如何从python中已知似然函数的最大似然估计中获得参数误差？

Question

我有一些数据，并希望适合给定的心理测量功能p。 我也参与了拟合参数和误差。使用scipy包中的curve_fit函数的“经典”方法，很容易得到p的参数和错误。但是我想使用最大似然估计（MLE）来做同样的事情。从输出和图中可以看出，两种方法都提供了略微不同的参数。实现MLE不是问题，但我不知道如何使用此方法获取错误。有一个简单的方法来获得它们吗？我的似然函数L是：我无法适应这里描述的代码http://rlhick.people.wm.edu/posts/estimating-custom-mle.html，但这可能是一个解决方案。我该如何实现呢？或者这还有其他方式吗？

这里使用scipy stats模型拟合了类似的功能：https://stats.stackexchange.com/questions/66199/maximum-likelihood-curve-model-fitting-in-python。但是，也不计算参数的误差。

负对数似然函数是正确的，因为它提供了正确的参数，但我想知道这个函数是否依赖于y数据？负对数似然函数l显然是l = -ln（L）。这是我的代码：

#!/usr/bin/env python
# -*- coding: utf-8 -*- 

## libary
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
from scipy.optimize import minimize


def p(x,x50,s50):
    """return y value of psychometric function p"""
    return 1./(1+np.exp(4.*s50*(x50-x)))

def initialparams(x,y):
    """return initial fit parameters for function p with given dataset"""
    midpoint = np.mean(x)
    slope = (np.max(y)-np.min(y))/(np.max(x)-np.min(x))
    return [midpoint, slope]

def cfit_error(pcov):
    """return errors of fir from covariance matrix"""
    return np.sqrt(np.diag(pcov))

def neg_loglike(params):
    """analytical negative log likelihood function. This function is dependend on the dataset (x and y) and the two parameters x50 and s50."""
    x50 = params[0]
    s50 = params[1]
    i = len(xdata)
    prod = 1.
    for i in range(i):
        #print prod
        prod *= p(xdata[i],x50,s50)**(ydata[i]*5) * (1-p(xdata[i],x50,s50))**((1.-ydata[i])*5)
    return -np.log(prod)


xdata = [0.,-7.5,-9.,-13.500001,-12.436171,-16.208617,-13.533123,-12.998025,-13.377527,-12.570075,-13.320075,-13.070075,-11.820075,-12.070075,-12.820075,-13.070075,-12.320075,-12.570075,-11.320075,-12.070075]
ydata = [1.,0.6,0.8,0.4,1.,0.,0.4,0.6,0.2,0.8,0.4,0.,0.6,0.8,0.6,0.2,0.6,0.,0.8,0.6]

intparams = initialparams(xdata, ydata)## guess some initial parameters


## normal curve fit using least squares algorithm
popt, pcov = curve_fit(p, xdata, ydata, p0=intparams)
print('scipy.optimize.curve_fit:')
print('x50 = {:f} +- {:f}'.format(popt[0], cfit_error(pcov)[0]))
print('s50 = {:f} +- {:f}\n'.format(popt[1], cfit_error(pcov)[1]))



## fitting using maximum likelihood estimation
results = minimize(neg_loglike, initialparams(xdata,ydata), method='Nelder-Mead')
print('MLE with self defined likelihood-function:')
print('x50 = {:f}'.format(results.x[0]))
print('s50 = {:f}'.format(results.x[1]))
#print results


## ploting the data and results
xfit = np.arange(-20,1,0.1)

fig = plt.figure()
ax = fig.add_subplot(1,1,1)
ax.plot(xdata, ydata, 'xb', label='measured data')
ax.plot(xfit, p(xfit, *popt), '-r', label='curve fit')
ax.plot(xfit, p(xfit, *results.x), '-g', label='MLE')
plt.legend()
plt.show()

输出是：

scipy.optimize.curve_fit:
x50 = -12.681586 +- 0.252561
s50 = 0.264371 +- 0.117911

MLE with self defined likelihood-function:
x50 = -12.406544
s50 = 0.107389

这里可以看到拟合和测量数据：我的Python版本在Debian Stretch上是2.7。谢谢您的帮助。

Answer 1

最后，Rob Hicks（http://rlhick.people.wm.edu/posts/estimating-custom-mle.html）描述的方法得以实现。安装numdifftools之后，我可以从粗麻布矩阵计算估计参数的误差。

使用su权限在Linux上安装numdifftools：

apt-get install python-pip
pip install numdifftools

上面我的程序的完整代码示例如下：

#!/usr/bin/env python
# -*- coding: utf-8 -*- 

## libary
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import curve_fit
from scipy.optimize import minimize
import numdifftools as ndt



def p(x,x50,s50):
    """return y value of psychometric function p"""
    return 1./(1+np.exp(4.*s50*(x50-x)))

def initialparams(x,y):
    """return initial fit parameters for function p with given dataset"""
    midpoint = np.mean(x)
    slope = (np.max(y)-np.min(y))/(np.max(x)-np.min(x))
    return [midpoint, slope]

def cfit_error(pcov):
    """return errors of fir from covariance matrix"""
    return np.sqrt(np.diag(pcov))

def neg_loglike(params):
    """analytical negative log likelihood function. This function is dependend on the dataset (x and y) and the two parameters x50 and s50."""
    x50 = params[0]
    s50 = params[1]
    i = len(xdata)
    prod = 1.
    for i in range(i):
        #print prod
        prod *= p(xdata[i],x50,s50)**(ydata[i]*5) * (1-p(xdata[i],x50,s50))**((1.-ydata[i])*5)
    return -np.log(prod)


xdata = [0.,-7.5,-9.,-13.500001,-12.436171,-16.208617,-13.533123,-12.998025,-13.377527,-12.570075,-13.320075,-13.070075,-11.820075,-12.070075,-12.820075,-13.070075,-12.320075,-12.570075,-11.320075,-12.070075]
ydata = [1.,0.6,0.8,0.4,1.,0.,0.4,0.6,0.2,0.8,0.4,0.,0.6,0.8,0.6,0.2,0.6,0.,0.8,0.6]



intparams = initialparams(xdata, ydata)## guess some initial parameters


## normal curve fit using least squares algorithm
popt, pcov = curve_fit(p, xdata, ydata, p0=intparams)
print('scipy.optimize.curve_fit:')
print('x50 = {:f} +- {:f}'.format(popt[0], cfit_error(pcov)[0]))
print('s50 = {:f} +- {:f}\n'.format(popt[1], cfit_error(pcov)[1]))



## fitting using maximum likelihood estimation
results = minimize(neg_loglike, initialparams(xdata,ydata), method='Nelder-Mead')
## calculating errors from hessian matrix using numdifftools
Hfun = ndt.Hessian(neg_loglike, full_output=True)
hessian_ndt, info = Hfun(results.x)
se = np.sqrt(np.diag(np.linalg.inv(hessian_ndt)))

print('MLE with self defined likelihood-function:')
print('x50 = {:f} +- {:f}'.format(results.x[0], se[0]))
print('s50 = {:f} +- {:f}'.format(results.x[1], se[1]))

生成以下输出：

scipy.optimize.curve_fit:
x50 = -18.702375 +- 1.246728
s50 = 0.063620 +- 0.041207

MLE with self defined likelihood-function:
x50 = -18.572181 +- 0.779847
s50 = 0.078935 +- 0.028783

但是，在使用numdifftools计算粗糙矩阵时会出现一些RuntimeErrors。有一些除零。这可能是因为我自定义了neg_loglike功能。最后有一些错误的结果。使用“扩展Statsmodels”的方法可能更优雅，但我无法弄清楚。