prior for a family of distributions is typically proper, meaning it integrates to a finite value and can be used for Bayesian inference.
The Jeffreys prior for a family of distributions is proper if it integrates to a finite value over the parameter space, and improper if it does not. To determine if the Jeffreys prior for the family of distributions \{\textsf{Ber}(p)\}_{p\in (0,1)} is proper or improper, we need to calculate the integral of the prior.
The Jeffreys prior is defined as:
\pi_j(p) \propto \sqrt{I(p)},
where I(p) is the Fisher information of the distribution.
For the Bernoulli distribution, the Fisher information is given by:
I(p) = \frac{1}{p(1-p)}.
Therefore, the Jeffreys prior for the Bernoulli distribution is:
\pi_j(p) \propto \sqrt{\frac{1}{p(1-p)}}.
To determine if this prior is proper or improper, we need to check if the integral of this expression over the parameter space (p \in (0,1)) is finite or not.
Integrating this expression, we get:
\int_{0}^{1} \sqrt{\frac{1}{p(1-p)}} dp.
To determine whether this integral is finite or not, we evaluate it.