디리클레 분포는 일반화된 베타 분포이다. 베타 분포 유도 과정
fY1,...Yk(y1,...,yk)=Γ(α1+...+αk+1)Γ(α1)...Γ(αk+1)y1α1−1...ykαk−1(1−y1...−yk)αk+1−1f_{Y_1,...Y_k} (y_1,...,y_k) = \frac{\Gamma(\alpha_1 + ... + \alpha_{k+1})}{\Gamma(\alpha_1)...\Gamma(\alpha_{k+1})} y_1^{\alpha_1 - 1}...y_k^{\alpha_k - 1}(1-y_1...-y_k)^{\alpha_{k+1} - 1}fY1,...Yk(y1,...,yk)=Γ(α1)...Γ(αk+1)Γ(α1+...+αk+1)y1α1−1...ykαk−1(1−y1...−yk)αk+1−1 0<yi<1(i=1,...,k)0<y_i<1(i=1,...,k)0<yi<1(i=1,...,k) with y1+...+yk<1y_1+...+y_k<1y1+...+yk<1 αi>0(i=1,...,k+1)\alpha_i>0(i=1,...,k+1)αi>0(i=1,...,k+1) (Y1,...,Yk)∼Dirichlet(α1,α2,...,αk+1)(Y_1,...,Y_k) \sim Dirichlet(\alpha_1, \alpha_2, ..., \alpha_{k+1})(Y1,...,Yk)∼Dirichlet(α1,α2,...,αk+1)
trivariate case) Xi∼Gamma(αi,β),i=1,2,3X_i \sim Gamma( \alpha_i, \beta),i=1,2,3Xi∼Gamma(αi,β),i=1,2,3 and X1,X2,X3:indep.X_1,X_2,X_3: indep.X1,X2,X3:indep. Let Y1=X1X1+X2+X3,Y2=X2X1+X2+X3,Y3=X1+X2+X3Y_1 = \frac{X_1}{X_1+X_2+X_3},Y_2 = \frac{X_2}{X_1+X_2+X_3}, Y_3 = X_1+X_2+X_3Y1=X1+X2+X3X1,Y2=X1+X2+X3X2,Y3=X1+X2+X3 X1=Y1Y3, X2=Y2Y3, X3=Y3(1−Y1−Y2)X_1=Y_1 Y_3,\space X_2=Y_2Y_3, \space X_3=Y_3(1-Y_1-Y_2)X1=Y1Y3, X2=Y2Y3, X3=Y3(1−Y1−Y2) J=∣y30y10y3y2−y3−y3(1−y1−y2)∣=y32=∣J∣J= \begin{vmatrix} y_3 & 0 & y_1 \\ 0 & y_3 & y_2 \\ -y_3 & -y_3 & (1-y_1-y_2) \\ \end{vmatrix} = y_3^2=|J|J=∣∣∣∣∣∣∣y30−y30y3−y3y1y2(1−y1−y2)∣∣∣∣∣∣∣=y32=∣J∣ support of (Y1,Y2,Y3)(Y_1,Y_2,Y_3)(Y1,Y2,Y3) 0<y1y30<y_1 y_30<y1y3, 0<y2y30<y_2 y_30<y2y3, 0<y3(1−y1−y2)0<y_3(1-y_1-y_2)0<y3(1−y1−y2) => 0<y1+y2<1, y1>0, y2>0, y3>00 < y_1 + y_2 < 1, \space y_1>0, \space y_2>0, \space y_3>00<y1+y2<1, y1>0, y2>0, y3>0 pdf of (Y1,Y2,Y3)(Y_1,Y_2,Y_3)(Y1,Y2,Y3) fX1,X2,X3(x1,x2,x3)=1Γ(α1)Γ(α2)Γ(α3)βα1+α2+α3x1α1−1x2α2−1x3α3−1e−x1+x2+x3βf_{X_1,X_2,X_3} (x_1,x_2,x_3) = \\ \frac{1}{\Gamma(\alpha_1) \Gamma(\alpha_2) \Gamma(\alpha_3) \beta^{\alpha_1 + \alpha_2 + \alpha_3}} x_1^{\alpha_1-1} x_2^{\alpha_2-1} x_3^{\alpha_3-1} e^{-\frac{x_1+x_2+x_3}{\beta}}fX1,X2,X3(x1,x2,x3)=Γ(α1)Γ(α2)Γ(α3)βα1+α2+α31x1α1−1x2α2−1x3α3−1e−βx1+x2+x3 fY1,Y2,Y3(y1,y2,y3)=y32Γ(α1)Γ(α2)Γ(α3)βα1+α2+α3(y1y3)α1−1(y2y3)α2−1(y3(1−y1−y2))α3−1e−y3β=Γ(α1+α2+α3)Γ(α1)Γ(α2)Γ(α3)y1α1−1y2α2−1(1−y1−y2)α3−11Γ(α1+α2+α3)βα1+α2+α3y3α1+α2+α3−1e−y3βf_{Y_1,Y_2,Y_3} (y_1,y_2,y_3) = \\ \frac{y_3^2}{\Gamma(\alpha_1) \Gamma(\alpha_2) \Gamma(\alpha_3) \beta^{\alpha_1 + \alpha_2 + \alpha_3}} (y_1y_3)^{\alpha_1 - 1} (y_2y_3)^{\alpha_2 - 1} (y_3(1-y_1-y_2))^{\alpha_3-1} e^{-\frac{y_3}{\beta}} = \\ \frac{\Gamma(\alpha_1 + \alpha_2 + \alpha_3)}{\Gamma(\alpha_1) \Gamma(\alpha_2) \Gamma(\alpha_3)} y_1^{\alpha_1 - 1} y_2^{\alpha_2 - 1} (1- y_1 - y_2)^{\alpha_3-1} \frac{1}{\Gamma(\alpha_1 + \alpha_2 + \alpha_3) \beta^{\alpha_1 + \alpha_2 + \alpha_3}} y_3^{\alpha_1 + \alpha_2 + \alpha_3 - 1} e^{-\frac{y_3}{\beta}}fY1,Y2,Y3(y1,y2,y3)=Γ(α1)Γ(α2)Γ(α3)βα1+α2+α3y32(y1y3)α1−1(y2y3)α2−1(y3(1−y1−y2))α3−1e−βy3=Γ(α1)Γ(α2)Γ(α3)Γ(α1+α2+α3)y1α1−1y2α2−1(1−y1−y2)α3−1Γ(α1+α2+α3)βα1+α2+α31y3α1+α2+α3−1e−βy3 => (Y1,Y2) and Y3:indep.(Y_1,Y_2) \space and \space Y_3: indep.(Y1,Y2) and Y3:indep. fY1,Y2(y1,y2)=Γ(α1+α2+α3)Γ(α1)Γ(α2)Γ(α3)y1α1−1y2α2−1(1−y1−y2)α3−1f_{Y_1,Y_2} (y_1,y_2) = \frac{\Gamma(\alpha_1 + \alpha_2 + \alpha_3)}{\Gamma(\alpha_1) \Gamma(\alpha_2) \Gamma(\alpha_3)} y_1^{\alpha_1 - 1} y_2^{\alpha_2 - 1} (1- y_1 - y_2)^{\alpha_3-1}fY1,Y2(y1,y2)=Γ(α1)Γ(α2)Γ(α3)Γ(α1+α2+α3)y1α1−1y2α2−1(1−y1−y2)α3−1 => (Y1,Y2)∼Dirichlet(α1,α2,α3)(Y_1,Y_2) \sim Dirichlet(\alpha_1,\alpha_2,\alpha_3)(Y1,Y2)∼Dirichlet(α1,α2,α3) fY3(y3)=1Γ(α1+α2+α3)βα1+α2+α3y3α1+α2+α3−1e−y3βf_{Y_3} (y_3) = \frac{1}{\Gamma(\alpha_1 + \alpha_2 + \alpha_3) \beta^{\alpha_1 + \alpha_2 + \alpha_3}} y_3^{\alpha_1 + \alpha_2 + \alpha_3 - 1} e^{-\frac{y_3}{\beta}}fY3(y3)=Γ(α1+α2+α3)βα1+α2+α31y3α1+α2+α3−1e−βy3 => Y3∼Gamma(α1+α2+α3,β)Y_3 \sim Gamma(\alpha_1 + \alpha_2 + \alpha_3, \beta)Y3∼Gamma(α1+α2+α3,β)
trivariate case) Xi∼Gamma(αi,β),i=1,2,3X_i \sim Gamma( \alpha_i, \beta),i=1,2,3Xi∼Gamma(αi,β),i=1,2,3 and X1,X2,X3:indep.X_1,X_2,X_3: indep.X1,X2,X3:indep. Let Y1=X1X1+X2+X3,Y2=X2X1+X2+X3,Y3=X1+X2+X3Y_1 = \frac{X_1}{X_1+X_2+X_3},Y_2 = \frac{X_2}{X_1+X_2+X_3}, Y_3 = X_1+X_2+X_3Y1=X1+X2+X3X1,Y2=X1+X2+X3X2,Y3=X1+X2+X3 X1=Y1Y3, X2=Y2Y3, X3=Y3(1−Y1−Y2)X_1=Y_1 Y_3,\space X_2=Y_2Y_3, \space X_3=Y_3(1-Y_1-Y_2)X1=Y1Y3, X2=Y2Y3, X3=Y3(1−Y1−Y2) J=∣y30y10y3y2−y3−y3(1−y1−y2)∣=y32=∣J∣J= \begin{vmatrix} y_3 & 0 & y_1 \\ 0 & y_3 & y_2 \\ -y_3 & -y_3 & (1-y_1-y_2) \\ \end{vmatrix} = y_3^2=|J|J=∣∣∣∣∣∣∣y30−y30y3−y3y1y2(1−y1−y2)∣∣∣∣∣∣∣=y32=∣J∣
