Properties
Exponential families have a large number of properties that make them extremely useful for statistical analysis. In many cases, it can be shown that, except in a few exceptional cases, only exponential families have these properties. Examples:
- Exponential families have sufficient statistics that can summarize arbitrary amounts of independent identically distributed data using a fixed number of values.
- Exponential families have conjugate priors, an important property in Bayesian statistics.
- The posterior predictive distribution of an exponential-family random variable with a conjugate prior can always be written in closed form (provided that the normalizing factor of the exponential-family distribution can itself be written in closed form). Note that these distributions are often not themselves exponential families. Common examples of non-exponential families arising from exponential ones are the Student's t-distribution, beta-binomial distribution and Dirichlet-multinomial distribution.
- In the mean-field approximation in variational Bayes (used for approximating the posterior distribution in large Bayesian networks), the best approximating posterior distribution of an exponential-family node with a conjugate prior is in the same family as the node.
Read more about this topic: Exponential Family
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)