Generalized Dirichlet distribution

Generalized Dirichlet distribution

In statistics, the generalized Dirichlet distribution (GD) is a generalization of the Dirichlet distribution with a more general covariance structure and twice the number of parameters. Random variables with a GD distribution are neutral [R. J. Connor and J. E. Mosiman 1969. "Concepts of independence for proportions with a generalization of the Dirichlet distibution". Journal of the American Statistical Association, volume 64, pp194--206] .

The density function of p_1,ldots,p_{k-1} is

:left [prod_{i=1}^{k-1}B(a_i,b_i) ight] ^{-1}p_k^{b_{k-1}-1}prod_{i=1}^{k-1}left [p_i^{a_i-1}left(sum_{j=i}^kp_j ight)^{b_{i-1}-(a_i+b_i)} ight] where we define p_k= 1- sum_{i=1}^{k-1}p_i. Here B(x,y) denotes the Beta function. This reduces to the standard Dirichlet distribution if b_{i-1}=a_i+b_i for 2leqslant ileqslant k-1 (b_0 is arbitrary).

Wong [T.-T. Wong 1998. "Generalized Dirichlet distribution in Bayesian analysis". Applied Mathematics and Computation, volume 97, pp165-181] gives the slightly more concise form for x_1+cdots +x_kleqslant 1

:prod_{i=1}^kfrac{x_i^{alpha_i-1}left(1-x_1-ldots-x_i ight)^{gamma_i{B(alpha_i,eta_i)}where gamma_i=eta_j-alpha_{j+1}-eta_{j+1} for 1leqslant jleqslant k-1 and gamma_k=eta_k-1. Note that Wong defines a distribution over a k dimensional space (implicitly defining x_{k+1}=1-sum_{i=1}^kx_i) while Connor and Mosiman use a k-1 dimensional space with x_k=1-sum_{i=1}^{k-1}x_i. The remainder of this article will use Wong's notation.

General moment function

If X=left(X_1,ldots,X_k ight)sim GD_kleft(alpha_1,ldots,alpha_k;eta_1,ldots,eta_k ight), then

:Eleft [X_1^{r_1}X_2^{r_2}cdots X_k^{r_k} ight] =prod_{j=1}^kfrac{ Gammaleft(alpha_j+eta_j ight) Gammaleft(alpha_j+r_j ight) Gammaleft(eta_j+delta_j ight)}{ Gammaleft(alpha_j ight) Gammaleft(eta_j ight) Gammaleft(alpha_j+eta_j+r_j+delta_j ight)} where delta_j=r_{j+1}+r_{j+2}+cdots +r_k. Thus:Eleft(X_j ight)=frac{alpha_j}{alpha_j+eta_j}prod_{m=1}^{j-1}frac{eta_m}{alpha_m+eta_m}.

Reduction to standard Dirichlet distribution

As stated above, if b_{i-1}=a_i+b_i for 2leqslant ileqslant k then the distribution reduces to a standard Dirichlet. This condition is different from the usual case in which the new parameters being equal to zero gives the original distribution. However, in the case of the GDD attempting to do this results in a very complicated density function.

Bayesian analysis

Suppose X=left(X_1,ldots,X_k ight)sim GD_kleft(alpha_1,ldots,alpha_k;eta_1,ldots,eta_k ight) is generalized Dirichlet, and that Y|X is multinomial with n trials (here Y=left(Y_1,ldots,Y_k ight)). Writing Y_j=y_j for 1leqslant 1leqslant k and y_{k+1}=n-sum_{i=1}^ky_i the joint posterior of X|Y is a generalized Dirichlet distribution with

:X|Ysim GD_kleft({alpha'}_1,ldots,{alpha'}_k;{eta'}_1,ldots,{eta'}_k ight)

where {alpha'}_j=alpha_j+y_j and {eta'}_j=eta_j+sum_{i=j+1}^{k+1}y_i for 1leqslant k.

ampling experiment

Wong gives the following system as an example of how the Dirichlet and generalized Dirichlet distributions differ. He posits that a large urn contains balls of k+1 different colours. The proportion of each colour is unknown. Write X=(X_1,ldots,X_k) for the proportion of the balls with colour j in the urn.

