Suppose that I have a prior distribution $\pi_0(\theta)$ on $\mathbb{R}^k$ and a "naively chosen" sample distribution $f_0(x|\theta)$ on $\mathbb{R}^n$. I would like to online-update both of these quantities.
I know from that from the Bayes' theorem we can update $\pi_0$ to the posterior distribution $\pi_1$ by $$ \pi_1(\theta) \propto f_0(x|\theta) \pi_0(\theta). $$ But how do I update $f_0$?
Background on Question:
I have a dynamic context in mind, as is implied by my notation, and I would like to think of $\theta$ as my observations and $x$ as the quantities I want to predict. So, for example, I'm intersted in continually improving my conditional mean (point) estimate of $x$ given $\theta$ while information about both comes in.
Disclaimer
It's possible that I'm ommiting some key information by accident. So I also mention that I recieve data $\{x_i\}_i$ and $\{\theta_i\}$ as time moves on. Somehow shouldn't these be used for updates?
You have $\{X_i\}_{i=1}^n$, but you want to learn the posterior of the parameters $\theta$ online as new information comes in. Assuming your likelihood stays the same $f(x|\theta)$ does not evolve, you can obtain a recursive form for the posterior distribution:
\begin{equation} \pi_n(\theta) \propto f(x_n | \theta) \pi_{n-1}(\theta) \end{equation}
So for $X_1$ and $\pi_0$, you use the formula to obtain $\pi_1(\theta) \propto f(X_1| \theta)\pi_0(\theta)$. Then you take the new posterior and use it as a prior for $X_2$ $\Rightarrow$ $\pi_2(\theta) \propto f(X_2| \theta) \pi_1(\theta)$.
Repeating this routine would incorporate all of the information in the samples, in an online fashion, learning the full posterior over time.
If your sampling distribution changes over time you can still do the same routine, just modifying it so that for every data point $X_i$ you update your posterior for that sample as $\pi_{i} \propto f_i(X_i|\theta)\pi_{i-1}(\theta)$.,