Converting notebooks using Pelican

IPython notebook files are a useful way of presenting information. This entry was created in IPython notebook and then converted into a GitHub pages page using Pelican.

Converting links using definitions within external_links.md

To make it easier to both enter and use external URLs, I extended add_links.py to substitute in links found in external_links.md. All link keys in square brackets within this notebook that are also found in external_links.md are changed into appropriate markdown link text.

Running Python code:

import matplotlib.pyplot as plt
import numpy

Plotting with matplotlib:

%matplotlib inline


x = numpy.linspace(0, 3 * numpy.pi, 500)
plt.plot(x, numpy.sin(x**2))
plt.title('A simple chirp');

Mathematics

Let us start with Bayes' theorem:

$$ P(\mu~|~D) = \frac{P(D~|~\mu)P(\mu)}{P(D)} $$

We'll use a flat prior on $\mu$ (i.e. $P(\mu) \propto 1$ over the region of interest) and use the likelihood

$$ P(D~|~\mu) = \prod_{i=1}^N \frac{1}{\sqrt{2\pi\sigma_x^2}}\exp\left[\frac{(\mu - x_i)^2}{2\sigma_x^2}\right] $$

Computing this product and manipulating the terms, it's straightforward to show that this gives

$$ P(\mu~|~D) \propto \exp\left[\frac{-(\mu - \bar{x})^2}{2\sigma_\mu^2}\right] $$

which is recognizable as a normal distribution with mean $\bar{x}$ and standard deviation $\sigma_\mu$.

That is, the Bayesian posterior on $\mu$ in this case is exactly equal to the frequentist sampling distribution for $\mu$.