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A "proper" prior or posterior distribution is defined as a distribution for which the PDF integrates to 1 (or in practice, if we're working with a known distribution, one for which the PDF without normalizing constants integrates to a constant). However, in the case of a known distribution, one can often tell just by looking at the PDF whether it is defined for a given set of parameters (e.g., if the posterior is a Normal, then the PDF is defined iff the posterior variance > 0).

Am I correct in thinking that: the parameter space over which a known PDF is defined is equivalent to the parameter space over which the distribution is proper?

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This is unrelated and demonstrates some confusion between sampling and parameter spaces.

Both proper and improper priors are defined as measures over the parameter space, $\Theta$, which is a set for which the sampling probability density $f(\cdot|\theta)$ is a well-defined pdf meaning in particular that $$\int_\mathcal{X} f(x|\theta)\,\text{d}x=1\qquad\forall\theta\in\Theta$$

For instance, for an Exponential $\mathcal{E}(\theta)$ family of sampling distributions, the parameter space is $\Theta=\mathbb{R}^*_+=(0,\infty)$.

The difference between proper and improper priors $\pi$ is whether or not the integral$$\int_\Theta \text{d}\pi(\theta)$$is finite.

For instance, for an Exponential $\mathcal{E}(\theta)$ family of sampling distributions, the prior $$\pi(\theta)=\exp\{-\theta\}\,\mathbb{I}_{\theta>0}$$ is proper and the prior $$\pi(\theta)=\theta^{-1}\,\mathbb{I}_{\theta>0}$$ is improper.

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    $\begingroup$ Hit the nail on the head with these edits! Thank you. $\endgroup$ – half-pass Feb 15 '16 at 15:33

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