## Principles of Probability

*N. Thompson Hobbs and Mevin B. Hooten*

### in Bayesian Models: A Statistical Primer for Ecologists

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691159287
- eISBN:
- 9781400866557
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159287.003.0003
- Subject:
- Biology, Ecology

This chapter describes the rules of probability as well as probability distributions. Because models are inherently, deliberately approximate, there comes a need to understand the approximation ... More

## Introduction to Probability and Random Variables

*M. Vidyasagar*

### in Hidden Markov Processes: Theory and Applications to Biology

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691133157
- eISBN:
- 9781400850518
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691133157.003.0001
- Subject:
- Mathematics, Probability / Statistics

This chapter provides an introduction to probability and random variables. Probability theory is an attempt to formalize the notion of uncertainty in the outcome of an experiment. For instance, ... More

## Causal Models

*Steven Sloman*

### in Causal Models: How People Think about the World and Its Alternatives

- Published in print:
- 2005
- Published Online:
- January 2007
- ISBN:
- 9780195183115
- eISBN:
- 9780199870950
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195183115.003.0004
- Subject:
- Philosophy, Philosophy of Mind

According to human perception, the world is full of causal systems composed of autonomous mechanisms that generate events as effects of other events. This chapter attempts to make this idea more ... More

## Introduction to Information Theory

*M. Vidyasagar*

### in Hidden Markov Processes: Theory and Applications to Biology

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691133157
- eISBN:
- 9781400850518
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691133157.003.0002
- Subject:
- Mathematics, Probability / Statistics

This chapter provides an introduction to some elementary aspects of information theory, including entropy in its various forms. Entropy refers to the level of uncertainty associated with a random ... More

## Data Depth: Multivariate Statistics and Geometry

*Cascos Ignacio*

### in New Perspectives in Stochastic Geometry

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199232574
- eISBN:
- 9780191716393
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199232574.003.0012
- Subject:
- Mathematics, Geometry / Topology

This chapter presents several ways to measure the degree of centrality of a point with respect to a multivariate probability distribution or a data cloud. Such degree of centrality is called depth, ... More

## Markov Processes

*M. Vidyasagar*

### in Hidden Markov Processes: Theory and Applications to Biology

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691133157
- eISBN:
- 9781400850518
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691133157.003.0004
- Subject:
- Mathematics, Probability / Statistics

This chapter deals with Markov processes. It first defines the “Markov property” and shows that all the relevant information about a Markov process assuming values in a finite set of cardinality n ... More

## What Is a Quantum State?

*Gary E. Bowman*

### in Essential Quantum Mechanics

- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780199228928
- eISBN:
- 9780191711206
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199228928.003.0003
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter first discusses statistical techniques and concepts — including probability, probability distribution, average and expectation value, and uncertainty — that are relevant to quantum ... More

## Likelihood

*N. Thompson Hobbs and Mevin B. Hooten*

### in Bayesian Models: A Statistical Primer for Ecologists

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691159287
- eISBN:
- 9781400866557
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159287.003.0004
- Subject:
- Biology, Ecology

This chapter is an overview of likelihood and maximum likelihood. Likelihood forms the fundamental link between models and data in the Bayesian framework. In addition, maximum likelihood is a widely ... More

## What is a random process

*Melvin Lax, Wei Cai, and Min Xu*

### in Random Processes in Physics and Finance

- Published in print:
- 2006
- Published Online:
- January 2010
- ISBN:
- 9780198567769
- eISBN:
- 9780191718359
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567769.003.0002
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

A random or stochastic process is a random variable that evolves in time by some random mechanism (of course, the time variable can be replaced by a space variable, or some other variable, in ... More

## Introduction to Large Deviation Theory

*M. Vidyasagar*

### in Hidden Markov Processes: Theory and Applications to Biology

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691133157
- eISBN:
- 9781400850518
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691133157.003.0005
- Subject:
- Mathematics, Probability / Statistics

This chapter provides an introduction to large deviation theory. It begins with an overview of the motivatio n for the problem under study, focusing on probability distributions and how to construct ... More

## Quantum Statistics of Fields

*STEPHEN M. BARNETT and PAUL M. RADMORE*

### in Methods in Theoretical Quantum Optics

- Published in print:
- 2002
- Published Online:
- February 2010
- ISBN:
- 9780198563617
- eISBN:
- 9780191714245
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198563617.003.0004
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter considers methods for describing the quantum statistics of a single mode of the electromagnetic field. Some of these methods may be extended to multimode fields, and this is discussed in ... More

