Simple probability distribution model

In the following we learn the mean and covariance of a univariate gaussian distribution from data.

Initially we have to import the necessary modules.

import numpy as np
import matplotlib.pyplot as plt
from UQpy.inference import DistributionModel, MLE
from UQpy.distributions import Normal
from UQpy.inference import MinimizeOptimizer

First, for the sake of this example, we generate fake data from a gaussian distribution with mean 0 and standard deviation 1.

mu, sigma = 0, 0.1  # true mean and standard deviation
data_1 = np.random.normal(mu, sigma, 1000).reshape((-1, 1))
print('Shape of data vector: {}'.format(data_1.shape))

count, bins, ignored = plt.hist(data_1, 30, density=True)
plt.plot(bins, 1 / (sigma * np.sqrt(2 * np.pi)) * np.exp(- (bins - mu) ** 2 / (2 * sigma ** 2)),
         linewidth=2, color='r')
plt.title('Histogram of the data')
plt.show()

Shape of data vector: (1000, 1)

Create an instance of the class Model. The user must define the number of parameters to be estimated, in this case 2 (mean and standard deviation), and set those parameters to be learnt as None when instantiating the Distribution object. For maximum likelihood estimation, no prior pdf is required.

# set parameters to be learnt as None
dist = Normal(loc=None, scale=None)
candidate_model = DistributionModel(n_parameters=2, distributions=dist)

ml_estimator = MLE(inference_model=candidate_model, data=data_1, n_optimizations=3)
print('ML estimates of the mean={0:.3f} (true=0.) and std. dev={1:.3f} (true=0.1)'.format(
    ml_estimator.mle[0], ml_estimator.mle[1]))

ML estimates of the mean=-0.001 (true=0.) and std. dev=0.102 (true=0.1)

We can also fix one of the parameters and learn the remaining one

d = Normal(loc=0., scale=None)
candidate_model = DistributionModel(n_parameters=1, distributions=d)

optimizer = MinimizeOptimizer(bounds=[[0.0001, 2.]])
ml_estimator = MLE(inference_model=candidate_model, data=data_1,
                   n_optimizations=1)
print('ML estimates of the std. dev={0:.3f} (true=0.1)'.format(ml_estimator.mle[0]))

ML estimates of the std. dev=0.102 (true=0.1)