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The beta distribution prior density is proportional to $p^{a-1}(1-p{)}^{b-1}$. By Bayes' theorem, its product with the likelihood function gives the posterior densityThis is again a beta distribution, namelyThus the beta family is a conjugate prior family for the geometric distribution. The update is an instance of beta-geometric conjugacy: counts of observed successes and failures are added to the two beta parameters.