In Bayesian probability theory, if the posterior distributions p(θ|x) are in
the same family as the prior probability distribution p(θ), the prior and
posterior are then called conjugate distributions, and the prior is called a
conjugate prior for the likelihood function. For example, the Gaussian
family is conjugate to itself with respect to a Gaussian likelihood
function: if the likelihood function is Gaussian, choosing a Gaussian prior over
the mean will ensure that the posterior distribution is also Gaussian. This
means that the Gaussian distribution is a conjugate prior for the likelihood
that is also Gaussian. The concept, as well as the term "conjugate prior", were
introduced by Howard Raiffa and Robert Schlaifer in their work on Bayesian
decision theory. A similar concept had been discovered independently by George
Alfred Barnard. Consider the general problem of
inferring a distribution for a parameter θ given some datum or data x. From
Bayes' theorem, the posterior distribution is equal to the product of
the likelihood function and prior , normalized by the probability of the
data : Let the likelihood function be
considered fixed; the likelihood function is usually well-determined from
a statement of the data-generating process. It is clear that different
choices of the prior distribution p(θ) may make the integral more or less
difficult to calculate, and the product p(x|θ) × p(θ) may take one algebraic
form or another. For certain choices of the prior, the posterior has the same
algebraic form as the prior. Such a choice is a conjugate prior.
A conjugate prior is an algebraic convenience, giving a closed-form
expression for the posterior; otherwise a difficult numerical integration may be
necessary. Further, conjugate priors may give intuition, by more transparently
showing how a likelihood function updates a prior distribution.
All members of the exponential family have conjugate priors.
Example The form of the conjugate prior can
generally be determined by inspection of the probability density or probability
mass function of a distribution. For example, consider a random variable
which consists of the number of successes in n Bernoulli trials with
unknown probability of success q in [0,1]. This random variable will follow
the binomial distribution, with a probability mass function of the form
Expressed as a function of , this has the form
for some constants and . Generally, this functional form will have an
additional multiplicative factor ensuring that the function is a
probability distribution, i.e. the integral over the entire range is 1.
This factor will often be a function of and , but never of .
In fact, the usual conjugate prior is the beta distribution with parameters:
where and are chosen to reflect any existing belief or information and Β(, )
is the Beta function acting as a normalising constant.
In this context, and are called hyperparameters, to distinguish them
from parameters of the underlying model. It is a typical characteristic of
conjugate priors that the dimensionality of the hyperparameters is one greater
than that of the parameters of the original distribution. If all parameters
are scalar values, then this means that there will be one more hyperparameter
than parameter; but this also applies to vector-valued and matrix-valued
parameters. If we then sample this random variable
and get s successes and f failures, we have
which is another Beta distribution with parameters. This posterior distribution
could then be used as the prior for more samples, with the hyperparameters simply
adding each extra piece of information as it comes.
Pseudo-observations It is often useful to think of the
hyperparameters of a conjugate prior distribution as corresponding to having
observed a certain number of pseudo-observations with properties
specified by the parameters. For example, the values and of a beta
distribution can be thought of as corresponding to successes and
failures if the posterior mode is used to choose an optimal parameter setting,
or successes and failures if the posterior mean is used to choose an
optimal parameter setting. In general, for nearly all conjugate prior
distributions, the hyperparameters can be interpreted in terms of
pseudo-observations. This can help both in providing an intuition behind the
often messy update equations, as well as to help choose reasonable
hyperparameters for a prior. Interpretations
= Analogy with eigenfunctions= Conjugate priors are analogous to
eigenfunctions in operator theory, in that they are distributions on which the
"conditioning operator" acts in a well-understood way, thinking of the
process of changing from the prior to the posterior as an operator.
In both eigenfunctions and conjugate priors, there is a finite-dimensional
space which is preserved by the operator: the output is of the same form
as the input. This greatly simplifies the analysis, as it otherwise considers
an infinite-dimensional space. However, the processes are only
analogous, not identical: conditioning is not linear, as the space of
distributions is not closed under linear combination, only convex combination,
and the posterior is only of the same form as the prior, not a scalar
multiple. Just as one can easily analyze how a
linear combination of eigenfunctions evolves under application of an
operator, one can easily analyze how a convex combination of conjugate priors
evolves under conditioning; this is called using a hyperprior, and
corresponds to using a mixture density of conjugate priors, rather than a
single conjugate prior. = Dynamical system=
One can think of conditioning on conjugate priors as defining a kind of
dynamical system: from a given set of hyperparameters, incoming data updates
these hyperparameters, so one can see the change in hyperparameters as a kind
of "time evolution" of the system, corresponding to "learning". Starting at
different points yields different flows over time. This is again analogous with
the dynamical system defined by a linear operator, but note that since different
samples lead to different inference, this is not simply dependent on time,
but rather on data over time. For related approaches, see Recursive
Bayesian estimation and Data assimilation.
Table of conjugate distributions Let n denote the number of observations.
In all cases below, the data is assumed to consist of n points .
If the likelihood function belongs to the exponential family, then a conjugate
prior exists, often also in the exponential family; see Exponential
family: Conjugate distributions. = Discrete distributions=
= Continuous distributions= See also
Beta-binomial distribution Notes
References