= Natural conjugate credibility identity
{title2=$\mathbb E[m(\Theta)\mid x]=(k\mu+\sum_i x_i)/(k+n)$}
For a positive-support <exponential family> $f(x\mid\theta)=p(x)e^{-\theta x}/q(\theta)$, normalization gives $m(\theta)=-q'(\theta)/q(\theta)$. 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> $n/(n+k)$.
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