Beta函数与Gamma函数及其与Beta分布的关系

相关函数在scipy.special

import scipy.special as ssss.beta(x1, x2)
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相关分布（概率密度）在scipy.stats

import scipy.stats as stss.beta(a, b)           # 随机变量
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β Function：

B(x,y)=∫10tx−1(1−t)y−1dt

主要性质： 

B(x,y)=B(y,x)B(x,y)=(x−1)!(y−1)!(x+y−1)!

Γ function

Γ(x)=∫∞0tx−1e−tdt

其递归形式如下： 

Γ(x+1)=xΓ(x)

关系

B(x,y)=Γ(x)Γ(y)Γ(x+y)

scipy中的相关实现

>>> from scipy.special import beta>>> from scipy.special import gamma>>> beta(6, 7) == gamma(6)*gamma(7)/gamma(6+7)True
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Beta distribution 及其与Beta function and Gamma function的关系

Beta分布的概率密度函数（pdf）为： 

f(x;α,β)===C⋅xα−1(1−x)β−1Γ(x+y)Γ(x)Γ(y)xα−1(1−x)β−11Beta(x,y)xα−1(1−x)β−1

f(x;2,2)=x(1−x)Beta(2,2)

>>> st.beta(2, 2).pdf(.5) == .5*.5/ss.beta(2, 2)True
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beta distribution是一种在[0, 1]区间上的连续分布，在[0, 1]区间内概率密度值可以为 ∞，再次证明了概率密度函数值不是密度。

import numpy as npimport scipy.special as ssfrom scipy import integrateimport matplotlib.pyplot as pltdef beta(x, a, b):    return x**(a-1)*(1-x)**(b-1)/ss.beta(a, b)def show_pdf():    x = np.arange(0, 1, .01)    for (a, b) in [(.5, .5), (5, 1), (1, 3), (2, 2), (2, 5)]:        plt.plot(x, beta_pdf(x, a, b), label='a={:.1f}, b={:.1f}'.format(a, b), lw=2)    plt.legend(loc='upper center', frameon=False)    plt.ylim([0, 3.5])    plt.ylabel('pdf')    plt.savefig('./imgs/pdf.png')    plt.show()def show_cdf():    x = np.arange(0+0.0001, 1, .0001)    for (a, b) in [(.5, .5), (5, 1), (1, 3), (2, 2), (2, 5)]:        plt.plot(x, [(integrate.quad(beta_pdf, 0, x, args=(a, b)))[0] for x in x],                 label='a={:.1f}, b={:.1f}'.format(a, b), lw=2)    plt.legend(loc='upper left', frameon=False)    plt.ylabel('cdf')    plt.savefig('./imgs/cdf.png')    plt.show()if __name__ == '__main__':    show_pdf()    show_cdf()
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Beta分布的峰值在 a−1a+b−2 处取得，这一点在概率密度函数的曲线中可清晰地看出。

先验为 Beta 分布，似然为二项分布时的后验分布

p(θ)=θa−1(1−θ)b−1B(a,b)p(X|θ)=(nk)θk(1−θ)n−kp(θ|X)=θa+k−1(1−θ)b+n−k−1B(a+k,b+n−k)

import numpy as npimport scipy.stats as stimport matplotlib.pyplot as pltn, k = 100, 61p = k/nrv = st.binom(n, p)a, b = 10, 10prior = st.beta(a, b)poster = st.beta(a+k, b+n-k)thetas = np.arange(0, 1, .001)plt.figure(figsize=(12, 9))plt.style.use('ggplot')plt.plot(thetas, prior.pdf(thetas), 'b', label='Prior')plt.plot(thetas, n*st.binom(n, thetas).pmf(k), 'g', label='Likelihood')plt.plot(thetas, poster.pdf(thetas), 'r', label='Poster')plt.axvline(p, c='g', ls='--', label='MLE', alpha=.4)plt.axvline((a+k-1)/(a+b+n-2), c='r', ls='--', label='MAP', alpha=.4)plt.legend(frameon=False)plt.xlabel(r'$\theta$', fontsize=14)plt.ylabel('Density', fontsize=16)plt.show()
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