 Natural conjugate prior

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.