9.11. Compare Gaussian noise sampling to lighthouse analysis

Here we observe that Gaussian noise sampling is carried out just like the radioactive lighthouse analysis from exercise notebook 📥 Radioactive lighthouse problem (which we assume you have worked through). Jump to the Bayesian approach in the exercise notebook 📥 Parameter estimation example: Gaussian noise and averages II. The goal is to sample a posterior \(p(\pars|D,I)\)

\[ p(\mu,\sigma | D, I) \leftrightarrow p(x_0,y_0|X,I) \]

where \(D\) on the left are the \(x\) points and \(D=X\) on the right are the \(\{x_k\}\) where scintillation flashes are detected.

What do we need? From Bayes’ theorem, we need

\[\begin{split}\begin{align} \text{likelihood:}& \quad p(D|\mu,\sigma,I) \leftrightarrow p(D|x_0,y_0,I) \\ \text{prior:}& \quad p(\mu,\sigma|I) \leftrightarrow p(x_0,y_0|I) \end{align}\end{split}\]

You are generalizing the functions for log PDFs and the plotting of posteriors that are in 📥 Radioactive lighthouse problem. Note the functions for log-prior and log-likelihood in 📥 Parameter estimation example: Gaussian noise and averages II. Here \(\pars = [\mu,\sigma]\) is a vector of parameters; cf. \(\pars = [x_0,y_0]\).

Let’s step through the essential set up for emcee.

• It is best to create an environment that will include emcee and corner.

  Hint

  Nothing in the emcee sampling part needs to change!

• Basically we are doing 50 random walks in parallel to explore the posterior. Where the walkers end up will define our samples of \(\mu,\sigma\) \(\Longrightarrow\) the histogram is an approximation to the (unnormalized) joint posterior.

• Plotting is also the same, once you change labels and mu_true, sigma_true to x0_true, y0_true. (And skip the maxlike part.)

Maximum likelihood here is the frequentist estimate \(\longrightarrow\) this is an optimization problem. And you can read off marginalized estimates for \(\mu\) and \(\sigma\).

Question

Are \(\mu\) and \(\sigma\) correlated or uncorrelated?

Answer

They are uncorrelated because the contour ellipses in the joint posterior have their major and minor axes parallel to the \(\mu\) and \(\sigma\) axes. Note that the fact that they look like circles is just an accident of the ranges chosen for the axes; if you changed the \(\sigma\) axis range by a factor of two, the circle would become flattened.

Bottom line: the two analyses are completely analogous.