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" 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.
Copies of lecture notes will appear here as the course proceeds. They are
for reference only, and should not be used to replace your own lecture notes.
You will find the course much harder if you don't create your own written version,
so please do not print these notes out in quantity.
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.
Pictures: [Bayes' grave]
Handouts: [ ]
Misc: [ The Bayesian Songbook]
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.
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. 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.
There are many books on probability theory, not all useful for astronomers. Here are some that are: