import numpy as np
import scipy.stats as sps
import matplotlib.pyplot as plt
from scipy.misc import logsumexp
#%matplotlib qt5
%matplotlib inline
plt.rcParams.update({'figure.figsize': (10.0, 6.0), 'font.size': 18})

Gaussian distribution¶

probability density finction

p (x ∣ μ, σ^{2}) = N (x ∣ μ, σ^{2}) = \frac{1}{\sqrt{2 π σ^{2}}} \exp {\frac{- (x - μ)^{2}}{2 σ^{2}}}

$\large \DeclareMathOperator{\Norm}{\mathcal{N}} \DeclareMathOperator{\Gam}{Gam} \DeclareMathOperator{\e}{exp} p(x \mid \mu, \sigma^2) = \Norm(x \mid \mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} \e\left\{ \frac{-(x - \mu)^2}{2\sigma^2} \right\}$

Maximum likelihood estimate of Gaussian density paramters

\begin{aligned} μ & = \frac{1}{N} \sum_{i} x_{i} \\ σ^{2} & = \frac{1}{N} \sum_{i} x_{i}^{2} \end{aligned}

$\large \begin{align} \mu &= \frac{1}{N} \sum_i x_i \\ \sigma^2 &= \frac{1}{N} \sum_i x_i^2 \end{align}$

# Plot gaussian distribution
mu = 0    # mean
sigma = 1 # standard deviation (std)
t = np.linspace(-10,10,1000)
plt.clf()
plt.plot(t, sps.norm.pdf(t, mu, sigma), 'r',lw=3)

# Genereate data from this distribution and plot corresponding ML estimated distribution
# Repeat few times to see different model estimates for the different training data sets
N=10 # number of training observations
for i in range(10):
    x = sps.norm.rvs(mu, sigma, N)
    p=plt.plot(t, sps.norm.pdf(t, np.mean(x), np.std(x)))
    plt.plot(x, np.zeros_like(x), '+', color=p[0].get_color())