# Is the posterior distribution random?

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.

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

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.