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Astro 193 : Feb 25
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Follow up
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reduced 2 : mean = 1 and variance = 2/

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non-linear least-squares fitting (contd.) Working through Homework 5 Bayesian Probability Theory MCMC basics Homework 6


Homework 5
1. Generate Poisson distributions using rejection sampling for =1,10,100 2. Compute means for 10 consecutive random deviates and demonstrate the Central Limit Theorem


Probability
Reading
T. Loredo (1990) monograph, chapters 1-3 From Laplace to Supernova 1987A: Bayesian Inference in Astrophysics Gelman et al., Bayesian Data Analysis, chapter 1


Probability Definitions

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Frequency of occurrence of an event Degree of belief in a proposition


Axioms of Probability Theory

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p(A or not A) = p(A) + p(not A) = 1 p(A and B) = p(B) p(A given B) p(A) p(B given A)


Axioms of Probability Theory

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p(A + A) = p(A) + p(A) = 1 p(A B) = p(B) p(A | B) p(A) p(B | A)


(Alt) Sum Rule
p(A+B) = 1 ­ p(AB) = 1 ­ p(A) p(B|A) = 1 ­ p(A) (1 ­ p(B|A) = 1 ­ p(A) + p(A) p(B|A) = p(A) + p(AB) = p(A) + p(B) p(A|B) = p(A) + p(B) (1 ­ p(A|B)) = p(A) + p(B) ­ p(B) p(A|B) = p(A) + p(B) ­ p(AB)


Bayes' Theorem
p(AB|C) = p(A|BC) p(B|C) = p(B|AC) p(A|C) p(A|BC) = p(B|AC) p(A|C) / p(B|C) p(|D I) = p(D| I) p(|I) / p(D|I)


prior, likelihood, posterior
p(|D I) = p(D| I) p(|I) / p(D|I) a priori distribution: p(|I) likelihood: p(D| I) a posteriori distribution: p(|D I)


prior
· Unfairly maligned as "subjective", but actually a mechanism to explicitly encode your assumptions · When your data are weak, your prior beliefs don't change; when your data are strong, your prior beliefs don't matter. · You update your prior belief with new data, using Bayes' Theorem. Lets you daisy-chain analyses. · When your prior is informative, takes more data to make a large change. · Technically, the biggest difference between likelihood analysis and Bayesian analysis: converts p(D| I) to p(|D I)


marginalization
· multi-dimensional parameters = {} · joint posterior: p()d · integrate over "nuisance" parameters p()d = p(,) d d


uncertainty
· p(|D) describes the uncertainty on · Usually reported as 68% or 90% central intervals (always say what they are!) · No guarantee of good coverage properties (because of priors), unlike frequentist confidence intervals ("the true value is contained 90% of the time for CIs calculated in this manner when the experiment is repeated")


frequentist vs Bayesian
Bayesian
Data are fixed and parameters have uncertainties Uncertainty is range in parameter values that encompass a certain probability Prior assumptions are explicit, but can be arbitrary Allows daisy-chaining of analyses No guarantee of good coverage

Frequentist
The true parameter is fixed, and data are realizations from different experiments Confidence interval overlaps the true value a certain fraction of the time You have to be aware of assumptions and limits of applicability Must be careful about stopping rules Confidence interval defined to produce good coverage


frequentist vs Bayesian
· use whichever technique as appropriate · Bayesian: non-repeatable experiments (many in Astronomy), make assumptions explicit via prior, daisychain analyses. hierarchical modeling, etc. · Frequentist: repeatable experiments, large datasets, vast toolkit for highly specific problems, non-parametric tests, hypothesis tests, least-squares fitting, goodnessof-fit, etc.


MCMC Basics