Existence of regular conditional probability measures

1966 ◽  
pp. 34-41
Author(s):  
J. Pfanzagl ◽  
W. Pierlo
Keyword(s):  
Conditional Probability ◽  
Probability Measures

Multivariate monotone likelihood ratio and uniform conditional stochastic order

1982 ◽  
Vol 19 (3) ◽  
pp. 695-701 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Conditional Probability ◽  
Likelihood Ratio ◽  
Probability Distributions ◽  
Stochastic Order ◽  
Total Positivity ◽  
Probability Measures ◽  
Monotone Likelihood Ratio ◽  
Fkg Inequality ◽  
The Fkg Inequality

Karlin and Rinott (1980) introduced and investigated concepts of multivariate total positivity (TP2) and multivariate monotone likelihood ratio (MLR) for probability measures on Rn These TP and MLR concepts are intimately related to supermodularity as discussed in Topkis (1968), (1978) and the FKG inequality of Fortuin, Kasteleyn and Ginibre (1971). This note points out connections between these concepts and uniform conditional stochastic order (ucso) as defined in Whitt (1980). ucso holds for two probability distributions if there is ordinary stochastic order for the corresponding conditional probability distributions obtained by conditioning on subsets from a specified class. The appropriate subsets to condition on for ucso appear to be the sublattices of Rn. Then MLR implies ucso, with the two orderings being equivalent when at least one of the probability measures is TP2.

A Comparison of Methods for Constructing Probability Measures on Infinite Product Spaces

1987 ◽  
Vol 30 (3) ◽  
pp. 282-285 ◽  
Author(s):  
Charles W. Lamb
Keyword(s):  
Probability Measure ◽  
Conditional Probability ◽  
Product Space ◽  
Infinite Product ◽  
Probability Measures ◽  
Classical Theorem ◽  
Comparison Of Methods ◽  
Product Spaces ◽  
Probability Functions ◽  
Finite Dimensional

AbstractThe construction, from a consistent family of finite dimensional probability measures, of a probability measure on a product space when the marginal measures are perfect is shown to follow from a classical theorem due to Ionescu Tulcea and known results on the existence of regular conditional probability functions.

Hybrid Probabilistic Models of Fuzzy and Rough Events

2003 ◽  
Vol 7 (3) ◽  
pp. 322-329
Author(s):  
Rolly Intan ◽  
◽  
Masao Mukaidono ◽  
Hung T. Nguyen ◽  
◽  
...  
Keyword(s):  
Conditional Probability ◽  
Rough Set ◽  
Fuzzy Set ◽  
Probabilistic Models ◽  
Evidence Theory ◽  
Probability Measures ◽  
The Relationship

This paper discusses the relationship between probability and fuzziness based on the process of perception. As a generalization of the crisp set, the fuzzy set is used to model fuzzy events as proposed by Zadeh. Similarly, we may consider the rough set to represent a rough event in terms of probability measures. Special attention will be given to conditional probability of fuzzy events as well as the conditional probability of rough events. Several combinations of formulation and properties are examined. In the relation to evidence theory, the probability of a rough event may be considered as a connecting bridge between belief-plausibility measures and the probability measures. Moreover, a generalized fuzzy-rough event is introduced to generalize both fuzzy and rough events.

Multivariate monotone likelihood ratio and uniform conditional stochastic order

1982 ◽  
Vol 19 (03) ◽  
pp. 695-701 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Conditional Probability ◽  
Likelihood Ratio ◽  
Probability Distributions ◽  
Stochastic Order ◽  
Total Positivity ◽  
Probability Measures ◽  
Monotone Likelihood Ratio ◽  
Fkg Inequality ◽  
The Fkg Inequality

Karlin and Rinott (1980) introduced and investigated concepts of multivariate total positivity (TP2) and multivariate monotone likelihood ratio (MLR) for probability measures on Rn These TP and MLR concepts are intimately related to supermodularity as discussed in Topkis (1968), (1978) and the FKG inequality of Fortuin, Kasteleyn and Ginibre (1971). This note points out connections between these concepts and uniform conditional stochastic order (ucso) as defined in Whitt (1980). ucso holds for two probability distributions if there is ordinary stochastic order for the corresponding conditional probability distributions obtained by conditioning on subsets from a specified class. The appropriate subsets to condition on for ucso appear to be the sublattices of Rn . Then MLR implies ucso, with the two orderings being equivalent when at least one of the probability measures is TP2.

Probability Spaces

2021 ◽  
pp. 147-153
Author(s):  
James Davidson
Keyword(s):  
Conditional Probability ◽  
General Context ◽  
Probability Measures ◽  
Product Spaces ◽  
Probability Spaces ◽  
Product Measures

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

scholarly journals Real Functions in Stochastic Dependence

2019 ◽  
Vol 74 (1) ◽  
pp. 17-34
Author(s):  
Dušana Babicová ◽  
Roman Frič
Keyword(s):  
Probability Theory ◽  
Conditional Probability ◽  
Probability Measures ◽  
Multivalued Logic ◽  
Stochastic Dependence ◽  
Categorical Methods ◽  
Measurable Functions ◽  
Random Events ◽  
Real Functions ◽  
Stochastic Channels

Abstract In a fuzzified probability theory, random events are modeled by measurable functions into [0,1] and probability measures are replaced with probability integrals. The transition from Boolean two-valued logic to Lukasiewicz multivalued logic results in an upgraded probability theory in which we define and study asymmetrical stochastic dependence/independence and conditional probability based on stochastic channels and joint experiments so that the classical constructions follow as particular cases. Elementary categorical methods enable us to put the two theories into a perspective.

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