Academic Journal of Science and Technology ISSN: 2771-3032 | Vol. 5, No. 2, 2023 121 Generalized Dirichlet Distribution Based on Confluent Hypergeometric Series Ruixin Zhao, Hongmei Liu, Yu Tang School of Science, Dalian Minzu University, Liaoning 116600, P.R.China Abstract: Dirichlet distribution is a kind of high-dimensional continuous probability distribution, which has important applications in the fields of statistics, machine learning and bioinformatics. In this paper, based on gamma distribution we study two two-dimensional random variables. Then we derive the properties of these two two-dimensional random variables by using the properties of non-central gamma distribution and confluent hypergeometric series. From these properties, we find the two random variables follow generalized Dirichlet distributions. Applying hypergeometric series to Dirichlet distribution broadens the research of Dirichlet distribution. Keywords: Dirichlet distribution, Confluent hypergeometric function, Gamma distribution, Non-central gamma distribution. 1. Introduction Dirichlet distribution is a kind of high-dimensional continuous probability distribution with positive simplex as the support set in the real fields, and is a generalization of the beta distribution in the high-dimensional case [1]. In Bayesian analysis, the Dirichlet distribution as a conjugate prior for multinomial distributions is used for parameter estimation of multinomial, binomial and type distributions [1]. In the field of machine learning, Dirichlet and generalized Dirichlet distributions are mostly applied to build mixture models to deal with unsupervised learning problems such as high- dimensional clustering and feature empowerment [2]. In natural language processing and bioinformatics research, the implicit Dirichlet distribution is used to identify potential subject word information in a document set [3][4]. In this article, we study the joint probability density, marginal density, and the distribution of sum, product and quotient for two two-dimensional random variables by using the properties of generalized Dirichlet distribution, non- central gamma distribution and confluent hypergeometric function. So we firstly introduce the generalized Dirichlet type 1 distribution and generalized Dirichlet type 2 distribution. Let ),( 21 ZZ be a two-dimensional random variable, if its p.d.f is 1,0,0, ),,( )1( ),,;,(1 2121 321 1 21 1 2 1 1 32121 321     zzzz aaaB zzzz aaazzD aaa , where )( )()()( ),,( 321 321 321 aaa aaa aaaB    , then ),( 21 ZZ follows the Dirichlet type 1 distribution. Let ),( 21 TT be a two-dimensional random variable, if its p.d.f is 0,0, ),,( )1( ),,;,(2 21 321 )( 21 1 2 1 1 32121 32121     tt aaaB tttt aaattD aaaaa , then ),( 21 TT follows the Dirichlet type 2 distribution. Random variable U has a non-central gamma distribution with shape parameter )0(k and non-central parameter  0 , denoted by );(~ kNCGU , the probability density is given by [5] ,0),;()exp()}({);;( 10 11   uuFuukuf k NCG  where     0 10 !)( 1 );( j j j j n m nmF . When 0 , the non- central gamma distribution reduces to the standard gamma distribution, denoted by )(~ GU . Then we introduce hypergeometric series. The definition of generalized hypergeometric function is given by [6]     0 1 1 11 !)()( )()( );;( k kqq k kpk qpqp kbb zaa zbbaaF … … …… (1.1) where Pochhammer symbol ，�3,2,1),1()1()(  nnaaaa n 1)( 0 a , the convergence condition of this series is shown in [7]. The confluent hypergeometric function is a special case of the generalized hypergeometric function, which is expressed as follows [6]:     0 11 !)( )( );;( k k k k kb xa xbaF . Its integral form is       1 0 11 11 )exp()1( )()( )( );;( dtxttt aba b xbaF aba . Let independent random variable 3,2,1, iX i follow gamma distributions with shape parameter 3,2,1, ivi , ))(,)((),( 3212321121 XXXXXXXXZZ  and ),(),( 323121 XXXXYY  , then ),( 21 ZZ and ),( 21 YY follow Dirichlet type 1 distribution and Dirichlet type 2 distribution respectively. Orozco-Castañeda, Nagar and Gupta [6] generalized the above distribution by hypergeometric series. Let the random variables WVU ,, be independent, U and V follow gamma distributions with shape parameter a and b , W follows the non-central gamma distribution with shape parameter c and non-central parameter  , )( WUUP  , )( WVVQ  , Gupta, Orozco-Castañeda and Nagar [5] studied the joint probability density of P and Q as well as other statistical properties. Based on the previous work, in this paper we define 122 ))(,)((),( WVUVWVUUYX  and ),(),( 11 WVWUYX  , then study their joint probability density, marginal probability density, and statistical properties of sum, product, quotient. From these properties, authors find that they follow generalized Dirichlet type 1 distribution and generalized Dirichlet type 2 distribution respectively. 