Experiment 1. Analyst 1 believes that Xsim D(alpha_1,ldots,alpha_k,alpha_{k+1}) (ie, X is Dirichlet with parameters alpha_i). The analyst then makes k+1 glass boxes and puts alpha_i marbles of colour i in box i (it is assumed that the alpha_i are integers geq 1). Then analyst 1 draws a ball from the urn, observes its colour (say colour j) and puts it in box j. He can do this because glass is transparent. The process continues until n balls have been drawn. The posterior distribution is then Dirichlet with parameters being the number of marbles in each box.

Experiment 2. Analyst 2 believes that X follows a generalized Dirichlet distribution: Xsim GD(alpha_1,ldots,alpha_k;eta_1,ldots,eta_k). All parameters are again assumed to be positive integers. The analyst makes k+1 wooden boxes. The boxes have two areas: one for balls and one for marbles. The balls are coloured but the marbles are not coloured. Then for j=1,ldots,k, he puts alpha_j balls of colour j, and eta_j marbles, in to box j. He then puts a ball of colour k+1 in box k+1. The analyst then draws a ball from the urn. Because the boxes are wood, the analyst cannot tell which box to put the ball in (as he could in experiment 1 above); he also has a poor memory and cannot remember which box contains which colour balls. He has to discover which box is the correct one to put the ball in. He does this by opening box 1 and comparing the balls in it to the drawn ball. If the colours differ, the box is the wrong one. The analyst puts a marble (sic) in box 1 and proceeds to box 2. He repeats the process until the balls in the box match the drawn ball, at which point he puts the ball (sic) in the box with the other balls of matching colour. The analyst then draws another ball from the urn and repeats until n balls are drawn. The posterior is then generalized Dirichlet with parameters alpha being the number of balls, and eta the number of marbles, in each box.

Note that in experiment 2, changing the order of the boxes has a non-trivial effect, unlike experiment 1.

ee also

* Multivariate Polya distribution

References


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Dirichlet distribution — Several images of the probability density of the Dirichlet distribution when K=3 for various parameter vectors α. Clockwise from top left: α=(6, 2, 2), (3, 7, 5), (6, 2, 6), (2, 3, 4). In probability and… …   Wikipedia

  • Multivariate Pólya distribution — The multivariate Pólya distribution, named after George Pólya, also called the Dirichlet compound multinomial distribution, is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution with parameter …   Wikipedia

  • Multivariate Polya distribution — The multivariate Pólya distribution, also called the Dirichlet compound multinomial distribution, is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution with parameter vector alpha, and a set… …   Wikipedia

  • Generalized Riemann hypothesis — The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so called global L functions, which… …   Wikipedia

  • Dirichlet kernel — In mathematical analysis, the Dirichlet kernel is the collection of functions It is named after Johann Peter Gustav Lejeune Dirichlet. The importance of the Dirichlet kernel comes from its relation to Fourier series. The convolution of Dn(x) with …   Wikipedia

  • Probability distribution — This article is about probability distribution. For generalized functions in mathematical analysis, see Distribution (mathematics). For other uses, see Distribution (disambiguation). In probability theory, a probability mass, probability density …   Wikipedia

  • Multinomial distribution — Multinomial parameters: n > 0 number of trials (integer) event probabilities (Σpi = 1) support: pmf …   Wikipedia

  • Chi-squared distribution — This article is about the mathematics of the chi squared distribution. For its uses in statistics, see chi squared test. For the music group, see Chi2 (band). Probability density function Cumulative distribution function …   Wikipedia

  • Normal distribution — This article is about the univariate normal distribution. For normally distributed vectors, see Multivariate normal distribution. Probability density function The red line is the standard normal distribution Cumulative distribution function …   Wikipedia

  • Conway–Maxwell–Poisson distribution — Conway–Maxwell–Poisson parameters: support: pmf: cdf …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”