Gaussian Distribution with Gaussian ErrorsΒΆ

Figure 5.7

The logarithm of the posterior probability density function for \mu and \sigma, L_p(\mu,\sigma), for a Gaussian distribution with heteroscedastic Gaussian measurement errors (sampled uniformly from the 0-3 interval), given by eq. 5.64. The input values are \mu = 1 and \sigma = 1, and a randomly generated sample has 10 points. Note that the posterior pdf is not symmetric with respect to the \mu = 1 line, and that the outermost contour, which encloses the region that contains 0.997 of the cumulative (integrated) posterior probability, allows solutions with \sigma = 0.

../../_images/fig_likelihood_gaussgauss_1.png

Code output:



# 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.
if "setup_text_plots" not in globals():
    from astroML.plotting import setup_text_plots
setup_text_plots(fontsize=8, usetex=True)


def gaussgauss_logL(xi, ei, mu, sigma):
    """Equation 5.63: gaussian likelihood with gaussian errors"""
    ndim = len(np.broadcast(sigma, mu).shape)

    xi = xi.reshape(xi.shape + tuple(ndim * [1]))
    ei = ei.reshape(ei.shape + tuple(ndim * [1]))

    s2_e2 = sigma ** 2 + ei ** 2
    return -0.5 * np.sum(np.log(s2_e2) + (xi - mu) ** 2 / s2_e2, 0)

#------------------------------------------------------------
# Define the grid and compute logL
np.random.seed(5)
mu_true = 1.
sigma_true = 1.
N = 10
ei = 3 * np.random.random(N)
xi = np.random.normal(mu_true, np.sqrt(sigma_true ** 2 + ei ** 2))

sigma = np.linspace(0.01, 5, 70)
mu = np.linspace(-3, 5, 70)

logL = gaussgauss_logL(xi, ei, mu, sigma[:, np.newaxis])
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.text(0.5, 0.93,
         (r'$L(\mu,\sigma)\ \mathrm{for}\ \bar{x}=1,\ '
          r'\sigma_{\rm true}=1,\ n=10$'),
         bbox=dict(ec='k', fc='w', alpha=0.9),
         ha='center', va='center', transform=plt.gca().transAxes)

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

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

plt.show()