Conjugate prior
In Bayesian probability theory, if the posterior distributions p are in the same probability distribution family as the prior probability distribution p, the prior and posterior are then called conjugate distributions, and the prior is called a conjugate prior for the likelihood function p. 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 may make the integral more or less difficult to calculate, and the product p × p 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 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 Bernoulli trials with unknown probability of success in . This random variable will follow the binomial distribution, with a probability mass function of the formThe 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.Practical example
Suppose a rental car service operates in your city. Drivers can drop off and pick up cars anywhere inside the city limits. You can find and rent cars using an app.Suppose you wish to find the probability that you can find a rental car within a short distance of your home address at any given time of day.
Over three days you look at the app at random times of the day and find the following number of cars within a short distance of your home address:
If we assume the data comes from a Poisson distribution, we can compute the maximum likelihood estimate of the parameters of the model which is Using this maximum likelihood estimate we can compute the probability that there will be at least one car available:
This is the Poisson distribution that is the most likely to have generated the observed data. But the data could also have come from another Poisson distribution, e.g. one with, or, etc. In fact there is an infinite number of poisson distributions that could have generated the observed data and with relatively few data points we should be quite uncertain about which exact poisson distribution generated this data. Intuitively we should instead take a weighted average of the probability of for each of those Poisson distributions, weighted by how likely they each are, given the data we've observed.
Generally, this quantity is known as the posterior predictive distribution where is a new data point, is the observed data and are the parameters of the model. Using Bayes' theorem we can expand such that Generally, this integral is hard to compute. However, if you choose a conjugate prior distribution, a closed form expression can be derived. This is the posterior predictive column in the tables below.
Returning to our example, if we pick the Gamma distribution as our prior distribution over the rate of the poisson distributions, then the posterior predictive is the negative binomial distribution as can be seen from the last column in the table below. The Gamma distribution is parameterized by two hyperparameters which we have to choose. By looking at plots of the gamma distribution we pick, which seems to be a reasonable prior for the average number of cars. The choice of prior hyperparameters is inherently subjective and based on prior knowledge.
Given the prior hyperparameters and we can compute the posterior hyperparameters and
Given the posterior hyperparameters we can finally compute the posterior predictive of
This much more conservative estimate reflect the uncertainty in the model parameters, which the posterior predictive takes into account.
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.