# The Statistical Eyeglasses

- The math behind scientific knowledge

- Edoardo Milotti

- November 2018

### Description

Science often deals with hard-to-see phenomena, and they only stand out and become real when viewed through the lens of complex statistical tools. This book is not a textbook about statistics applied to science – there are already many excellent books to choose from – rather, it gives an overview of the basic principles that physical scientists use to analyze their data and bring out the order of nature from the fog of background noise.### About Editors

Edoardo Milotti is a professor of physics at the University of Trieste, Italy. After working mostly in experimental particle physics, he has also authored papers on noise processes in physics, and on the physics of cancer. His long-time research interests are in the direction of the analysis of experimental data and in the modeling of complex phenomena. He has published more than 200 scientific papers in peer-reviewed scientific journals. He lives in Trieste with his wife Alessandra.### Table of Contents

Preface

1 Models of Nature

2 Randomness

2.1 What is random?

2.2 How does randomness show up in nature?

2.3 Random and deterministic signals

2.4 From noisy data to the likelihood function

3 Bayesian and frequentist approaches to scientific inference

3.1 Bayes' theorem

3.1.1 A game of boxes: a simple example of Bayesian inference

3.1.2 The initial prior: a critical issue in the Bayesian framework

3.1.3 Moving on to a discrete set of hypotheses

3.2 The same game, and a mysterious result

3.3 Statistical descriptors

4 The Principles of Inferential statistics

4.1 Bayes and the likelihood function

4.2 The "least informative prior"

4.2.1 Maximum Entropy (MaxEnt)

4.3 The principles of inferential statistics

5 Parametric Inference

5.1 Bayesian parametric inference

5.1.1 A fair coin?

5.1.2 Trying different priors

5.2 Frequentist parametric inference

5.2.1 Chebyshev's inequality and the weak law of large numbers

5.2.2 Frequentist estimate

5.2.3 Maximum likelihood approach

6 Prior distributions and equiprobable events in the physical sciences

6.1 Elementary Monte Carlo method

6.1.1 Uniformly distributed pseudo-random numbers

6.1.2 The acceptance-rejection method

6.2 Transformations of random variables by Monte Carlo

6.3 Bertrand's paradox

7 Conclusions: the statistical nature of scientific knowledge

Appendices

A Short review of some basic concepts

A.1 Probability and relative frequency

A.2 The basic rules of probability

A.2.1 Conditional probabilities

A.3 Probability distributions

A.3.1 The binomial distribution

A.3.2 The Poisson distribution

A.3.3 The normal distribution

A.3.4 Mean, variance and other moments of a probability distribution

A.4 Sample mean and sample variance

B Abbreviations

Index

### Bibliographic

Paperback ISBN: 9780750329491

Ebook ISBN: 9781643271491

DOI: 10.1088/2053-2571/aada8d

Publisher: Morgan & Claypool Publishers