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import numpy as np

import math

def poisson_lambda_mle(d):

"""

Computes the Maximum Likelihood Estimate for a given 1D training

dataset from a Poisson distribution.

"""

return sum(d) / len(d)

def likelihood_poisson(x, lam):

"""

Computes the class-conditional probability for an univariate

Poisson distribution

"""

if x // 1 != x:

likelihood = 0

else:

likelihood = math.e**(-lam) * lam**(x) / math.factorial(x)

return likelihood

if __name__ == "__main__":

# Plot Probability Density Function

from matplotlib import pyplot as plt

training_data = [0, 1, 1, 3, 1, 0, 1, 2, 1, 2, 2, 1, 2, 0, 1, 4]

mle_poiss = poisson_lambda_mle(training_data)

true_param = 1.0

x_range = np.arange(0, 5, 0.1)

y_true = [likelihood_poisson(x, true_param) for x in x_range]

y_mle = [likelihood_poisson(x, mle_poiss) for x in x_range]

plt.figure(figsize=(10,8))

plt.plot(x_range, y_true, lw=2, alpha=0.5, linestyle='--', label='true parameter ($\lambda={}$)'.format(true_param))

plt.plot(x_range, y_mle, lw=2, alpha=0.5, label='MLE ($\lambda={}$)'.format(mle_poiss))

plt.title('Poisson probability density function for the true and estimated parameters')

plt.ylabel('p(x|theta)')

plt.xlim([-1,5])

plt.xlabel('random variable x')

plt.legend()

plt.show()