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Understanding Poisson Analysis: Mastering Probability and Likelihood Techniques

Dive into the world of Poisson analysis and probability with this comprehensive guide. Learn how to apply Poisson Likelihood, Bayes' Theorem, and the gamma distribution to real-world data analysis. Discover how to effectively marginalize source intensity and background noise, derive essential probability equations, and understand the importance of priors in Bayesian analysis. This resource is perfect for anyone looking to enhance their statistical skills, providing clear explanations and practical examples to ensure you can analyze data effectively for a lifetime.

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Understanding Poisson Analysis: Mastering Probability and Likelihood Techniques

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  1. POISSON TALES FISH TALES If you give a man Cash he will analyze for a day. If you teach a man Poisson, he will analyze for a lifetime. If you give a man a fish he will eat for a day. If you teach a man to fish he will eat for a lifetime.

  2. It is all about Probabilities! • The Poisson Likelihood • Probability Calculus – Bayes’ Theorem • Priors – the gamma distribution • Source intensity and background marginalization • Hardness Ratios (BEHR)

  3. 0 1 N,t®¥ Deriving the Poisson Likelihood

  4. marginalization :: p(a|D) = p(ab|D) db Probability Calculus A or B :: p(A+B) = p(A) + p(B) - p(AB) A and B :: p(AB) = p(A|B) p(B) = p(B|A) p(A) Bayes’ Theorem :: p(B|A) = p(A|B) p(B) / p(A) p(Model |Data) = p(Model) p(Data|Model) / p(Data) prior distribution posterior distribution likelihood normalization

  5. Priors

  6. Priors • Incorporate known information • Forced acknowledgement of bias • Non-informative priors • – flat • – range • – least informative (Jeffrey’s) • – gamma

  7. Priors the gamma distribution

  8. Panning for Gold: Source and Background

  9. Hardness Ratios Simple, robust, intuitive summary Proxy for spectral fitting Useful for large samples Most needed for low counts

  10. BEHR http://hea-www.harvard.edu/AstroStat/BEHR/

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