Bonferroni inequalities and deviations of discrete distributions

1996 ◽  
Vol 33 (1) ◽  
pp. 115-121 ◽  
Author(s):  
Tamás F. Móri
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Probability Distributions ◽  
Type Inequality ◽  
Discrete Distributions ◽  
Maximum Distance ◽  
Discrete Probability ◽  
Bonferroni Inequalities ◽  
Discrete Probability Distributions

In the paper we first show how to convert a generalized Bonferroni-type inequality into an estimation for the generating function of the number of occurring events, then we give estimates for the deviation of two discrete probability distributions in terms of the maximum distance between their generating functions over the interval [0, 1].

Bonferroni inequalities and deviations of discrete distributions

1996 ◽  
Vol 33 (01) ◽  
pp. 115-121 ◽  
Author(s):  
Tamás F. Móri
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Probability Distributions ◽  
Type Inequality ◽  
Discrete Distributions ◽  
Maximum Distance ◽  
Discrete Probability ◽  
Bonferroni Inequalities ◽  
Discrete Probability Distributions

In the paper we first show how to convert a generalized Bonferroni-type inequality into an estimation for the generating function of the number of occurring events, then we give estimates for the deviation of two discrete probability distributions in terms of the maximum distance between their generating functions over the interval [0, 1].

scholarly journals Accuracy of Approximation for Discrete Distributions

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Tamás F. Móri
Keyword(s):  
Total Variation ◽  
Generating Functions ◽  
Probability Distributions ◽  
Discrete Distributions ◽  
Discrete Probability ◽  
Order Of Magnitude ◽  
Variation Distance ◽  
The Difference ◽  
Discrete Probability Distributions ◽  
Accuracy Of Approximation

The paper is a contribution to the problem of estimating the deviation of two discrete probability distributions in terms of the supremum distance between their generating functions over the interval [0,1]. Deviation can be measured by the difference of the kth terms or by total variation distance. Our new bounds have better order of magnitude than those proved previously, and they are even sharp in certain cases.

scholarly journals On the Quantum versus Classical Learnability of Discrete Distributions

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 417
Author(s):  
Ryan Sweke ◽  
Jean-Pierre Seifert ◽  
Dominik Hangleiter ◽  
Jens Eisert
Keyword(s):  
Probability Distributions ◽  
Explicit Construction ◽  
Discrete Distributions ◽  
Discrete Probability ◽  
Diffie Hellman ◽  
Original Distribution ◽  
Modelling Problem ◽  
Probably Approximately Correct ◽  
Generative Modelling ◽  
Discrete Probability Distributions

Here we study the comparative power of classical and quantum learners for generative modelling within the Probably Approximately Correct (PAC) framework. More specifically we consider the following task: Given samples from some unknown discrete probability distribution, output with high probability an efficient algorithm for generating new samples from a good approximation of the original distribution. Our primary result is the explicit construction of a class of discrete probability distributions which, under the decisional Diffie-Hellman assumption, is provably not efficiently PAC learnable by a classical generative modelling algorithm, but for which we construct an efficient quantum learner. This class of distributions therefore provides a concrete example of a generative modelling problem for which quantum learners exhibit a provable advantage over classical learning algorithms. In addition, we discuss techniques for proving classical generative modelling hardness results, as well as the relationship between the PAC learnability of Boolean functions and the PAC learnability of discrete probability distributions.

scholarly journals Classroom note: An inductive derivation of Stirling numbers of the second kind and their applications in statistics

1997 ◽  
Vol 1 (2) ◽  
pp. 151-157 ◽  
Author(s):  
Anwar H. Joarder ◽  
Munir Mahmood
Keyword(s):  
Probability Distributions ◽  
Stirling Numbers ◽  
Higher Order ◽  
Inductive Method ◽  
Higher Order Moments ◽  
Discrete Probability ◽  
Discrete Probability Distributions

An inductive method has been presented for finding Stirling numbers of the second kind. Applications to some discrete probability distributions for finding higher order moments have been discussed.

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