Natural conjugate prior
= Natural conjugate prior
For an <exponential family> with density proportional to $\exp\{\theta^TT(x)-A(\theta)\}$, its natural conjugate prior has density proportional to $\exp\{\theta^T\lambda_1-\lambda_2A(\theta)\}$. Observations update $\lambda_1$ by adding sufficient statistics and update $\lambda_2$ by adding sample size.