# Bayesian parameter estimation for discrete data spectra

Wang, Li (2017-12)

Thesis (MSc)--Stellenbosch University, 2017

Thesis

ENGLISH ABSTRACT : Discrete spectra are ubiquitous in physics; for example nuclear physics, laser physics and experimental high energy physics measure integer counts in the form of particles in dependence of angle, wavelength, energy etc. Bayesian parameter estimation ( tting a function with free parameters to the data) is a sophisticated framework which can handle cases of sparse data as well as input of pertinent background information into the data analysis in the form of a prior probability. Bayesian comparison of competing models and functions takes into account all possible parameter values rather than just the best t values. We rst review the general statistical basis of data analysis, focusing in particular on the Poisson, Negative Binomial and associated distributions. After introducing the conceptual shift and basic relations of the Bayesian approach, we show how these distributions can be combined with arbitrary model functions and data counts to yield two general discrete likelihoods. While we keep an eye on the asymptotic behaviour as useful analytical checks, we then introduce and review the theoretical basis for Markov Chain Monte Carlo numerical methods and show how these are applied in practice in the Metropolis-Hastings and Nested Sampling algorithms. We proceed to apply these to a number of simple situations based on simulation of a background plus two or three Gaussian peaks with both Poisson and Negative Binomial likelihoods, and discuss how to select models based on numerical outputs.

AFRIKAANSE OPSOMMING : Diskrete spektra is 'n algemene verskynsel in sika: kern sika, laser sika en eksperimentele hoë-energie sika meet byvoorbeeld heelgetalle in die vorm van deeltjies as 'n funksie van hoek, gol engte, energie ens. Bayesiese parameterberaming (die passing van 'n funksie met vrye parameters op die data) is 'n geso stikeerde raamwerk wat gevalle van lae tellings asook pertinente agtergrondinligting as inligting vir die data-analise in die vorm van prior-waarskynlikhede kan hanteer. Bayesiese vergelyking van kompeterende modelle en modelfunksies neem alle moontlike parameterwaardes in ag eerder as net die enkele beste waardes daarvan. Ons gee eerstens 'n oorsig van die algemene statistiese basis van dataanalise met 'n besondere fokus op die Poisson-, Negative Binomial- en verwante verdelings. Die konseptuele omwenteling wat Bayes impliseer en die basiese vergelykings word bespreek, waarna ons wys hoe hierdie verdelings met willekeurige modelfunksies en datatellings gekombineer kan word om twee algemene diskrete likelihood-waarskynlikhede te skep. Terwyl ons 'n oog hou op die asimptotiese gedrag as nuttige analitiese verwysings, gee ons daarna 'n inleiding tot en sit ons die teoretiese basis van Markovketting Monte Carlo numeriese metodes uiteen en wys hoe hulle in die vorm van die Metropolis-Hastings en Nested Sampling algoritmes toegepas word. Ons pas hierdie algoritmes op 'n aantal eenvoudige situasies gebaseer op simulasies van 'n agtergrond plus twee of drie Gaussiese pieke toe met sowel Poisson asook Negative Binomial waarskynlikhede, en bespreek hoe om modelle te kies gebaseer op numeriese uitsette.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/102822

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