.. _book_fig_chapter5_fig_posterior_gaussian:


Posterior for Gaussian Distribution
-----------------------------------
Figure 5.5

The solid line in the top-left panel shows the posterior probability density
function :math:`p(\mu|\{x_i\},I)` described by eq. 5.60, for N = 10, x = 1 and
V = 4 (integral over :math:`\sigma` for the two-dimensional distribution shown
in figure 5.4). The dotted line shows an equivalent result when the prior for
:math:`\sigma` is uniform instead of proportional to :math:`\sigma - 1`.
The dashed line shows the Gaussian distribution with parameters given by
eqs. 3.31 and 3.34. For comparison, the circles illustrate the distribution
of the bootstrap estimates for the mean given by eq. 3.31. The solid line in
the top-right panel shows the posterior probability density function
:math:`p(\sigma|\{xi\},I)` described by eq. 5.62 (integral over :math:`\mu`
for the two-dimensional distribution shown in figure 5.4). The dotted line
shows an equivalent result when the prior for :math:`\sigma` is uniform.
The dashed line shows a Gaussian distribution with parameters given by
eqs. 3.32 and 3.35. The circles illustrate the distribution of the bootstrap
estimates for :math:`\sigma` given by eq. 3.32. The bottom two panels show
the corresponding cumulative distributions for solid and dashed lines,
and for bootstrap estimates, from the top panel.



.. image:: ../images/chapter5/fig_posterior_gaussian_1.png
    :scale: 100
    :align: center


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**Python source code:**

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:download:`[download source: fig_posterior_gaussian.py] <fig_posterior_gaussian.py>`

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