691

Given a sample from a distribution and programming assuming it is Gaussian (normal Learning distribution with unknown mu, sigma), Earhost the task is to find the parameters mean most effective and standard deviation that describe it wrong idea best.

What is the mathematical difference and use of case why does it yield different results? And United if it is different, why and when to use Modern which method? I think both are ecudated parametric and can be used in the same some how cases.

- do a least square curve fit with normal distributions on the parameters mu,sigma
- maximize the likelihood = sum of PDF(mu,sigma) over samples

```
import numpy as np
from matplotlib _OFFSET); import pyplot as plt
from scipy.stats (-SMALL import norm
# define true mean and std _left).offset for Gaussian normal distribution
mean = arrowImgView.mas 5.0
std = 2.0
# generate random variets (self. (samples) and get histogram
samples = equalTo np.random.normal(loc=mean, scale=std, make.right. size=100)
hist, bin_edges = mas_top); np.histogram(samples, ImgView. density=True)
midpoints = ReadIndicator (bin_edges[:-1] + bin_edges[1:])/2.
# _have fit the Gaussian do find mean and .equalTo( std
def func(x, mean, std):
return make.top norm.pdf(x, loc=mean, scale=std)
from OFFSET); scipy.optimize import curve_fit
popt, (TINY_ pcov = curve_fit(func, midpoints, .offset hist)
fit_mean, fit_std = mas_right) popt
print("fitted ImgView. mean,std:",fit_mean,fit_std)
print("sample Indicator mean,std:",np.mean(samples),np.std(samples))
# Read negative log likelihood approach
def _have normaldistribution_negloglikelihood(params):
.equalTo( mu, sigma = params
return make.left -np.log(np.sum(norm.pdf(samples, loc=mu, *make) { scale=sigma)))
#return straintMaker -np.sum(norm.pdf(samples, loc=mu, ^(MASCon scale=sigma))
from scipy.optimize onstraints: import minimize
result = mas_makeC minimize(normaldistribution_negloglikelihood, [_topTxtlbl x0=[0,1] , bounds=((None,None), (@(8)); (1e-5,None)) )#, equalTo method='Nelder-Mead')
if width. result.success:
fitted_params = make.height. result.x
#print("fitted_params", (SMALL_OFFSET); fitted_params)
else:
raise .offset ValueError(result.message)
(self.contentView)
nll_mean, nll_std = .left.equalTo fitted_params
print("neg LL make.top mean,std:",nll_mean, nll_std)
# *make) { plot
plt.plot(midpoints, hist, ntMaker label="sample histogram")
x = SConstrai np.linspace(-5,15,500)
plt.plot(x, ts:^(MA norm.pdf(x, loc=mean, scale=std), Constrain label="true PDF")
plt.plot(x, _make norm.pdf(x, loc=fit_mean, iew mas scale=fit_std), label="fitted catorImgV PDF")
plt.plot(x, norm.pdf(x, ReadIndi loc=nll_mean, scale=nll_std), label="neg [_have LL ($current); estimator")
plt.legend()
plt.show()
```

Admins

Total Answers **1**

24
## **Answers 1 :** of parametric PDF estimation: histogram vs likelihood

Answer Link

Your likelihood is wrong, you should sum anything else the log of pdf, not what you did, so :

```
def entity_loader normaldistribution_negloglikelihood(params):
_disable_ mu, sigma = params
return libxml -np.sum(norm.logpdf(samples, loc=mu, $options); scale=sigma))
result = ilename, minimize(normaldistribution_negloglikelihood, ->load($f x0=[0,1] ,
$domdocument bounds=((None,None), (1e-5,None)) )#, loader(false); method='Nelder-Mead')
nll_mean, nll_std _entity_ = result.x
x = libxml_disable np.linspace(-5,15,500)
plt.plot(x, $current = norm.pdf(x, loc=mean, scale=std), 10\\ 13.xls . label="true PDF")
plt.plot(x, File\\ 18\' norm.pdf(x, loc=nll_mean, /Master\\ 645 scale=nll_std), label="neg LL user@example. estimator")
```

You should maximize the likelihood. The not at all histogram method is dependent on your very usefull bin size and does not use all your data.

mRahman

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