In
Bayesian statistics
Bayesian statistics ( or ) is a theory in the field of statistics based on the Bayesian interpretation of probability, where probability expresses a ''degree of belief'' in an event. The degree of belief may be based on prior knowledge about ...
, 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
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1
The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
or (0, 1) in terms of two positive Statistical parameter, parameters, denoted by ''alpha'' (''α'') an ...
to model the distribution of the parameter ''p'' of a
Bernoulli distribution
In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, is the discrete probability distribution of a random variable which takes the value 1 with probability p and the value 0 with pro ...
, then:
* ''p'' is a parameter of the underlying system (Bernoulli distribution), and
* ''α'' and ''β'' are parameters of the prior distribution (beta distribution), hence ''hyper''parameters.
One may take a single value for a given hyperparameter, or one can iterate and take a probability distribution on the hyperparameter itself, called a
hyperprior.
Purpose
One often uses a prior which comes from a
parametric family of probability distributions – this is done partly for explicitness (so one can write down a distribution, and choose the form by varying the hyperparameter, rather than trying to produce an arbitrary function), 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 (and various statistics of it, such as
credible intervals
In Bayesian statistics, a credible interval is an interval used to characterize a probability distribution. It is defined such that an unobserved parameter value has a particular probability \gamma to fall within it. For example, in an experime ...
) 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.
[Giulio D'Agostini]
Purely subjective assessment of prior probabilities
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Hyperpriors
Instead of using a single value for a given hyperparameter, one can instead consider a probability distribution of the hyperparameter itself; this is called a " hyperprior." In principle, one may iterate this, calling parameters of a hyperprior "hyperhyperparameters," and so forth.
See also
* Empirical Bayes method
Empirical Bayes methods are procedures for statistical inference in which the prior probability distribution is estimated from the data. This approach stands in contrast to standard Bayesian methods, for which the prior distribution is fixed ...
References
Further reading
*
*
* {{cite book , last=Kruschke , first=J. K. , author-link=John K. Kruschke , year=2010 , title=Doing Bayesian Data Analysis: A Tutorial with R and BUGS , publisher=Academic Press , isbn=978-0-12-381485-2 , pages=241–264 , url=https://books.google.com/books?id=ZRMJ-CebFm4C&pg=PA241
Bayesian statistics
Sensitivity analysis