Suppose that, at first, I am trying to estimate the mean and standard deviation of some data that I assume to be normally distributed. My prior is gaussian with mean $\mu_0$ and variance $\sigma^2_0$. I observe data $D_0$ and update my prior according to Bayes' rule, leading to a gaussian posterior with mean $\mu_1$ and variance $\sigma^2_1$. Next (and bear with me, this is weird), suppose that I sample from the posterior to generate data $D_1$. Again, I update by Bayes' rule, leading to a new posterior ($\mu_3,\sigma^2_3$). Rinse and repeat. This defines a random walk over $u$ and $\sigma^2$.
Has this sort of thing been studied? Can anyone point me towards something relevant?