= Beta-geometric conjugacy
{title2=$p\mid D\sim\operatorname{Beta}(a+d,b+F)$}
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 $\operatorname{Beta}(a,b)$ gives the <posterior distribution> $\operatorname{Beta}(a+d,b+F)$. With complete waiting times $x_1,\ldots,x_n$, 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.
Back to article page
