Contents
Acronyms and Abbreviations xi
Notation xiii
Foreword xv
1 Introduction 1
1.1 A Little History 1
1.2 Sensors, Measurements, and Problem Definition 3
1.3 How This Book Is Organized 4
1.4 Relationship to Other Books 5
Part I Estimation Machinery 7
2 Primer on Probability Theory 9
2.1 Probability Density Functions 9
2.1.1 Definitions 9
2.1.2 Bayes’ Rule and Inference 10
2.1.3 Moments 11
2.1.4 Sample Mean and Covariance 12
2.1.5 Statistically Independent, Uncorrelated 12
2.1.6 Normalized Product 13
2.1.7 Shannon and Mutual Information 14
2.1.8 Cram´er-Rao Lower Bound and Fisher Information 14
2.2 Gaussian Probability Density Functions 15
2.2.1 Definitions 15
2.2.2 Isserlis’ Theorem 16
2.2.3 Joint Gaussian PDFs, Their Factors, and Inference 18
2.2.4 Statistically Independent, Uncorrelated 20
2.2.5 Linear Change of Variables 20
2.2.6 Normalized Product of Gaussians 22
2.2.7 Sherman-Morrison-Woodbury Identity 23
2.2.8 Passing a Gaussian through a Nonlinearity 24
2.2.9 Shannon Information of a Gaussian 28
2.2.10 Mutual Information of a Joint Gaussian PDF 30
2.2.11 Cram´er-Rao Lower Bound Applied to Gaussian PDFs 30
2.3 Gaussian Processes 32
2.4 Summary 33
2.5 Exercises 33
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