Likelihood principle

All evidence in observed data about a parameter is contained in its likelihood function; proportional likelihoods carry the same evidence. If a stopping rule contributes only a positive factor independent of the parameter to the complete observed-sample likelihood, that factor does not change likelihood-based inference. Recording only a coarsened part of the sample produces a different likelihood and is not covered by the same invariance claim.

For independent exponential observations of rate $\theta$, stopping after observation $x$ with known probability $\alpha (x)\in (0,1)$ gives a complete-data likelihood proportional to ${\theta }^N{e}^{-\theta S}$. Its minimal sufficient statistic is $(N,S)$ by the parameter-free likelihood-ratio criterion. If only the last value $Y$ is recorded, its density instead is $\alpha (y)e^{-\theta y}/{\int }_0^{\infty }\alpha (x)e^{-\theta x}dx$. The stopping rule therefore changes the coarsened-data inference even though it adds no complete-data likelihood factor involving $\theta$.