Reading conjugacy from a kernel

Topic: Exponential families and conjugacy · more questions on this topic
‹ A normal statistic Three different “flat” Poisson priors ›

Given a one-parameter exponential-family likelihood

L(\theta) \propto \frac{\exp\{c(\theta)^{\mathsf T}S({\boldsymbol{x}})\}} {a(\theta)^n},

and prior kernel

p(\theta\mid\alpha,\beta) \propto \frac{\exp\{c(\theta)^{\mathsf T}\alpha\}} {a(\theta)^\beta},

derive the update from (\alpha,\beta) to (\alpha+S({\boldsymbol{x}}),\beta+n). Identify the two sufficient summaries of the data that enter the posterior.

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