高斯分布

\[\mathcal{N}(x|\mu,\sigma ^2)=\frac{1}{(2\pi\sigma ^2)^{1/2}}exp\left\{-\frac{1}{2\sigma ^2}(x-\mu)^2\right\}\]

多维高斯分布

\[\mathcal{N}(\mathbf{x}|\mathbf{\mu},\mathbf{\Sigma})=\frac{1}{(2\pi)^{D/2}|\mathbf{\Sigma}|^{1/2}}exp\left\{-\frac{1}{2}(\mathbf{x}-\mathbf{\mu})^{T}\mathbf{\Sigma}^{-1}(\mathbf{x}-\mathbf{\mu})\right\}\]

\(\mathbf{x}=(x_1,\ldots ,x_N)^T\)是对变量\(x\)的\(N\)个观测，每个观测都是独立从同一高斯分布中抽样的，独立同分布，所以得到一个观测的概率可以表示为

\[p(\mathbf{x}|\mu,\sigma ^2)=\prod_{n=1}^N\mathcal{N}(x_n|\mu,\sigma ^2)\]

这是似然函数，是关于\(\mu,\sigma ^2\)的函数，通过极大化似然函数可以求得参数，极大似然法寻找一组参数使得所观测的概率最大，挺奇怪的

x轴上的点是观测值，绿色的线长度表示观测值的概率密度，所有绿线的长度相乘，得到观测值的概率密度，通过调整参数\(\mu,\sigma ^2\)使得这个乘积最大，这是极大似然法的思想

python+matplotlib+tkinter实现演示：

1 import numpy as np 2 import matplotlib 3 import matplotlib.pyplot as plt 4 import random 5 from Tkinter import * 6 from matplotlib.backends.backend_tkagg import FigureCanvasTkAgg 7 from matplotlib.figure import Figure 8  9 N = 100    #采样点数目10 mu = 0.5    #采样均值11 sigma = 1    #采样标准差12 13 x0 =np.linspace(-2,3,200,endpoint=True)    #高斯分布画图点14 x = np.random.normal(loc=mu, scale=sigma, size=N)    #高斯分布采样15 t1 = np.mean(x)    #最大似然均值16 t2 = np.sqrt(np.mean(np.square(x-t1)))    #最大似然标准差17 leftlim = -2    #横坐标左极限18 rightlim = 3    #横坐标右极限19 20 #高斯分布21 def gaussian(x,mu,sigma):22     return 1/(2*np.pi*sigma**2)**(0.5)*np.exp(-1/(2*sigma**2)*(x-mu)**2)23 #滑块sc1和sc2回调函数24 def draw_pic(self):25     #获得均值和标准差参数26     try:mu = float(v1.get())27     except:28         mu = 0.529         sc1.set(mu)30     try:sigma = float(v2.get())31     except:32         sigma = 133         sc2.set(sigma)        34     #清除图像35     draw_pic.f.clf()36     #画图37     draw_pic.a = draw_pic.f.add_subplot(111)38     draw_pic.a.set_xlim(leftlim,rightlim)39     draw_pic.a.set_ylim(0,1)40     draw_pic.a.set_xlabel(r'$x$')41     draw_pic.a.set_ylabel(r'$p(x)$')42     draw_pic.a.set_xticks(np.arange(leftlim,rightlim,0.5).tolist())43     x0 =np.linspace(leftlim,rightlim,200,endpoint=True)44     draw_pic.a.plot(x0,gaussian(x0,mu,sigma),color='r',linewidth=2)45     draw_pic.a.vlines(x.tolist(),[0],gaussian(x,mu,sigma).tolist(),color='g')46     draw_pic.canvas.show()47     #设置标签48     lb1.configure(text = "mu_ML:"+t1.astype('str')+ \49                   "\nsigma_ML:"+t2.astype('str')50                   +"\nSum of log of gaussian of x_n: "\51                   +np.sum(np.log(gaussian(x,mu,sigma))).astype('|S10'))52     53     54 if __name__ == '__main__':55     56     matplotlib.use('TkAgg')57     root = Tk()58     59     draw_pic.f = Figure(figsize=(5,4),dpi=100)60     draw_pic.canvas = FigureCanvasTkAgg(draw_pic.f,master = root)61     draw_pic.canvas.show()62     draw_pic.canvas.get_tk_widget().grid(row = 0,columnspan = 3)63     #标签64     lb1 = Label(root, text="")65     lb1.grid(row=1,column=0)66     #滑块1,用于设定均值67     v1 = StringVar()68     sc1 = Scale(root,69       from_ = 0,         70       to = 1,             71       resolution = 0.001,       72       orient = HORIZONTAL,  73                 variable = v1 ,        74       label = 'mu:'    ,75       command = draw_pic76       )77     sc1.grid(row=1,column=1)78     sc1.set(0.5)79     #滑块2,用于设定标准差80     v2= StringVar()81     sc2 = Scale(root,82       from_ = 0.000001,     83       to = 3,             84       resolution = 0.001,       85       orient = HORIZONTAL,  86       variable = v2 ,        87       label = 'sigma:'    ,88       command = draw_pic89       )90     sc2.grid(row=1,column=2)91     sc2.set(1)92      93     root.mainloop()