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Can a multinomial be used as a Dirichlet distribution?
However, (i) whatever prior you put on the weights of the multinomial is a legitimate answer at the subjective Bayes level and (ii) in case of prior information being available there is no reason it simplifies into a Dirichlet distribution. Note also that mixtures and convolutions of Dirichlet distributions can be used as priors.
Which is the conjugate prior of a Dirichlet distribution?
Dirichlet distributions are commonly used as prior distributions in Bayesian statistics, and in fact the Dirichlet distribution is the conjugate prior of the categorical distribution and multinomial distribution.
When does a data point have a Dirichlet distribution?
This means that if a data point has either a categorical or multinomial distribution, and the prior distribution of the distribution’s parameter (the vector of probabilities that generates the data point) is distributed as a Dirichlet, then the posterior distribution of the parameter is also a Dirichlet.
How are Dirichlet distributions used in Bayesian inference?
Dirichlet distributions are very often used as prior distributions in Bayesian inference. The simplest and perhaps most common type of Dirichlet prior is the symmetric Dirichlet distribution, where all parameters are equal.
Is the Dirichlet distribution a compound or conjugate distribution?
Dirichlet-multinomial as a compound distribution. The Dirichlet distribution is a conjugate distribution to the multinomial distribution. This fact leads to an analytically tractable compound distribution.
Can a Dirichlet distribution be collapsed in a Bayesian network?
In a Bayesian network. In a larger Bayesian network in which categorical (or so-called “multinomial”) distributions occur with Dirichlet distribution priors as part of a larger network, all Dirichlet priors can be collapsed provided that the only nodes depending on them are categorical distributions.
Is the Dirichlet multinomial an urn model?
Dirichlet-multinomial as an urn model. The Dirichlet-multinomial distribution can also be motivated via an urn model for positive integer values of the vector α, known as the Polya urn model. Specifically, imagine an urn containing balls of K colors numbering for the ith color, where random draws are made.