In Bayesian probability theory, a class of prior probability distributionsp(?) is said to be conjugate to a class of likelihood functionsp(x|?) if the resulting posterior distributionsp(?|x) are in the same family as p(?). For example, the Gaussian family is conjugate to itself (or self-conjugate): if the likelihood function is Gaussian, choosing a Gaussian prior will ensure that the posterior distribution 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.[1] A similar concept had been discovered independently by George Alfred Barnard.[2]
Consider the general problem of inferring a distribution for a parameter ? given some datum or data x. From Bayes' theorem, the posterior distribution is calculated from the prior p(?) and the likelihood function\theta \mapsto p(x\mid\theta)\! as
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(?) may make the integral more or less difficult to calculate, and the product p(x|?) × p(?) may take one algebraic form or another. For certain choices of the prior, the posterior has the same algebraic form as the prior (generally with different parameters). Such a choice is a conjugate prior.
A conjugate prior is an algebraic convenience: otherwise a difficult numerical integration may be necessary.
All members of the exponential family have conjugate priors. See Gelman et al.[3] for a catalog.
where \alpha and \beta are chosen to reflect any existing belief or information (\alpha = 1 and \beta = 1 would give a uniform distribution) and ?(\alpha, \beta) is the Beta function acting as a normalising constant.
If we then sample this random variable and get s successes and f failures, we have
which is another Beta distribution with a simple change to the parameters. This posterior distribution could then be used as the prior for more samples, with the parameters simply adding each extra piece of information as it comes.
\frac{\lambda \gamma+n\bar{x}}{\gamma+n},\, \gamma+n,\, \alpha+\frac{n}{2},\, \beta+\frac{nS^2}{2}+\frac{n\gamma(\bar{x}-\lambda)^2}{2(n+\gamma)}, where \bar{x} is sample mean and S^2 is the sample variance.
\left(\Sigma_0^{-1} + n\Sigma^{-1}\right)^{-1}\left( \Sigma_0^{-1}\mu_0 + n \Sigma^{-1} \bar{x} \right),\, \left(\Sigma_0^{-1} + n\Sigma^{-1}\right)^{-1}, where \bar{x} is the sample mean.
\left(\Lambda_0 + n\Lambda\right)^{-1}\left( \Lambda_0\mu_0 + n \Lambda \bar{x} \right),\, \left(\Lambda_0 + n\Lambda\right)^{-1}, where \bar{x} is the sample mean.
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