Probability and measurements

324 Pages·2001·6.76 MB·English

by  Tarantola A.

AALLBBEERRTT TTAARRAANNTTOOLLAA
ttoo bbee ppuubblliisshheedd bbyy ......

Probability
and
Measurements 1
Albert Tarantola
Universit´e 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 (Bartolom´e Coll, Georges Jobert, Klaus Mosegaard,
´
Miguel Bosch, Guillaume Evrard, John Scales, Christophe Barnes, Fr´ed´eric 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