Sequential Bühlmann credibility update (${\hat{m}}_{n+1}={\hat{m}}_n+(x_{n+1}-{\hat{m}}_n)/(n+1+K)$)

With fixed structural parameters and $K=v/a$, the Bühlmann credibility estimate has form $({\sum }_{i=1}^n{x}_i+K{m}_0)/(n+K)$. The next observation updates it by the displayed rule. Consequently the estimate decreases exactly when the new observation is less than the previous credibility estimate, with strict equality handled separately.