Table Of Content

AALLBBEERRTT TTAARRAANNTTOOLLAA ttoo bbee ppuubblliisshheedd bbyy ...... Probability and Measurements 1 Albert Tarantola Université de Paris, Institut de Physique du Globe 4, place Jussieu; 75005 Paris; France E-mail: [email protected] December 3, 2001 1(cid:1)c A. Tarantola, 2001. ii iii To the memory of my father. To my mother and my wife. iv v Preface In this book, I attempt to reach two goals. The first is purely mathematical: to clarify some of the basic concepts of probability theory. The second goal is physical: to clarify the methods to be used when handling the information brought by measurements, in order to understand how accurate are the predictions we may wish to make. Probability theory is solidly based on Kolmogorov axioms, and there is no problem when treating discrete probabilities. But I am very unhappy with the usual way of extending the theorytocontinuousprobabilitydistributions. Inthistext, Iintroducethenotionof‘volumetric probability’ different from the more usual notion of ‘probability density’. I claim that some of the more basic problems of the theory of continuous probability distributions can only ne solved within this framework, and that many of the well known ‘paradoxes’ of the theory are fundamental misunderstandings, that I try to clarify. I start the book with an introduction to tensor calculus, because I choose to develop the probability theory considering metric manifolds. The second chapter deals with the probability theory per se. I try to use intrinsic notions everywhere, i.e., I only introduce definitions that make sense irrespectively of the particular coordinates being used in the manifold under investigation. The reader shall see that this leads to many develoments that are at odds with those found in usual texts. Inphysicalapplicationsonenotonlyneedstodefineprobabilitydistributionsover(typically) large-dimensional manifolds. One also needs to make use of them, and this is achieved by sampling the probability distributions using the ‘Monte Carlo’ methods described in chapter 3. There is no major discovery exposed in this chapter, but I make the effort to set Monte Carlo methods using the intrinsic point of view mentioned above. The metric foundation used here allows to introduce the important notion of ‘homogeneous’ probability distributions. Contrary to the ‘noninformative’ probability distributions common in the Bayesian literature, the homogeneity notion is not controversial (provided one has agreed ona given metric over the space of interest). After a brief chapter that explain what an ideal measuring instrument should be, the book enters in the four chapter developing what I see as the four more basic inference problems in physics: (i) problems that are solved using the notion of ‘sum of probabilities’ (just an elaborate way of ‘making histograms), (ii) problems that are solved using the ‘product of probabilities’ (and approach that seems to be original), (iii) problems that are solved using ‘conditional probabilities’ (these including the so-called ‘inverse problems’), and (iv) problems that are solved using the ‘transport of probabilities’ (like the typical [indirect] mesurement problem, but solved here transporting probability distributions, rather than just transporting ‘uncertainties). I am very indebted to my colleagues (Bartolomé Coll, Georges Jobert, Klaus Mosegaard, ´ Miguel Bosch, Guillaume Evrard, John Scales, Christophe Barnes, Frédéric Parrenin and Bernard Valette) for illuminating discussions. I am also grateful to my collaborators at what was the Tomography Group at the Institut de Physique du Globe de Paris. Paris, December 3, 2001 Albert Tarantola vi Contents 1 Introduction to Tensors 1 2 Elements of Probability 69 3 Monte Carlo Sampling Methods 153 4 Homogeneous Probability Distributions 169 5 Basic Measurements 185 6 Inference Problems of the First Kind (Sum of Probabilities) 207 7 Inference Problems of the Second Kind (Product of Probabilities) 211 8 Inference Problems of the Third Kind (Conditional Probabilities) 219 9 Inference Problems of the Fourth Kind (Transport of Probabilities) 287 vii viii