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A sufficient statistic is complete when every integrable function of it with expectation zero for every parameter is almost surely zero for every parameter. Full regular one-parameter exponential families with natural parameter ranging over an open interval provide a standard completeness result. Completeness makes an unbiased function of a sufficient statistic unique, giving the Lehmann–Scheffé theorem.

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.

For known size $\theta >0$, the negative binomial distribution is an exponential family with natural parameter $\eta =\log [\mu /(\theta +\mu )]<0$ and cumulant $b_{\theta }(\eta )=-\theta \log (1-e^{\eta })$. A generalized linear model can use a logarithmic link function for its mean, even though that link is not canonical. Jointly estimating the size changes the family and variance function, requiring additional updates beyond a single fixed-family GLM fit; this is what MASS::glm.nb does.

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 $n/(n+k)$.

For an exponential family with density proportional to $\exp \{{\theta }^{T}T(x)-A(\theta )\}$, its natural conjugate prior has density proportional to $\exp \{{\theta }^T{\lambda }_1-{\lambda }_{2}A(\theta )\}$. Observations update ${\lambda }_1$ by adding sufficient statistics and update ${\lambda }_2$ by adding sample size.