Information Loss Due to the Data Reduction of Sample Data from Discrete Distributions
In this paper, we study the information lost when a real-valued statistic is used to reduce or summarize sample data from a discrete random variable with a one-dimensional parameter. We compare the probability that a random sample gives a particular data set to the probability of the statistic’s val...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-09-01
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| Series: | Data |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2306-5729/5/3/84 |
| Summary: | In this paper, we study the information lost when a real-valued statistic is used to reduce or summarize sample data from a discrete random variable with a one-dimensional parameter. We compare the probability that a random sample gives a particular data set to the probability of the statistic’s value for this data set. We focus on sufficient statistics for the parameter of interest and develop a general formula independent of the parameter for the Shannon information lost when a data sample is reduced to such a summary statistic. We also develop a measure of entropy for this lost information that depends only on the real-valued statistic but neither the parameter nor the data. Our approach would also work for non-sufficient statistics, but the lost information and associated entropy would involve the parameter. The method is applied to three well-known discrete distributions to illustrate its implementation. |
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| ISSN: | 2306-5729 |