Bounding Discrete Distribution Averages via Category Support
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Solution Overview
Problem
Existing methods for determining bounds of population or distribution averages do not effectively utilize additional information about the underlying distribution, leading to suboptimal estimation of statistical bounds, particularly in big-data settings where data samples are from an unknown multinomial distribution.
Innovation Solution
The approach involves using a set of distributions that are likely to contain the generating distribution, identifying distributions with minimum or maximum means as lower and upper bounds, and employing binomial inversion and Bonferroni correction to compute probably approximately correct (PAC) bounds on the expectation of category values, leveraging the known category values to improve bound estimation.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If distribution-free concentration inequalities are used to bound statistics, then the method is universally applicable without requiring distributional assumptions, but the bounds are loose and not tight because no extra information about the underlying distribution is utilized
Solution Approach 1:
The patent changes the parameter of distributional assumptions from 'none' (distribution-free) to 'discrete-valued with known support' (multinomial distribution with known category values). This allows the use of tailored concentration inequalities that exploit the discrete nature and known support of the distribution, resulting in tighter bounds while maintaining reasonable complexity.
Solution Approach 2:
The patent segments the problem by considering each category value separately and deriving bounds for each category's probability. By using the known discrete category values to create separate concentration inequalities for each category, the method achieves tighter overall bounds on the mean compared to treating the distribution as completely unknown.
2Measurement precision
If traditional concentration inequalities are applied to empirical means, then the approach is simple and computationally efficient, but it fails to improve estimation by leveraging known characteristics of the data distribution
Solution Approach 1:
The patent performs preliminary action by incorporating known category values and discrete distribution characteristics into the bound estimation process before computing the final bounds. By pre-specifying the support of the distribution and using these known characteristics in the concentration inequalities, the method improves estimation accuracy without significantly complicating the overall approach.
3Measurement precision
If bounds are computed without utilizing discrete distribution characteristics, then the method is broadly applicable to continuous and discrete distributions, but the bounds are suboptimal for discrete-valued data where category information is available
Solution Approach 1:
The patent applies local quality by tailoring the concentration inequalities specifically to discrete-valued distributions with known category values. Instead of using a one-size-fits-all approach, the method customizes the bounds to exploit the specific structure of discrete distributions, achieving tighter bounds for this particular case while maintaining the ability to adapt to different discrete scenarios by changing the support values.
Data Source
AI summary
The present teaching relates to method, system, medium, and implementations for characterizing data with categorical classes and the number of observations for each of the categorical classes. Each categorical class is associated with a category value. The categorical classes are arranged in a first order based on category values. A total of observations is determined based on the numbers of observations for each categorical class. A bound of the average value of the data is estimated based on the categorical classes, the total of observations, and the numbers of observations for the categorical classes in accordance with a dot product of a probability vector and a categorical class vector comprising the category values of the categorical classes.