## Internal Consistency of Survey Respondents' Forecasts: Evidence Based on the Survey of Professional Forecasters *

*Michael P. Clements*

### in The Methodology and Practice of Econometrics: A Festschrift in Honour of David F. Hendry

- Published in print:
- 2009
- Published Online:
- September 2009
- ISBN:
- 9780199237197
- eISBN:
- 9780191717314
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199237197.003.0008
- Subject:
- Economics and Finance, Econometrics

This chapter asks whether the different types of forecasts made by individual survey respondents are mutually consistent, using the SPF survey data. It compares the point forecasts and central ... More

## Probability

*Stephen J. Blundell and Katherine M. Blundell*

### in Concepts in Thermal Physics

- Published in print:
- 2009
- Published Online:
- January 2010
- ISBN:
- 9780199562091
- eISBN:
- 9780191718236
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199562091.003.0003
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter defines some basic concepts in probability theory. It begins by stating that the probability of occurrence of a particular event, taken from a finite set of possible events, is zero if ... More

## Bayesians versus Frequentists

*Peter Coles*

### in From Cosmos to Chaos: The Science of Unpredictability

- Published in print:
- 2006
- Published Online:
- January 2010
- ISBN:
- 9780198567622
- eISBN:
- 9780191718250
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567622.003.0004
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

There are two competing interpretations of probability. One is the view that probabilities should be interpreted as frequencies in some large ensemble of repeated experiments under identical ... More

## Accuracy and stability

*J. C. Gower and G. B. Dijksterhuis*

### in Procrustes Problems

- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198510581
- eISBN:
- 9780191708961
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198510581.003.0012
- Subject:
- Mathematics, Probability / Statistics

This chapter discusses what little is known about probability models for Procrustes problems. For most practical purposes, it relies on data sampling procedures like permutation tests, jack-knifing, ... More

## Probability Distributions on Hilbert Space

*Heinz-Peter Breuer and Francesco Petruccione*

### in The Theory of Open Quantum Systems

- Published in print:
- 2007
- Published Online:
- February 2010
- ISBN:
- 9780199213900
- eISBN:
- 9780191706349
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213900.003.05
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter characterizes the statistical properties of a quantum mechanical ensemble in terms of a density matrix. However, if selective measurements of one or several observables are carried out ... More

## Commutators and Uncertainty Relations

*Gary E. Bowman*

### in Essential Quantum Mechanics

- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780199228928
- eISBN:
- 9780191711206
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199228928.003.0007
- Subject:
- Physics, Condensed Matter Physics / Materials

Commutators and uncertainty relations have sparked much debate and confusion as to their physical meaning. As such, this chapter focuses partly on historical and interpretational aspects. Because it ... More

## Statistics in Deterministic Sciences: An Introduction

*Gidon Eshel*

### in Spatiotemporal Data Analysis

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691128917
- eISBN:
- 9781400840632
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691128917.003.0007
- Subject:
- Environmental Science, Environmental Studies

This chapter focuses on the relevance of statistics in deterministic science. While most physical phenomena of concern to natural sciences are governed by fundamental, mostly known, physics, their ... More

## Benfordʼs Law Geometry

*Lawrence Leemis*

### in Benford's Law: Theory and Applications

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691147611
- eISBN:
- 9781400866595
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691147611.003.0004
- Subject:
- Mathematics, Probability / Statistics

This chapter switches from the traditional analysis of Benford's law using data sets to a search for probability distributions that obey Benford's law. It begins by briefly discussing the origins of ... More

## Introduction to Probability

*Therese M. Donovan and Ruth M. Mickey*

### in Bayesian Statistics for Beginners: a step-by-step approach

- Published in print:
- 2019
- Published Online:
- July 2019
- ISBN:
- 9780198841296
- eISBN:
- 9780191876820
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198841296.003.0001
- Subject:
- Biology, Biomathematics / Statistics and Data Analysis / Complexity Studies

In this chapter, the concept of probability is introduced. The rolling of a die is an example of a random process: the face that comes up is subject to chance. In probability, the goal is to quantify ... More