Bayesian Analysis in Kaon Photoproduction

Angular distributions of differential cross sections from the latest CLAS data sets ~\cite{bradford}, for the reaction ${\gamma}+p

{\rightarrow} K^{+} + {\Lambda}$ have been analyzed using associated Legendre polynomials. This analysis is based upon theoretical calculations in Ref.~\cite{fasano} where all sixteen observables in kaon photoproduction can be classified into four Legendre classes. Each observable can be described by an expansion of associated Legendre polynomial functions. One of the questions to be addressed is how many associated Legendre polynomials are required to describe the data. In this preliminary analysis, we used data models with different numbers of associated Legendre polynomials. We then compared these models by calculating posterior probabilities of the models. We found that the CLAS data set needs no more than four associated Legendre polynomials to describe the differential cross section data. In addition, we also show the extracted coefficients of the best model.

Introduction
Significant information on the structure of the nucleon can be obtained by studying its excitation spectrum. Over the last few decades, a large amount information about the spectrum of the nucleon has been collected. Most of this information has been extracted from pion-induced and pion photoproduction reactions. However, pionic reactions may have biased the information on the existence of certain resonances. Constituent quark model calculations predict a much richer resonance spectrum than has been observed in pion production experiments~\cite{capstick}. Predicted resonances which have not been observed are called "missing" resonances. Instead, the constituent quark model also predicts that these "missing" resonances may couple strongly to K$\Lambda$ and K$\Sigma$ channels or other final states involving vector mesons~\cite{capstick,mart1,mart2}. Since performing kaon-hyperon, kaon-nucleon or hyperon-nucleon scattering experiments is a daunting task, kaon photoproduction on the nucleon appears to be a good alternative solution~\cite{mart1,mart2}.

Experiments on kaon photoproduction and electroproduction started in the 1960s. However the old experimental data are often inconsistent and have large error bars. In recent years a large amount of data for kaon photoproduction has been collected. High statistics data from CLAS, for differential cross sections, recoil polarization, $C_{x}$ and $C_{z}$ double polarizations for the reaction $\gamma + p \rightarrow K^{+} + \Lambda$ have been published~\cite{bradford,bradfor2}. Additional experimental data have also been measured by SAPHIR~\cite{glander,tran,glander2}, LEPS~\cite{sumihama,zegers} and GRAAL~\cite{leres}.

Several previous analyses have been applied to the results of these experiments, such as Isobar models~\cite{mart1,mart2,ireland,janssen,janssen2} and Coupled channel models~\cite{shyklar,usov,penner}. However different theoretical model calculations often produce very different predictions. In Ref.\cite{fasano} all sixteen observables in kaon photoproduction were shown to be classified into the classes ${\cal L}_0(\hat{{\bf I}};\hat{{\bf E}};\hat{{\bf C_{z'}}};\hat{{\bf
L_{z'}}})$, ${\cal L}_{1a}(\hat{{\bf P}}; \hat{{\bf H}}; \hat{{\bf C_{x'}}}; \hat{{\bfL_{x'}}})$, ${\cal L}_{1b}(\hat{{\bf T}}; \hat{{\bf F}}; \hat{{\bf O_{x'}}};\hat{{\bf T_{z'}}})$ and ${\cal L}_2(\hat{{\bf {\Sigma}}}; \hat{{\bf G}}; \hat{{\bf O_{z'}}}; \hat{{\bf T_{x'}}})$, where each class is an expansion in a different set of associated Legendre polynomials. What is not apparent is how many terms in each expansion are required. This work attempts to address the issue by examining data models with different numbers of terms, and calculating which one has the greatest posterior probability. In this article we only focus on the differential cross section observables, which are described by the associated Legendre class ${\cal L}_0$. (PTPH)

Who was the Reverend Thomas Bayes?

Bayes, Thomas (b. 1702, London - d. 1761, Tunbridge Wells, Kent), mathematician who first used probability inductively and established a mathematical basis for probability inference (a means of calculating, from the number of times an event has not occured, the probability that it will occur in future trials).

He set down his findings on probability in "Essay Towards Solving a Problem in the Doctrine of Chances" (1763), published posthumously in the Philosophical Transactions of the Royal Society of London.

The only works he is known to have published in his lifetime are Divine Benevolence, or an Attempt to Prove That the Principal End of the Divine Providence and Government is the Happiness of His Creatures (1731) and An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians Against the Objections of the Author of the Analyst (1736) which countered attacks by Bishop Berkeley on the logical foundations of Newton's calculus.

