<abstract xmlns="http://www.w3.org/1999/xhtml">Estimation of probabilities from empirical data samples has drawn close attention in the scientific community and has been identified as a crucial phase in many machine learning and knowledge discovery research projects and applications. In addition to trivial and straightforward estimation with relative frequency, more elaborated probability estimation methods from small samples were proposed and applied in practice (e.g., Laplace’s rule, the <italic>m</italic>-estimate). Piegat and Landowski (2012) proposed a novel probability estimation method from small samples <italic>Ep</italic>h√2 that is optimal according to the mean absolute error of the estimation result. In this paper we show that, even though the articulation of Piegat’s formula seems different, it is in fact a special case of the <italic>m</italic>-estimate, where <italic>p</italic>a =1<italic>/</italic>2 and <italic>m</italic> = <italic>√2</italic>. In the context of an experimental framework, we present an in-depth analysis of several probability estimation methods with respect to their mean absolute errors and demonstrate their potential advantages and disadvantages. We extend the analysis from single instance samples to samples with a moderate number of instances. We define small samples for the purpose of estimating probabilities as samples containing either less than four successes or less than four failures and justify the definition by analysing probability estimation errors on various sample sizes.</abstract>

Estimation of probabilities from empirical data samples has drawn close attention in the scientific community and has been identified as a crucial phase in many machine learning and knowledge discovery research projects and applications. In addition to trivial and straightforward estimation with relative frequency, more elaborated probability estimation methods from small samples were proposed and applied in practice (e.g., Laplace’s rule, the m-estimate). Piegat and Landowski (2012) proposed a novel probability estimation method from small samples Eph√2 that is optimal according to the mean absolute error of the estimation result. In this paper we show that, even though the articulation of Piegat’s formula seems different, it is in fact a special case of the m-estimate, where pa =1/2 and m = √2. In the context of an experimental framework, we present an in-depth analysis of several probability estimation methods with respect to their mean absolute errors and demonstrate their potential advantages and disadvantages. We extend the analysis from single instance samples to samples with a moderate number of instances. We define small samples for the purpose of estimating probabilities as samples containing either less than four successes or less than four failures and justify the definition by analysing probability estimation errors on various sample sizes.

Revisiting the Optimal Probability Estimator from Small Samples for Data Mining

Department of Knowledge Technologies Jožef Stefan Institute

International Journal of Applied Mathematics and Computer Science

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"Revisiting the Optimal Probability Estimator from Small Samples for Data Mining"}

Estimation of probabilities from empirical data samples has drawn close attention in the scientific community and has been identified as a crucial phase in...

Revisiting the Optimal Probability Estimator from Small Samples for Data Mining

Published Online: Dec 31, 2019

Page range: 783 - 796

Received: Dec 15, 2018

Accepted: Apr 23, 2019

DOI: https://doi.org/10.2478/amcs-2019-0058

Keywords
probability estimation, small samples, minimal error, -estimate

© 2019 Bojan Cestnik, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Revisiting the Optimal Probability Estimator from Small Samples for Data Mining

Published Online: Dec 31, 2019

Page range: 783 - 796

Received: Dec 15, 2018

Accepted: Apr 23, 2019

DOI: https://doi.org/10.2478/amcs-2019-0058

Keywordsprobability estimation, small samples, minimal error, -estimate

© 2019 Bojan Cestnik, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
probability estimation, small samples, minimal error, -estimate