2. Generalized Dirichlet Type 1 Distribution Theorem 2.1: Let VU , and W be independent random variables , )(~),(~ bGVaGU and ),c(~ NCGW , define )( WVUUX  , )( WVUVY  , then the joint density of X and Y is give by   1,0,0 ,)1(;; ),,( )1( 11 111    yxyxyxccbaF cbaB eyxyx cba   (2.1) The p.d.f of WVUS  is given by );( )( 10 1 scbaF cba se cbas      . Proof: Since VU , and W are independent random variables, the joint density is given by .0,0,0 ,);( )()()( 10 111    wvuwcF cba wvue cbawvu   (2.2) Making the transformation )(),( WVUVYWVUUX  and WVUS  with the Jacobian 2),,,,( ssyxwvuJ  , then we obtain the joint p.d.f of X ,Y and S as:   , )1(; )()()( )1( 10 1 111 yxscFse cba eyxyx cbas cba        (2.3) where 1,0,0  yxyx and 0s , integrating s , then the joint p.d.f of ),( YX is given by   )1(s; )()()( )1( 100 1 111 dsyxcFse cba eyxyx cbas cba          . !)( )()1( )()()( )1( !)( )1( )()()( )1( 0 111 0 1 0 111                   j j jjcba jcbas j j jjcba jc jcbayx cba eyxyx dsse jc yx cba eyxyx     From )()()( aaka k , (2.1) is proved. Further, integrating x and y in (2.3), we can get the marginal density of S : .) 1 1()1( )( )( )()()( )( )1( )()()( )1( 1 0 1 0 1 111 0 1 0 11 0 1 0 111 dxdy x y yxx c s cba se dxdy c yxs se cba yxyx x jc bjca j j jcbas j j jjj cbasx cba                                Setting )1( xyu  , from the definition of the beta function, the above formula can be calculated as: . )()()( )()()()()( )()()( )( )()( )1( )( )( )()()( 0 1 1 0 11 0 1                   j j jcbas jbca j j jcbas jcbajcbc jbcajcbs cba se dx jcb jcb xx c s cba se     From )()()( aaka k , the p.d.f of S is proved. From Theorem 2.1, we find that ),( YX follows the generalized Dirichlet type 1 distribution, WVUS  follows the non-central gamma distribution, denoted by );(~ cbaNCGS  . Theorem 2.2: If the p.d.f of ),( YX is given by Theorem 2.1, then the marginal density of X is  )1(;; ),( )1( 11 11 xcbcbaF cbaB xxe cba       . Proof: Integrating y in (2.1), one obtains  dyyxccbaFyxyx cbaB e x cba )1(;;)1( ),,( 11 1 0 111       . Inserting the series form of 11F into this formula, the marginal density of X is . 1 1)1( !)( )( ),,( )1( !)( )( ),,( 1 0 1 11 0 1 1 0 11 0 1                          x jc bjc j j j j a x jcb j j j j a dy x y yx jc cba cbaB xe dyyxy jc cba cbaB xe     Setting )1( xyt  , then the marginal density of X is )( )()( )1( !)( )( ),,( 1 0 1 jcb jcb x jc cba cbaB xe bjc j j j j a            )1(;; ),( )1( 11 11 xcbcbaF cbaB xxe cba        . From Theorem 2.2, we find that X follows the generalized Beta type 1 distribution. Theorem 2.3: If the p.d.f of ),( YX is given by Theorem 2.1, then the density of YXZ  is   10,)1(;; ),( )1( 11 11     zzccbaF bacB ezz cba   . Proof: The p.d.f (2.1) of ),YX（ is known. By using the convolution formula, the probability density of YXZ  is dxzccbaFezxzx cbaB z cba ))1(;;()1()( ),,( 1 110 111    .1 !)( )1()( ),,( )1( )( !)( )1()( ),,( )1( 0 1 11 0 1 0 11 0 1 dx z x xz jc zcba cbaB ez dxxzx jc zcba cbaB ez z b ab j j jj j c z ba j j jj j c                            Similar to the proof of the previous theorem, (2.3) is proved. Theorem 2.4: If the p.d.f of ),( YX is given by Theorem 2.1, then the density of XYZ  is           0 0 12 111 )1(!)( )1()1()1()( ),,( j k jkj jkk kjcjcj j b cbajc cbajczcba cbaB ez  . Proof: According to the probability density formula of two- dimensional random variable product: 123 dx x z xf x zfXY          , 1 )( , similar to the proof of the previous theorem, (2.4) is proved. Theorem 2.5: If the p.d.f of ),( YX is given by Theorem 2.1, then the density of XYZ  is         0 0 1 )(!!)( )1()1()( ),,( j k j k k j j b kbakjc zjccba cbaB ez  . Proof: According to probability density formula of two- dimensional random variable quotient ： dxxzxfxzf X Y ),()(     , similar to the proof of the previous theorem, (2.5) is proved. 