*James Davidson*

- Published in print:
- 1994
- Published Online:
- November 2003
- ISBN:
- 9780198774037
- eISBN:
- 9780191596117
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198774036.003.0007
- Subject:
- Economics and Finance, Econometrics

This chapter defines probability measures and probability spaces in a general context, as a case of the concepts introduced in Ch. 3. The axioms of probability are explained, and the important ...
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This chapter defines probability measures and probability spaces in a general context, as a case of the concepts introduced in Ch. 3. The axioms of probability are explained, and the important concepts of conditional probability and independence introduced, and linked to the role of product spaces and product measures.Less

This chapter defines probability measures and probability spaces in a general context, as a case of the concepts introduced in Ch. 3. The axioms of probability are explained, and the important concepts of conditional probability and independence introduced, and linked to the role of product spaces and product measures.

*Heinz-Peter Breuer and Francesco Petruccione*

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

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213900.003.01
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter contains a survey of classical probability theory and stochastic processes. It starts with a description of the fundamental concepts of probability space and Kolmogorov axioms. These ...
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This chapter contains a survey of classical probability theory and stochastic processes. It starts with a description of the fundamental concepts of probability space and Kolmogorov axioms. These concepts are then used to define random variables and stochastic processes. The mathematical formulation of the special class of Markov processes through classical master equations is given, including deterministic processes (Liouville equation), jump processes (Pauli master equation), and diffusion processes (Fokker–Planck equation). Special stochastic processes which play an important role in the developments of the following chapters, such as piecewise deterministic processes and Lévy processes, are described in detail together with their basic physical properties and various mathematical formulations in terms of master equations, path integral representation, and stochastic differential equations.Less

This chapter contains a survey of classical probability theory and stochastic processes. It starts with a description of the fundamental concepts of probability space and Kolmogorov axioms. These concepts are then used to define random variables and stochastic processes. The mathematical formulation of the special class of Markov processes through classical master equations is given, including deterministic processes (Liouville equation), jump processes (Pauli master equation), and diffusion processes (Fokker–Planck equation). Special stochastic processes which play an important role in the developments of the following chapters, such as piecewise deterministic processes and Lévy processes, are described in detail together with their basic physical properties and various mathematical formulations in terms of master equations, path integral representation, and stochastic differential equations.

*Martin Smith*

- Published in print:
- 2016
- Published Online:
- March 2016
- ISBN:
- 9780198755333
- eISBN:
- 9780191816635
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198755333.003.0010
- Subject:
- Philosophy, Metaphysics/Epistemology, Philosophy of Science

A number of epistemologists have attempted to refine the risk minimisation conception of justification, proposing more complex probabilistic rules for justification that enable one to avoid the ...
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A number of epistemologists have attempted to refine the risk minimisation conception of justification, proposing more complex probabilistic rules for justification that enable one to avoid the lottery paradox without giving up multiple premise closure. These attempts to refine the risk minimisation conception often appear ad hoc, and are also beset by formal difficulties. An important result, proved by Douven and Williamson (in 2006), comes close to showing that the ambition behind these refinements cannot be realised. In this chapter it is shown that a significant limitation of Douven and Williamson’s proof—namely, its assumption of a finite, uniform probability space—can be overcome, and the result is available for infinite probability spaces as well. While a normic theory of justification is able to accommodate multiple premise closure within a fallibilist framework, it is concluded that no purely probabilistic theory of justification can do so.Less

A number of epistemologists have attempted to refine the risk minimisation conception of justification, proposing more complex probabilistic rules for justification that enable one to avoid the lottery paradox without giving up multiple premise closure. These attempts to refine the risk minimisation conception often appear ad hoc, and are also beset by formal difficulties. An important result, proved by Douven and Williamson (in 2006), comes close to showing that the ambition behind these refinements cannot be realised. In this chapter it is shown that a significant limitation of Douven and Williamson’s proof—namely, its assumption of a finite, uniform probability space—can be overcome, and the result is available for infinite probability spaces as well. While a normic theory of justification is able to accommodate multiple premise closure within a fallibilist framework, it is concluded that no purely probabilistic theory of justification can do so.