Inverted Dirichlet distribution

In statistics, the inverted Dirichlet distribution is a multivariate generalization of the beta prime distribution, and is related to the Dirichlet distribution. It was first described by Tiao and Cuttman in 1965.{{cite journal|author-link=George Tiao| last=Tiao| first=George| title=The inverted Dirichlet distribution with applications| journal=Journal of the American Statistical Association| year=1965| volume=60| number=311| pages=793–805| doi=10.1080/01621459.1965.10480828}}

The distribution has a density function given by

:$$

p\left(x_1,\ldots, x_k\right) = \frac{\Gamma\left(\nu_1+\cdots+\nu_{k+1}\right)}{\prod_{j=1}^{k+1}\Gamma\left(\nu_j\right)}

x_1^{\nu_1-1}\cdots x_k^{\nu_k-1}\times\left(1+\sum_{i=1}^k x_i\right)^{-\sum_{j=1}^{k+1}\nu_j},\qquad x_i>0.

The distribution has applications in statistical regression and arises naturally when considering the multivariate Student distribution. It can be characterized{{cite journal| last=Ghorbel| first=M.| title=On the inverted Dirichlet distribution| journal=Communications in Statistics - Theory and Methods| year=2010| volume=39|pages=21–37| doi=10.1080/03610920802627062| s2cid=122956752}} by its mixed moments:

:$$

E\left[\prod_{i=1}^kx_i^{q_i}\right] = \frac{\Gamma\left(\nu_{k+1}-\sum_{j=1}^k q_j\right)}{\Gamma\left(\nu_{k+1}\right)}\prod_{j=1}^k\frac{\Gamma\left(\nu_j+q_j\right)}{\Gamma\left(\nu_j\right)}

provided that $q\_j>-\backslash nu\_j,\; 1\backslash leqslant\; j\backslash leqslant\; k$ and $\backslash nu\_\{k+1\}>q\_1+\backslash ldots+q\_k$.

The inverted Dirichlet distribution is conjugate to the negative multinomial distribution if a generalized form of odds ratio is used instead of the categories' probabilities- if the negative multinomial parameter vector is given by $p$, by changing parameters of the negative multinomial to $x\_i\; =\; \backslash frac\{p\_i\}\{p\_0\},\; i\; =\; 1\backslash ldots\; k$ where $p\_0\; =\; 1\; -\; \backslash sum\_\{i=1\}^\{k\}\; p\_i$.

T. Bdiri et al. have developed several models that use the inverted Dirichlet distribution to represent and model non-Gaussian data. They have introduced finite {{cite journal|last1=Bdiri|first1=Taoufik|last2=Nizar|first2=Bouguila|title=Positive vectors clustering using inverted Dirichlet finite mixture models|journal=Expert Systems with Applications|date=2012|volume=39|issue=2|pages=1869–1882|doi=10.1016/j.eswa.2011.08.063}}{{cite book|last1=Bdiri|first1=Taoufik|last2=Bouguila|first2=Nizar|title=Rough Sets, Fuzzy Sets, Data Mining and Granular Computing |chapter=Learning Inverted Dirichlet Mixtures for Positive Data Clustering |series=Lecture Notes in Computer Science |volume=6743|date=2011|pages=265–272|doi=10.1007/978-3-642-21881-1_42|isbn=978-3-642-21880-4}} and infinite {{cite book|last1=Bdiri|first1=Taoufik|last2=Bouguila|first2=Nizar|title=Neural Information Processing |chapter=An Infinite Mixture of Inverted Dirichlet Distributions |date=2011|volume=7063|pages=71–78|doi=10.1007/978-3-642-24958-7_9|series=Lecture Notes in Computer Science|isbn=978-3-642-24957-0}} mixture models of inverted Dirichlet distributions using the Newton–Raphson technique to estimate the parameters and the Dirichlet process to model infinite mixtures.

T. Bdiri et al. have also used the inverted Dirichlet distribution to propose an approach to generate Support Vector Machine kernels {{cite journal|last1=Bdiri|first1=Taoufik|last2=Nizar|first2=Bouguila|title=Bayesian learning of inverted Dirichlet mixtures for SVM kernels generation|journal=Neural Computing and Applications|date=2013|volume=23|issue=5|pages=1443–1458|doi=10.1007/s00521-012-1094-z|s2cid=254025619 |url=https://spectrum.library.concordia.ca/975145/1/Bouguila2012.pdf}} basing on Bayesian inference and another approach to establish hierarchical clustering.{{cite journal|last1=Bdiri|first1=Taoufik|last2=Bouguila|first2=Nizar|last3=Ziou|first3=Djemel|title=Object clustering and recognition using multi-finite mixtures for semantic classes and hierarchy modeling|journal=Expert Systems with Applications|date=2014|volume=41|issue=4|pages=1218–1235|doi=10.1016/j.eswa.2013.08.005}}{{cite book|last1=Bdiri|first1=Taoufik|last2=Bouguila|first2=Nizar|last3=Ziou|first3=Djemel|title=2013 IEEE 25th International Conference on Tools with Artificial Intelligence |chapter=Visual Scenes Categorization Using a Flexible Hierarchical Mixture Model Supporting Users Ontology |date=2013|pages=262–267|doi=10.1109/ICTAI.2013.48|isbn=978-1-4799-2972-6|s2cid=1236111 }}