 Beta-geometric conjugacy

For independent geometric random variables with success probability $p$, observing $d$ terminal events and $F$ preceding failed Bernoulli trials gives likelihood function proportional to $p^d(1-p{)}^F$. Multiplication by a beta distribution prior $\mathrm{Beta}(a,b)$ gives the posterior distribution $\mathrm{Beta}(a+d,b+F)$. With complete waiting times $x_1,\dots ,x_n$, take $d=n$ and $F={\sum }_i(x_i-1)$.

A waiting time that is right-censored after $c$ failed trials contributes $(1-p{)}^c$: increase $F$ by $c$ without increasing $d$. This requires the censoring mechanism to be ignorable, as with an externally fixed observation horizon. Counting a censored waiting time as an observed event changes the likelihood function and biases the update.