Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2025 / Paper 1 / 6 / d

For fixed prior parameters, divide numerator and denominator by $n$:under independent sampling from a geometric distribution with true success probability $p_0>0$. The law of large numbers gives $S/n\to 1/p_0$. In a large finite sample the posterior mean is close to $n/S=1/\bar{x}$, the maximum-likelihood estimate. The fixed prior's influence vanishes, and the estimate is consistent.