Conditional problem for objective probability

Kříž, Otakar. "Conditional problem for objective probability." Kybernetika 34.1 (1998): [27]-40. <http://eudml.org/doc/33332>.

@article{Kříž1998,
abstract = {Marginal problem (see [Kel]) consists in finding a joint distribution whose marginals are equal to the given less-dimensional distributions. Let’s generalize the problem so that there are given not only less-dimensional distributions but also conditional probabilities. It is necessary to distinguish between objective (Kolmogorov) probability and subjective (de Finetti) approach ([Col,Sco]). In the latter, the coherence problem incorporates both probabilities and conditional probabilities in a unified framework. Different algorithms available for its solution are described e. g. in ([Gil,Col,Vic]). In the context of the former approach, it will be shown that it is possible to split the task into solving the marginal problem independently and to subsequent solving pure “conditional" problem as certain type of optimization. First, an algorithm (Conditional problem) that generates a distribution whose conditional probabilities are equal to the given ones is presented. Due to the multimodality of the criterion function, the algorithm is only heuristical. Due to the computational complexity, it is efficient for small size problems e. g. 5 dichotomical variables. Second, a method is mentioned how to unite marginal and conditional problem to a more general consistency problem for objective probability. Due to computational complexity, both algorithms are effective only for limited number of variables and conditionals. The described approach makes possible to integrate in the solution of the consistency problem additional knowledge contained e. g. in an empirical distribution.},
author = {Kříž, Otakar},
journal = {Kybernetika},
keywords = {marginal problem; algorithm; marginal problem; algorithm},
language = {eng},
number = {1},
pages = {[27]-40},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Conditional problem for objective probability},
url = {http://eudml.org/doc/33332},
volume = {34},
year = {1998},
}

TY - JOUR
AU - Kříž, Otakar
TI - Conditional problem for objective probability
JO - Kybernetika
PY - 1998
PB - Institute of Information Theory and Automation AS CR
VL - 34
IS - 1
SP - [27]
EP - 40
AB - Marginal problem (see [Kel]) consists in finding a joint distribution whose marginals are equal to the given less-dimensional distributions. Let’s generalize the problem so that there are given not only less-dimensional distributions but also conditional probabilities. It is necessary to distinguish between objective (Kolmogorov) probability and subjective (de Finetti) approach ([Col,Sco]). In the latter, the coherence problem incorporates both probabilities and conditional probabilities in a unified framework. Different algorithms available for its solution are described e. g. in ([Gil,Col,Vic]). In the context of the former approach, it will be shown that it is possible to split the task into solving the marginal problem independently and to subsequent solving pure “conditional" problem as certain type of optimization. First, an algorithm (Conditional problem) that generates a distribution whose conditional probabilities are equal to the given ones is presented. Due to the multimodality of the criterion function, the algorithm is only heuristical. Due to the computational complexity, it is efficient for small size problems e. g. 5 dichotomical variables. Second, a method is mentioned how to unite marginal and conditional problem to a more general consistency problem for objective probability. Due to computational complexity, both algorithms are effective only for limited number of variables and conditionals. The described approach makes possible to integrate in the solution of the consistency problem additional knowledge contained e. g. in an empirical distribution.
LA - eng
KW - marginal problem; algorithm; marginal problem; algorithm
UR - http://eudml.org/doc/33332
ER -