Correlation Polytopes and the Geometry of Limit Laws in Probability
Let be n events in a probability space, and suppose that we have only partial information about the distribution: The probabilites of the events themselves, and their pair intersections. With this partial information we cannot, usually, deternine the probability of an event B in the algebra generated by the 's, but we can obtain lower and upper bounds. This is done by a linear program related to the correlation polytope c(n), a structure introduced in , . In the first part of the paper I demonstrate how laws of large numbers (for sequences of events which are not necessarily independent) can be proved, using only the duality theorem of linear programming. These include the weak law of large numbers (necessary and sufficient condition) and various sufficient conditions for strong laws. The connection between these laws and the facet structure of the correlation polytope is established. In the second part of the paper I consider a more general case. Assume that our information consists of the values of the probabilities of all intersections of the 's up to size k, k < n. The techniques of linear programming lead naturally to an application of the theory of polynomial approximation in estimating the size of various events. In particular, I prove an approximate version of the central limit theorem.
|Keywords||No keywords specified (fix it)|
|Categories||categorize this paper)|
References found in this work BETA
No references found.
Citations of this work BETA
No citations found.
Similar books and articles
The Limits of Probability Modelling: A Serendipitous Tale of Goldfish, Transfinite Numbers, and Pieces of String. [REVIEW]Ranald R. Macdonald - 2000 - Mind and Society 1 (2):17-38.
Justification by Infinite Loops.David Atkinson & Jeanne Peijnenburg - 2010 - Notre Dame Journal of Formal Logic 51 (4):407-416.
The Covering Law Model of Historical Explanation.Stanley Paluch - 1968 - Inquiry : An Interdisciplinary Journal of Philosophy 11 (1-4):368 – 387.
Honderich on Mental Events and Psychoneural Laws.Jaegwon Kim - 1989 - Inquiry : An Interdisciplinary Journal of Philosophy 32 (March):29-48.
Added to index2010-12-22
Total downloads26 ( #196,918 of 2,171,976 )
Recent downloads (6 months)1 ( #326,556 of 2,171,976 )
How can I increase my downloads?