Here is some more information about Bayes taken from the book The Official Guide to Bunhill Fields. Bunhill Fields is a park in London, England where Bayes is buried (see The Burial Place of Bayes below).

He was a Presbyterian minister in Tunbridge Wells from 1731, son of the Rev. Joshua Bayes, a Nonconformist minister. It is thought that his election to the Royal Society might have been based on a tract of 1736 in which Bayes defended the views and philosophy of Sir Isaac Newton. A notebook of his exists, and includes a method of finding the time and place of conjunction of two planets, notes on weights and measures, a method of differentiation, and logarithms.

Thomas Bayes' contributions are immortalized by naming a fundamental proposition in probability, called Bayes Rule, after him. (The following is quoted from the Encyclopaedia Britannica)

Bayesian Analysis

What is Bayesian Analysis?

What we now know as Bayesian statistics has not had a clear run since 1763. Although Bayes's method was enthusiastically taken up by Laplace and other leading probabilists of the day, it fell into disrepute in the 19th century because they did not yet know how to handle prior probabilities properly. The first half of the 20th century saw the development of a completely different theory, now called frequentist statistics. But the flame of Bayesian thinking was kept alive by a few thinkers such as Bruno de Finetti in Italy and Harold Jeffreys in England. The modern Bayesian movement began in the second half of the 20th century, spearheaded by Jimmy Savage in the USA and Dennis Lindley in Britain, but Bayesian inference remained extremely difficult to implement until the late 1980s and early 1990s when powerful computers became widely accessible and new computational methods were developed. The subsequent explosion of interest in Bayesian statistics has led not only to extensive research in Bayesian methodology but also to the use of Bayesian methods to address pressing questions in diverse application areas such as astrophysics, weather forecasting, health care policy, and criminal justice.

Scientific hypotheses typically are expressed through probability distributions for observable scientific data. These probability distributions depend on unknown quantities called parameters. In the Bayesian paradigm, current knowledge about the model parameters is expressed by placing a probability distribution on the parameters, called the "prior distribution", often written as

When new data y become available, the information they contain regarding the model parameters is expressed in the "likelihood," which is proportional to the distribution of the observed data given the model parameters, written as
This information is then combined with the prior to produce an updated probability distribution called the "posterior distribution," on which all Bayesian inference is based. Bayes' Theorem, an elementary identity in probability theory, states how the update is done mathematically: the posterior is proportional to the prior times the likelihood, or more precisely,

In theory, the posterior distribution is always available, but in realistically complex models, the required analytic computations often are intractable. Over several years, in the late 1980s and early 1990s, it was realized that methods for drawing samples from the posterior distribution could be very widely applicable.

There are many reasons for adopting Bayesian methods, and their applications appear in diverse fields. Many people advocate the Bayesian approach because of its philosophical consistency. Various fundamental theorems show that if a person wants to make consistent and sound decisions in the face of uncertainty, then the only way to do so is to use Bayesian methods. Others point to logical problems with frequentist methods that do not arise in the Bayesian framework. On the other hand, prior probabilities are intrinsically subjective -- your prior information is different from mine -- and many statisticians see this as a fundamental drawback to Bayesian statistics. Advocates of the Bayesian approach argue that this is inescapable, and that frequentist methods also entail subjective choices, but this has been a basic source of contention between the `fundamentalist' supporters of the two statistical paradigms for at least the last 50 years. In contrast, it is more the pragmatic advantages of the Bayesian approach that have fuelled its strong growth over the last 20 years, and are the reason for its adoption in a rapidly growing variety of fields. Powerful computational tools allow Bayesian methods to tackle large and complex statistical problems with relative ease, where frequentist methods can only approximate or fail altogether. Bayesian modelling methods provide natural ways for people in many disciplines to structure their data and knowledge, and they yield direct and intuitive answers to the practitioner's questions.

There are many varieties of Bayesian analysis. The fullest version of the Bayesian paradigm casts statistical problems in the framework of decision making. It entails formulating subjective prior probabilities to express pre-existing information, careful modelling of the data structure, checking and allowing for uncertainty in model assumptions, formulating a set of possible decisions and a utility function to express how the value of each alternative decision is affected by the unknown model parameters. But each of these components can be omitted. Many users of Bayesian methods do not employ genuine prior information, either because it is insubstantial or because they are uncomfortable with subjectivity. The decision-theoretic framework is also widely omitted, with many feeling that statistical inference should not really be formulated as a decision. So there are varieties of Bayesian analysis and varieties of Bayesian analysts. But the common strand that underlies this variation is the basic principle of using Bayes' theorem and expressing uncertainty about unknown parameters probabilistically. (quoted from http://www.bayesian.org/ ).