An introduction to Bayesian methods for analyzing chemistry data: Part 1: An introduction to Bayesian theory and methods

In this tutorial paper, we outline the application of Bayesian theory and methods for analysing experimental chemistry data. We provide an overview of the background theory and the essential rules necessary for manipulating conditional probabilities and density functions (pdfs) i.e. the product and marginalisation rules. Drawing on these rules we demonstrate, using a variety of examples from chemistry, how Bayes theorem can be adapted to analyse and interpret experimental data for a wide range of typical chemistry experiments, including basic model selection (i.e. hypothesis testing), parameter estimation, peak refinement and advanced model selection. An outline of the steps and underlying assumptions are presented, while necessary mathematics are also discussed.

[1]  Stephen F. Gull,et al.  Bayesian Data Analysis: Straight-line fitting , 1989 .

[2]  V. Dose.  Bayesian inference in physics: case studies , 2003 .

[3]  E. Jaynes.  Probability theory : the logic of science , 2003 .

[4]  Ng Armstrong,et al.  Bayesian analysis of band-broadening models used in high performance liquid chromatography , 2006 .

[5]  B. Bakshi,et al.  Toward Bayesian chemometrics--a tutorial on some recent advances. , 2007, Analytica chimica acta.

[6]  David Brynn Hibbert,et al.  An introduction to Bayesian methods for analyzing chemistry data Part II: A review of applications of Bayesian methods in chemistry , 2009 .

[7]  John Skilling,et al.  Maximum Entropy and Bayesian Methods , 1989 .

[8]  J. Skilling,et al.  Maximum entropy image reconstruction: general algorithm , 1984 .

[9]  R. Bryan.  SOLVING OVERSAMPLED DATA PROBLEMS BY MAXIMUM ENTROPY. , 1990 .

[10]  J. Justin Gooding,et al.  Electrochemical detection of lead ions via the covalent attachment of human angiotensin I to mercaptopropionic acid and thioctic acid self-assembled monolayers , 2005 .

[11]  David B. Dunson,et al.  Bayesian Data Analysis , 2010 .

[12]  Philip C. Gregory,et al.  Bayesian Logical Data Analysis for the Physical Sciences: Acknowledgements , 2005 .

[13]  Richard M. Bailey,et al.  A Bayesian Approach to the Evaluation of Equivalent Doses in Sediment Mixtures for Luminescence Dating , 2004 .

[14]  J. Skilling.  Quantified Maximum Entropy , 1990 .

[15]  I. Good.  When batterer turns murderer , 1995, Nature.

[16]  David J. C. MacKay,et al.  Information Theory, Inference, and Learning Algorithms , 2004, IEEE Transactions on Information Theory.

[17]  K OrJ.  Numerical Bayesian methods applied to signal processing , 1996 .

[18]  W von der Linden,et al.  Signal and background separation. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[19]  R. T. Cox.  Probability, frequency and reasonable expectation , 1990 .

[20]  Stephen F. Gull,et al.  Developments in Maximum Entropy Data Analysis , 1989 .

[21]  John Skilling,et al.  Data analysis : a Bayesian tutorial , 1996 .

[22]  D. Sivia,et al.  Molecular spectroscopy and Bayesian spectral analysis—how many lines are there? , 1992 .

[23]  R. Bryan,et al.  Maximum entropy analysis of oversampled data problems , 1990, European Biophysics Journal.

[24]  G. D'Agostini.  Bayesian inference in processing experimental data: principles and basic applications , 2003 .

[25]  Kenneth M. Hanson,et al.  Posterior sampling with improved efficiency , 1998, Medical Imaging.

[26]  W. von der Linden.  Maximum-entropy data analysis , 1995 .