贝塔分布¶
有两个形状参数 \(a,b > 0\) 支持的是 \(x\in[0,1]\) 。
\BEGIN{eqnarray*}f\Left(x；a，b\Right)&=&\frac{\Gamma\left(a+b\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}x^{a-1}\left(1-x\right)^{b-1}\\
F\Left(x；a，b\right)&=&\int_{0}^{x}f\Left(y；a，b\right)dy=i\Left(x；a，b\right)\\
g\Left(q；a，b\right)&=&i^{-1}\Left(q；a，b\right)\\
M\Left(t\Right)&=&\frac{\Gamma\left(a\right)\Gamma\left(b\right)}{\Gamma\left(a+b\right)}\，_{1}F_{1}\Left(a；a+b；t\Right)\\
\m&=&\frac{a}{a+b}\\
\MU_{2}&=&\frac{ab\left(a+b+1\right)}{\left(a+b\right)^{2}}\\
\Gamma_{1}&=&2\frac{b-a}{a+b+2}\sqrt{\frac{a+b+1}{ab}}\\
\Gamma_{2}&=&\frac{6\left(a^{3}+a^{2}\left(1-2b\right)+b^{2}\left(b+1\right)-2ab\left(b+2\right)\right)}{ab\left(a+b+2\right)\left(a+b+3\right)}\\
m_{d}&=&\frac{\Left(a-1\right)}{\Left(a+b-2\right)}\，a+b\neq2\end{eqnarray*}
哪里 \(I\left(x;a,b\right)\) 是正则化的不完全Beta函数。 \(f\left(x;a,1\right)\) 也称为幂函数分布。
\[L_{\mathbf{x}}\Left(a，b\right)=-N\log\Gamma\left(a+b\right)+N\log\Gamma\left(a\right)+N\log\Gamma\left(b\right)-N\left(a-1\right)\overline{\log\mathbf{x}}-N\left(b-1\right)\overline{\log\left(1-\mathbf{x}\right)}\]
实施： scipy.stats.beta