all exponential family distributions may be written in a form that depends upon a set of fixed dimension sufficient statistics.
https://en.wikipedia.org/wiki/Exponential_family
these sufficient statistics have the additive form described in this article (this is a consequence of i.i.d. sampling).
it is common to exploit this structure when implementing efficient inference in, for example, mixture models.
if you combine this property with a bayesian analysis, and put a conjugate prior on the parameters of an exponential family distribution, then the posterior distribution, and the marginal likelihood depend upon on the data only through these sufficient statistics and everything else is easily computed. in this form, one of the sufficient statistics often has an interpretation as a "pseudo-count"; how many effective samples are encoded in your prior?
exponential family distributions include: poisson, exponential, bernoulli, multinomial, gaussian, negative binomial, etc.
if you combine this property with a bayesian analysis, and put a conjugate prior on the parameters of an exponential family distribution, then the posterior distribution, and the marginal likelihood depend upon on the data only through these sufficient statistics and everything else is easily computed. in this form, one of the sufficient statistics often has an interpretation as a "pseudo-count"; how many effective samples are encoded in your prior?
exponential family distributions include: poisson, exponential, bernoulli, multinomial, gaussian, negative binomial, etc.