Log-likelihood for Gaussian Distribution

Figure5.4 An illustration of the logarithm of the posterior probability density function for and , (see eq. 5.58) for data drawn from a Gaussian distribution and N = 10, x = 1, and V = 4. The maximum of is renormalized to 0, and color coded as shown in the legend. The maximum value of is at and . The contours enclose the regions that contain 0.683, 0.955, and 0.997 of the cumulative (integrated) posterior probability.

Code output:

Python source code:

# Author: Jake VanderPlas
# License: BSD
#   The figure produced by this code is published in the textbook
#   "Statistics, Data Mining, and Machine Learning in Astronomy" (2013)
#   For more information, see http://astroML.github.com
#   To report a bug or issue, use the following forum:
#    https://groups.google.com/forum/#!forum/astroml-general
import numpy as np
from matplotlib import pyplot as plt
from astroML.plotting.mcmc import convert_to_stdev

#----------------------------------------------------------------------
# This function adjusts matplotlib settings for a uniform feel in the textbook.
# Note that with usetex=True, fonts are rendered with LaTeX.  This may
# result in an error if LaTeX is not installed on your system.  In that case,
# you can set usetex to False.
from astroML.plotting import setup_text_plots
setup_text_plots(fontsize=8, usetex=True)


def gauss_logL(xbar, V, n, sigma, mu):
    """Equation 5.57: gaussian likelihood"""
    return (-(n + 1) * np.log(sigma)
            - 0.5 * n * ((xbar - mu) ** 2 + V) / sigma ** 2)

#------------------------------------------------------------
# Define the grid and compute logL
sigma = np.linspace(1, 5, 70)
mu = np.linspace(-3, 5, 70)
xbar = 1
V = 4
n = 10

logL = gauss_logL(xbar, V, n, sigma[:, np.newaxis], mu)
logL -= logL.max()

#------------------------------------------------------------
# Plot the results
fig = plt.figure(figsize=(5, 3.75))
plt.imshow(logL, origin='lower',
           extent=(mu[0], mu[-1], sigma[0], sigma[-1]),
           cmap=plt.cm.binary,
           aspect='auto')
plt.colorbar().set_label(r'$\log(L)$')
plt.clim(-5, 0)

plt.contour(mu, sigma, convert_to_stdev(logL),
            levels=(0.683, 0.955, 0.997),
            colors='k')

plt.text(0.5, 0.93, r'$L(\mu,\sigma)\ \mathrm{for}\ \bar{x}=1,\ V=4,\ n=10$',
         bbox=dict(ec='k', fc='w', alpha=0.9),
         ha='center', va='center', transform=plt.gca().transAxes)

plt.xlabel(r'$\mu$')
plt.ylabel(r'$\sigma$')

plt.show()

[download source: fig_likelihood_gaussian.py]