Hyperparameter (Bayesian statistics)
In Bayesian statistics, a hyperparameter is a parameter of a prior distribution; the term is used to distinguish them from parameters of the model for the underlying system under analysis.
For example, if one is using a beta distribution to model the distribution of the parameter p of a Bernoulli distribution, then:
- p is a parameter of the underlying system, and
- α and β are parameters of the prior distribution, hence hyperparameters.
Purpose
One often uses a prior which comes from a parametric family of probability distributions – this is done partly for explicitness, and partly so that one can vary the hyperparameter, particularly in the method of conjugate priors, or for ''sensitivity analysis.''Conjugate priors
When using a conjugate prior, the posterior distribution will be from the same family, but will have different hyperparameters, which reflect the added information from the data: in subjective terms, one's beliefs have been updated. For a general prior distribution, this is computationally very involved, and the posterior may have an unusual or hard to describe form, but with a conjugate prior, there is generally a simple formula relating the values of the hyperparameters of the posterior to those of the prior, and thus the computation of the posterior distribution is very easy.Sensitivity analysis
A key concern of users of Bayesian statistics, and criticism by critics, is the dependence of the posterior distribution on one's prior. Hyperparameters address this by allowing one to easily vary them and see how the posterior distribution vary: one can see how sensitive one's conclusions are to one's prior assumptions, and the process is called sensitivity analysis.Similarly, one may use a prior distribution with a range for a hyperparameter, thus defining a hyperprior, perhaps reflecting uncertainty in the correct prior to take, and reflect this in a range for final uncertainty.