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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


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


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


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


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


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


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


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