I am thinking of a model where dataset $X_t=(x_{t1},x_{t2},...,x_{tn})$ arrives sequentially at each $t$ period $(t=1,2,...,T)$, and $x_{ti}$ in a dataset $X_t$ are $iid\sim f(\theta)$ where $\theta$ is an unknown parameter. $f$ is supported on a bounded interval, say, $[0,1]$.
I want to derive an explicit expression, given a specific $f$, for the posterior of $\theta$ at the last period ($t=T$). For example, I tried to assume that $f(\theta)$ is a uniform distribution on $[0,\theta]$, and the prior of $\theta$ be a Pareto distribution. It turns out that (if I am correct)
$\theta\sim Pareto(max[X_1,X_2,...,X_T,b],(T-1)n+K)$
It seems that, given a large $n$, new information is not very helpful because $E(\theta)=\frac{(t-1)n+K}{(t-1)n+K-1}$ updates very slowly after $t=2$.
My question is: Are there any other proposed distribution ($f$) and the prior of $\theta$, such that there's an explicit expression for posterior after $T$ periods, and the information updating process is not "too fast" (in the sense that sequential information does help)?
It would also be nice if there are related papers/literature about it!