Conjugate prior

In Bayesian probability theory, if the posterior distributions p(θ | x) are in the same probability distribution 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 (or self-conjugate) 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. In Bayesian probability theory, if the posterior distributions p(θ | x) are in the same probability distribution 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 (or self-conjugate) 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 (continuous) 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 θ ↦ p ( x ∣ θ ) {displaystyle heta mapsto p(xmid heta )!} and prior p ( θ ) {displaystyle p( heta )!} , normalized (divided) by the probability of the data p ( x ) {displaystyle p(x)!} : 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 (generally with different parameter values). Such a choice is a conjugate prior. A conjugate prior is an algebraic convenience, giving a closed-form expressionfor the posterior; otherwise 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. 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 s {displaystyle s} in n {displaystyle n} Bernoulli trials with unknown probability of success q {displaystyle q} in . This random variable will follow the binomial distribution, with a probability mass function of the form The usual conjugate prior is the beta distribution with parameters ( α {displaystyle alpha } , β {displaystyle eta } ): where α {displaystyle alpha } and β {displaystyle eta } are chosen to reflect any existing belief or information ( α {displaystyle alpha } = 1 and β {displaystyle eta } = 1 would give a uniform distribution) and Β( α {displaystyle alpha } ,  β {displaystyle eta } ) is the Beta function acting as a normalising constant. In this context, α {displaystyle alpha } and β {displaystyle eta } are called hyperparameters (parameters of the prior), to distinguish them from parameters of the underlying model (here q). 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. (See the general article on the exponential family, and consider also the Wishart distribution, conjugate prior of the covariance matrix of a multivariate normal distribution, for an example where a large dimensionality is involved.)