Difference between revisions of "Dirichlet"
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| − | = Dirichlet( alpha,I ) = | + | == Dirichlet(alpha, I) == |
| − | A Dirichlet distribution with parameters | + | A Dirichlet distribution with parameters «alpha»<sub>i</sub> > 0. |
| − | Each sample of a Dirichlet distribution produces a random vector whose elements sum to 1. It is commonly used to represent second order probability information. | + | Each sample of a [[Dirichlet]] distribution produces a random vector whose elements sum to 1. It is commonly used to represent second order probability information. |
The Dirichlet distribution has a density given by | The Dirichlet distribution has a density given by | ||
| − | + | :<code>k*Product(X^(alpha - 1), I)</code> | |
| + | |||
where k is a normalization factor equal to | where k is a normalization factor equal to | ||
| − | + | :<code>GammaFn(Sum(alpha, I))/Sum(GammaFn(alpha), I)</code> | |
| − | The parameters, alpha, can be interpreted as observation counts. The mean is given by the relative values of alpha (normalized to 1), but the variance narrows as the alphas get larger, just as your confidence in a distribution would narrow as you get more samples. | + | The parameters, alpha, can be interpreted as observation counts. The mean is given by the relative values of alpha (normalized to 1), but the [[variance]] narrows as the alphas get larger, just as your confidence in a distribution would narrow as you get more samples. |
| − | The Dirichlet lends itself to easy Bayesian updating. If you have a prior of | + | The [[Dirichlet]] lends itself to easy Bayesian updating. If you have a prior of «alpha0», and you observe ''N''. |
| − | = Library = | + | == Library == |
| + | Multivariate Distributions.ana | ||
| − | Multivariate | + | ==See Also== |
| + | * [[GammaFn]] | ||
| + | * [[Multivariate distributions]] | ||
| + | * [[Distribution Densities Library]] | ||
Revision as of 00:47, 28 January 2016
Dirichlet(alpha, I)
A Dirichlet distribution with parameters «alpha»i > 0.
Each sample of a Dirichlet distribution produces a random vector whose elements sum to 1. It is commonly used to represent second order probability information.
The Dirichlet distribution has a density given by
k*Product(X^(alpha - 1), I)
where k is a normalization factor equal to
GammaFn(Sum(alpha, I))/Sum(GammaFn(alpha), I)
The parameters, alpha, can be interpreted as observation counts. The mean is given by the relative values of alpha (normalized to 1), but the variance narrows as the alphas get larger, just as your confidence in a distribution would narrow as you get more samples.
The Dirichlet lends itself to easy Bayesian updating. If you have a prior of «alpha0», and you observe N.
Library
Multivariate Distributions.ana
See Also
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