3. Generalized Dirichlet Type 2 Distribution Theorem 3.1: Let VU , and W be independent random variables, )(~),(~ bGVaGU and ),c(~ NCGW , define WVYWUX  11 , , then the p.d.f of ),( 11 YX is 0,0, 1 ;; )1)(,,( 11 11 11 11 1 1 1 1            yx yx ccbaF yxcbaB yxe cba ba  (3.1) Proof: Making the transformation WWWVYWUX  ,, 11 with the Jacobian 2 11 ),,,,( wwyxwvuJ  in (2.2), then the joint p.d.f of 11,YX and W is 0,0);( )()()( 1110 11 1 1 1 )1( 11    yxwcF cba wyxe cbabawyx ，  . Integrating w in the above formula, we get the p.d.f of ),( 11 YX :                 0 1)1( 0 1 1 1 1 0 0 1)1( 1 1 1 1 11 11 !)()()()( !)( )( )()()( dwwe kccba yxe dw kc w we cba yxe kcbawyx k k kba k k k cbawyx ba     . !)( )()1( )1)(()()( 0 11 11 1 1 1 1          k k kk cba ba kc kcbayx yxcba yxe  After simplification, (3.1) is proved. From Theorem 3.1, we find that ),( 11 YX follows the generalized Dirichlet type 2 distribution. Theorem 3.2: If the p.d.f of ）（ 11,YX is given by Theorem 3.1, then the marginal density of 1X is            1 11 1 1 1 1 ;; )1)(,( x ccaF xcaB xe ca a  . Proof: Integrating 1y in (3.1), (3.2) is proved. From Theorem 3.2, we find that 1X follows the generalized Beta type 2 distribution.p; Theorem3.3: If the p.d.f of ）（ 11,YX is given by Theorem 3.1, then the density of 11S YX  is 0, 1 ;; )1)(,( e 11 1            s s ccbaF scbaB s cba ba  . Proof: By using the convolution formula, (3.3) is proved. 4. Conclusion This paper makes a preliminary study on two two- dimensional random variables based on confluent hypergeometic series. From the above theorems, we find that ),( YX and ),( 11 YX follow the generalized Dirichlet type 1 distribution and generalized Dirichlet type 2 distribution , which provides new methods for the future study of Dirichlet distribution. References [1] Kai Wang Ng, Guo-Liang Tian, Man-Lai Tang. Dirichlet and Related Distributions: Theory, Methods and Applications. New York: John Wiley & Sons, 2011. [2] Mohamed Maher Ben Ismaila, Hichem Friguib. Unsupervised clustering and feature weighting based on Generalized Dirichlet mixture modeling. Information Sciences, 2014, 274: 35-54. [3] David M. Blei, Andrew Y. Ng, Michael I. Jordan. Latent Dirichlet Allocation. Journal of Machine Learning Research, 2003, 3: 993-1022. [4] S. Shivashankar, S. Srivathsan, B. Ravindran, Ashish V. Tendulkar. Multi-view methods for protein structure comparison using latent dirichlet allocation. Bioinformatics, 2011, 27: i61-i68. [5] Arjun K. Gupta, Johanna Marcela Orozco-Castañeda, Daya K. Nagar. Non-central bivariate beta distribution. Springer, 2011, 52: 139-152. [6] Johanna Marcela Orozco-Castañeda, Daya K. Nagar, Arjun K. Gupta. Generalized bivariate beta distributions involving Appell’s hypergeometric function of the second kind. Computers and Mathematics with Applications, 2012, 64: 2507-2519. [7] Yudell L. Luke. The Special Functions and Their Approximations. Mathematics in Science and Engineering, 1969,53: 1-349. [8] D. K. Nagar, Johanna Marcela Orozco-Castañeda, Arjun K. Gupta. Product and quotient of correlated beta variables. Appl. Math. Lett., 2009, 22: 105-109. [9] Ingram Olkin, Ruixue Liu. A bivariate beta distribution. Statistics & Probability Letters, 2003, 62: 407-412. [10] Saralees Nadarajah, Samuel Kotz. Some bivariate beta distributions. Statistics, 2005, 39: 457-466. [11] James J. Chen, Melvin R. Novick, Bayesian Analysis for Binomial Models with Generalized Beta Prior Distributions. Journal of Educational and Behavioral Statistics, 1984, 9: 163- 175. [12] James M. Dickey. Multiple Hypergeometric Functions: Probabilistic Interpretations and Statistical Uses.Journal of the American Statistical Association, 1983, 78: 628-637. [13] Mohamed Abdalla, Muajebah Hidan. Analytical properties of the two-variables Jacobi matrix polynomials with applications. Demonstratio Mathematica, 2021, 54: 178-188. [14] Daya K. Nagar, Fabio Fabio Sepúlveda-Murillo. Properties of confluent hypergeometric function kind 1 distribution. Journal of Interdisciplinary Mathematics, 2008, 11: 495-504. [15] Jiayu Lin. On The Dirichlet Distribution by Jiayu Lin. Canada: Mathematics, 2016. [16] A. M. Mathai, R. K. Saxena. On a generalized hypergeometric distribution. Springer, 1967, 11: 127-132. [17] W. N. Bailey. Generalized hypergeometric series .London: Mathematics, 1953. 124 [18] Wentao Fan, Hassen Sallay, Nizar Bouguila. Online Learning of Hierarchical Pitman-Yor Process Mixture of Generalized Dirichlet Distributions With Feature Selection. IEEE Trans Neural Netw Learn Syst, 2017, 28: 2048-2061.