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The posterior distribution in a bayesian setting, $p(\theta \mid X)$, is random as it depends on the random observations $X$.
However, my question is, as seen as our belief over the parameter space $\Theta$, does it makes sense to refer to it as something that is random?
Since the prior distribution $p(\theta)$ is not random (if we exclude the case of hierarchical models) why would the posterior distribution, i.e. the new prior, be random?
Sorry in advance if this is off-topic.

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    $\begingroup$ What does "the prior distribution is not random" mean to you? $\endgroup$ Jan 23, 2020 at 17:09

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I think you are confusing the terms slightly. In the Bayesian framework, $ \theta $ is a random variable and hence, like all random variables, it is following a probability distribution (unlike the frequentist framework where $ \theta $ is constant). "Random distribution" is not something valid exactly.

Thus, before you gather the data, $ \theta $ is following the prior distribution and the posterior after you combine it with information (likelihood) from data. It always remains a random variable and it never changes.

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