A Bayesian estimate for rank I am not sure how to put this succinctly but here is the problem.
Consider a random variable $X$ which can take a finite number of non-negative values.  It takes each value with the same probability.
Now consider a fixed value $y$ which is one of the values $X$ can take.  We would like to estimate what proportion of the values in the range of $X$ are bigger than $y$.
Our prior belief is that it is 50% of them.  After sampling $n$ times from $X$, how should one update this belief in a Bayesian framework?

The prior belief should be uniform in the range, not simply 50% .

If your prior belief is just 50%, then you will never come to any other conclusion. What you need to do is equally weight all possible fractions:
Let $p_0(f_y)\sim \mathcal{U}(0,1)$ be your Bayesian prior density for the fraction of observations larger than a given $y$(i.e., $f_y$).
Now, lets get our $n$ observations of $X$, call this $S$. How do we modify our prior?
$p(f_y=z|S) \propto  p_0(z)P(S|f_y=z)=z^{\gamma}(1-z)^{\lambda}$ where $\gamma :=$ number of observations strictly greater than  $y$, and $\lambda :=$ number of observations less than or equal to $y$.
You simply normalize the above function to get your posterior density.
Unfortunately, this does not uniquely identify the bounds on the actual values of X.Y You need additional conditions to get a unique solution.
