The Statistical Eyeglasses

The math behind scientific knowledge

By (author) Edoardo Milotti

Publication date:

01 November 2018

Publisher

Morgan & Claypool Publishers

ISBN-13: 9781643271491

BOOK DESCRIPTION
TABLE OF CONTENTS

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.

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