Posterior equality for proportional likelihoods ($L_i^{\prime }=h{L}_i⟹{\pi }_i^{\prime }={\pi }_i$)

If every model's likelihood is multiplied by the same positive factor independent of its parameter or model index, Bayes theorem cancels that factor from the normalized posterior. Fixed-count Bernoulli sampling and sampling stopped at a specified positive head count give such proportional likelihoods for the same total heads and tails. This conclusion uses their actual sampling likelihoods; it is not a claim that every selection or stopping rule can be ignored.