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With a new count $x_{n+1}$, the updated Bühlmann credibility estimate is $(C_n+x_{n+1}+12)/(n+7)$. Subtracting the previous estimate gives the sequential Bühlmann credibility updateTherefore the new estimate is strictly smaller exactly whenBecause a claim count is a nonnegative integer, the equivalent set is $x_{n+1}\in \{0,1,\dots ,\lceil {\hat{m}}_n\rceil -1\}$. A count equal to the previous estimate leaves it unchanged; a larger count increases it. The comparison is with the previous credibility estimate, which already combines the prior population mean and the observed mean.