Generally, the uncertainty can be categorized into aleatory and epistemic according to whether it can be reduced or not. In Bayesian statistics, one "true fixed parameter" is presumed as discussions here, so it seems that the aleatory uncertainty in parameter is not considered.
To be more specific, let's see the Bayesian linear normal model. See Slide 16 here, the posterior variance of parameter $\beta$ is $\overline{V}=\frac{1}{V^{-1}+\sum x_{i}^{2}}$, in which $x_{i}$ is the independent variable. When more and more data $x_{i}$ are collected, the variance of $\beta$ can be arbitrarily small (Influence of $h$ is neglectable). Therefore, only epistemic uncertainty in $\beta$ is considered.
Here is my question: suppose $\beta$ has some aleatory uncertainty, for example $\beta$ can randomly change over time, is there a way to recover its aleatory uncertainty in Bayesian setting?
