Statistical Astronomy (STA1)

Prof. G. Woan
Starts September 2021

All this material is also available on the A345 Moodle site.

" The trouble is that what we [statisticians] call modern statistics was developed under strong pressure on the part of biologists. As a result, there is practically nothing done by us which is directly applicable to problems of astronomy." Jerzy Neyman, father of frequentist hypothesis testing.

Course details

These copies of the lecture notes are for reference only, and should not be used to replace your own notes. You will find the course much harder if you don't create your own written version, so please avoid just printing these notes out in quantity.  (There is a pdf of the whole course here if you really need it.)

Wikipedia is a particularly useful resource for understanding statistical inference, and the research community seems to have curated most of the entries well.  I've included some links below.

'Statistical Astronomy 1 in a nutshell' is a brief summary of the main points in the course and should be useful for revision.

Bayesian probability Deductive reasoning and Boolean algebra. Conditional probability and the extention to plausible reasoning. The idea of probability as a measure of plausibility of a statement. The sum and product rules in probability. Bayes' theorem and Bayesian Probability Theory.
lecture notes:  1  2  3  4  5  6  7  8  9  10  11

Pictures: [bad stats | Bayes' grave]
Notebooks: [at  https://jupyter.physics.gla.ac.uk/]  /examples/a345/STA1/logic.ipynb
Misc: [ Prosecutor's fallacy | Planck results paper | GW150914 parameter paper ]

Probability as a limit of relative frequency, and probability distributions Frequentist definition of probability; relative frequency; combinatorial probability; probability distributions and random variables; Poisson distribution as an example of a discrete distribution; continous distributions and pdfs; cumulative distribution functions; the uniform distribution; the Central (Normal) distribution, histograms; the Central distribution as a limiting distribution; measures and moments of a distribution - the mean, variance, standard deviation, median, mode, skewness and kurtosis; variable transforms; multivariate distributions; joint pdfs; marginal distributions; statistical independence; the bivariate normal distribution; samples and parents.  The central limit theorem.
lecture notes:  1  2  3  4  5  6  7  8  9  10  11  12  13
   14  15  16  17  18  19  20  21  22  23  24  25  26
Pictures: [ uniform distribution | Central distribution | histograms
Notebooks: [at  https://jupyter.physics.gla.ac.uk/]      /examples/a345/STA1/Poisson alarm.ipynb
                                                                                                                               multivariate normal distribution.ipynb
                                                                                                                                central limit theorem.ipynb
Handouts: [ Poisson statistics | The Bivariate Normal Distribution ]
Misc: [ central limit theorem demo ]

Bayesian parameter estimation The different approaches to parameter estimation in frequentist and Bayesian probability theory. Bayes' theorem as applied to parameter estimation and examples of its application. Priors, likelihoods and posterior distributions. The biased coin problem. Dependence (or otherwise) of posterior on choice of prior. General Bayesian parameter estimation. The idea of a 'model'. The universality of the posterior distribution. Best estimates and error bars. The Gaussian approximation to the posterior pdf. Shortest confidence intervals. Symmetric and asymmetric pdfs. The treatment of Gaussian noise, with uniform and non-uniform variance. Model fitting. More marginal distributions. Example of fitting to a weak spectral line (Poisson noise). The maximum likelihood and least-squares approximations. Fitting a straight line to data.
lecture notes:  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
  17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
  33 34 35 36 37                      
Pictures: [ ] 
Notebooks: [at  https://jupyter.physics.gla.ac.uk/]  /examples/a345/STA1/bayesian coin flips2.ipynb
Handouts: [ fair coin trials , and again | parameter estimation in the presence of Poisson noise]

Question Sheets


There are many books on probability theory, not all useful for astronomers. Here are some that are: