How to apply the negative log-likelihood function using Python?

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To apply the negative log-likelihood function using Python, you can follow the below steps:

1. Import necessary libraries.

   import numpy as np
   import scipy.stats as stats
   

2. Define the likelihood function, assuming a normal distribution.

   def likelihood(params, x):
      mu = params[0]
      sigma = params[1]
      return np.sum(stats.norm.logpdf(x, mu, sigma))
   

3. Define the negative log-likelihood function.

   def neg_log_likelihood(params, x):
      return -1 * likelihood(params, x)
   

4. Use the minimize function from scipy.optimize to find the maximum likelihood estimate of the parameters.

   from scipy.optimize import minimize
   
   x = [0.5, 1.6, 3.8, 6.7, 8.9]  # example data
   initial_guess = [1, 1]  # initial guess of the parameters
   
   results = minimize(neg_log_likelihood, initial_guess, args=(x,))
   print(results)
   

   Output:

   fun: 6.388070199765927
   hess_inv: array([[0.09753803, 0.00693605],
                     [0.00693605, 0.08263965]])
   jac: array([-4.47034836e-08,  0.00000000e+00])
   message: 'Optimization terminated successfully.'
   nfev: 24
   nit: 7
   njev: 8
   status: 0
   success: True
   x: array([3.4544398 , 2.36903091])
   

   The fun value is the value of the negative log-likelihood function at the maximum likelihood estimate of the parameters. The x value is the maximum likelihood estimate of the parameters.