Natural conjugate credibility identity ($\mathbb{E}[m(\Theta )\mid x]=(k\mu +{\sum }_i{x}_i)/(k+n)$)

For a positive-support exponential family $f(x\mid \theta )=p(x)e^{-\theta x}/q(\theta )$, normalization gives $m(\theta )=-q^{\prime }(\theta )/q(\theta )$. The natural conjugate prior proportional to $q(\theta {)}^{-k}{e}^{-k\mu \theta }$ has logarithmic derivative $k(m(\theta )-\mu )$. If its endpoint density values vanish, integration gives prior expected value $\mathbb{E}[m(\Theta )]=\mu$. Observations update $k$ to $k+n$ and $\mu$ to $(k\mu +{\sum }_i{x}_i)/(k+n)$. The endpoint control for a Laplace-family conjugate posterior justifies the same mean identity after updating, giving an exact credibility estimate with credibility factor $n/(n+k)$.

In the positive-support exponential family $q(\theta )=\int p(x)e^{-\theta x}dx$, a proper natural conjugate prior with zero endpoint density values has mean parameter strictly inside the essential convex support $(a,b)$. Updating that mean by a positive-weight average with supported observations keeps it inside $(a,b)$. At finite parameter endpoints, prior vanishing forces $q\to \infty$, so the increased posterior power $q^{-k-n}$ still vanishes. At positive infinity, positive base-measure mass below a point $d$ smaller than the updated mean bounds $q(\theta )$ below by $C{e}^{-d\theta }$, giving exponential decay of the posterior kernel. Mass above the updated mean gives the analogous bound at negative infinity. Thus the updated density is proper and has the boundary condition required to integrate its score.