Algorithm of Confidence Limits Calculation for the Probability of the Value 1 of a Monotone Boolean Function of Random Variables

1997 ◽  
pp. 123-133
Author(s):  
Alexander M. Andronov
Keyword(s):  
Boolean Function ◽  
Random Variables ◽  
Confidence Limits ◽  
Monotone Boolean Function

Permutation probabilities for gamma random variables

1983 ◽  
Vol 20 (4) ◽  
pp. 822-834 ◽  
Author(s):  
Robert J. Henery
Keyword(s):  
Order Statistics ◽  
Negative Binomial ◽  
Random Variables ◽  
Upper Bounds ◽  
Alternative Formulation ◽  
Confidence Limits ◽  
Lower And Upper Bounds ◽  
Sample Variances

The order statistics of a set of independent gamma variables, in general not identically distributed, may serve as a basis for ordering players in a hypothetical game. An alternative formulation in terms of negative binomial variables leads to an expression for the probability that the random gammas are in a given order. This expression may contain rather many terms and some approximations are discussed — firstly as the gamma parameters αi tend to equality with all ni the same, and secondly when the probability of an inversion is small. In another interpretation the probabilities discussed arise in the statement of confidence limits for the ratios of population variances, and here the inversion probability is small enough usually that lower and upper bounds may be given for the probability that the sample variances occur in their expected order. These bounds are calculated from the probability that two variables are in expected order, and for gamma variables this probability is obtained from the F-distribution.

scholarly journals Monotone Boolean Functions with s Zeros Farthest from Threshold Functions

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Kazuyuki Amano ◽  
Jun Tarui
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Learning Algorithm ◽  
Monotone Function ◽  
Threshold Function ◽  
Binary Representation ◽  
Threshold Functions ◽  
Monotone Boolean Function ◽  
Monotone Boolean Functions ◽  
International Audience

International audience Let $T_t$ denote the $t$-threshold function on the $n$-cube: $T_t(x) = 1$ if $|\{i : x_i=1\}| \geq t$, and $0$ otherwise. Define the distance between Boolean functions $g$ and $h$, $d(g,h)$, to be the number of points on which $g$ and $h$ disagree. We consider the following extremal problem: Over a monotone Boolean function $g$ on the $n$-cube with $s$ zeros, what is the maximum of $d(g,T_t)$? We show that the following monotone function $p_s$ maximizes the distance: For $x \in \{0,1\}^n$, $p_s(x)=0$ if and only if $N(x) < s$, where $N(x)$ is the integer whose $n$-bit binary representation is $x$. Our result generalizes the previous work for the case $t=\lceil n/2 \rceil$ and $s=2^{n-1}$ by Blum, Burch, and Langford [BBL98-FOCS98], who considered the problem to analyze the behavior of a learning algorithm for monotone Boolean functions, and the previous work for the same $t$ and $s$ by Amano and Maruoka [AM02-ALT02].

Permutation probabilities for gamma random variables

1983 ◽  
Vol 20 (04) ◽  
pp. 822-834 ◽  
Author(s):  
Robert J. Henery
Keyword(s):  
Order Statistics ◽  
Negative Binomial ◽  
Random Variables ◽  
Upper Bounds ◽  
Alternative Formulation ◽  
Confidence Limits ◽  
Lower And Upper Bounds ◽  
Sample Variances

The order statistics of a set of independent gamma variables, in general not identically distributed, may serve as a basis for ordering players in a hypothetical game. An alternative formulation in terms of negative binomial variables leads to an expression for the probability that the random gammas are in a given order. This expression may contain rather many terms and some approximations are discussed — firstly as the gamma parameters αi tend to equality with all ni the same, and secondly when the probability of an inversion is small. In another interpretation the probabilities discussed arise in the statement of confidence limits for the ratios of population variances, and here the inversion probability is small enough usually that lower and upper bounds may be given for the probability that the sample variances occur in their expected order. These bounds are calculated from the probability that two variables are in expected order, and for gamma variables this probability is obtained from the F-distribution.

Sign in / Sign up

Export Citation Format

Share Document