Bayesian Methods for the Physical Sciences

A.A. 2009/2010

by Stefano Andreon

A.A. 2009/2010

by Stefano Andreon

You need in advance: a) to be able of drawing plots and make simple operations on numbers and vectors, and b) that JAGS run on your computer. JAGS user manual is here. See my notes here. Files, to test your reading of JAGS output, are here: CODAchain1.txt and CODAindex.txt.

Lecture 1)

Probability axioms (inclusive of marginalization & the bad it may occurs following other rules). Carla Bruni & neutrino mass. Neutrino mass upper limit. Introduction to JAGS.

Student work: re-compute of p(T) of the most distant cluster known, JKCS041, (alias marginalization; p(T,nh) is here), compute neutrino mass upper limit, with informative prior, and with a sloppy mass prior. Small role of the prior where the likelihood is very near to zero.

Lecture 2)

Poisson (count) model. Combining different measurements. Binomial model. Again on the prior: the precise shape does not matter, but its shape does, alias the Malmquist/Eddington bias (i.e. the Bayes theorem re-discovered two hundred years later). The bad of forgetting to account for the existence of boundaries.

Student work: use JAGS to compute the posterior for a Poisson and a Binomial model (dry merger rate).

Lecture 3)

Multi-parameters models. Source plus background (both Poisson). Combining different measurements of a source flux with different background levels. Measuring the intrinsic scatter, from data with heteroscedastic (different from point to point) errors. Comparison with state-of-art standard, Robust. Accounting for a contaminating population. Overdispersed version of the source plus background model, alias "cosmic variance".

Student work: use JAGS to compute the posterior for source plus background model, for combining different measurements, for computing the cluster velocity dispersion. Data for the last exercise. How to compute (and plot) the error on the model (shading).

Lecture 4)

Regression. Remember selection effects! Do you regress for predicting, to infer the parameters, or for establishing if a trend is there?

Starting easy: no error on predictor variable, heteroscedastic errors on y, and instrinsic scatter.

Student work: test teacher statements, by computing yourself E(x|y), E(y|x) from this sample, generated with JAGS. Its CODAindex is here. Has the constant of fine structure changed? Data courtesy of Molaro et al. (2010, in preparation).

Lecture 5)

Regression, full problem: heteroscedastic errors on x, y, and instrinsic scatter (x is a difference of two Poisson variate!). Mixture of regressions (i.e. regressions in presence of a contaminating population).

Student work: compute how the mass of clusters depends on richness, using data from Andreon & Hurn (2010, MNRAS, in press). Compute the predicted mass (predictive posterior) of a cluster without a known mass. Finally, make the analysis of my last paper, Andreon (2010, MNRAS, in press). Did that alone, no hints, no suggestions, just data and data! Compute the stellar and gas baryon fraction, its dependence with mass, and the stellar+gas baryon fraction. Intrinsic scatter is there, heteroscedastic errros, different sample for different measurements, i.e. the routine astronomical job.

Homeworks.

Give a look to my Bayesian primer for astronomers, to my Inference Page, to my book , and maybe to my homepage.

The lectures have been hold in April-May 2010 to PhD students and post-doc of the Universita' degli Studi di Padova, Universita' dell'Insubria, with, as guests, PhD students of the Universita' degli Studi di Milano-Bicocca.

While the computer is running the model, you may listen some music from my jukebox.