What is a conjugate prior distribution for a multinomial likelihood?
The Dirichlet distribution is a conjugate prior for the multinomial distribution. This means that if the prior distribution of the multinomial parameters is Dirichlet then the posterior distribution is also a Dirichlet distribution (with parameters different from those of the prior).
What is the Dirichlet distribution used for?
A Dirichlet distribution (pronounced Deer-eesh-lay) is a way to model random probability mass functions (PMFs) for finite sets. It is also sometimes used as a prior in Bayesian statistics.
When is a posterior distribution called a conjugate distribution?
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.
Is the likelihood function a conjugate prior?
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 . This section needs additional citations for verification.
When do you call a posterior a conjugate prior?
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.
How are hyperparameters used in conjugate prior distributions?
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.