Bayesian Methods for the Physical Sciences

A.A. 2014/2015

by Stefano Andreon

A.A. 2014/2015

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 runs 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.

Working program.

Lecture 1)

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

Student work: 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). Multi-parameters models. Source plus background (both Poisson). Combining different measurements of a source flux with different background levels.

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

Lecture 3)

Measuring the intrinsic scatter, from data with heteroscedastic (different from point to point) errors. Comparison with state-of-art standard, Robust. Introduction to regression. Remember selection effects! Do you regress for predicting, to infer the parameters, or for establishing if a trend is there?

Student work: 1) compute the posterior for source plus background model by combining different measurements; 2) compute the cluster velocity dispersion using these Data. 3) test teacher statements, computing by yourself E(x|y), E(y|x) from this sample, generated with JAGS. Its CODAindex is here.

Lecture 4)

Regression. 1) Starting easy: no error on predictor, no intrinsic scatter: completeness. 2) No error on predictor variable, heteroscedastic errors on y, and instrinsic scatter: Has the constant of fine structure changed? Data courtesy of Molaro et al. (2010).

Lecture 5)

Regression, full problem: heteroscedastic errors on x, y, and instrinsic scatter: Magorrian relation, using these data. Mixture of regressions (i.e. regressions in presence of a contaminating population). Final exercise, to be performed completely alone (no hints, no suggestions): make the analysis of my paper, Andreon (2010, MNRAS, 407, 263) using the data. Compute the stellar baryon fraction and its dependence with mass. Intrinsic scatter is there, heteroscedastic errros, different sample for different measurements, i.e. the routine astronomical job.

Student work: Compute how the mass of clusters depends on richness, using data from Andreon & Hurn (2010, MNRAS, 404, 1922).

Homeworks.

Give a look to my book , and maybe to my homepage.

The lectures will hold during March-May 2015 to PhD students, post-doc and stuff of the Universita' degli Studi di Padova, Universita' degli Studi di Bologna and open to all people if registered (mail to Stefano Andreon).