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Bayesian maximum entropy approach and its applications: a review

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Abstract

The present paper reviews the conceptual framework and development of the Bayesian Maximum Entropy (BME) approach. BME has been considered as a significant breakthrough and contribution to applied stochastics by introducing an improved, knowledge-based modeling framework for spatial and spatiotemporal information. In this work, one objective is the overview of distinct BME features. By offering a foundation free of restrictive assumptions that limit comparable techniques, an ability to integrate a variety of prior knowledge bases, and rigorous accounting for both exact and uncertain data, the BME approach was coined as introducing modern spatiotemporal geostatistics. A second objective is to illustrate BME applications and adoption within numerous different scientific disciplines. We summarize examples and real-world studies that encompass the perspective of science of the total environment, including atmosphere, lithosphere, hydrosphere, and ecosphere, while also noting applications that extend beyond these fields. The broad-ranging application track suggests BME as an established, valuable tool for predictive spatial and space–time analysis and mapping. This review concludes with the present status of BME, and tentative paths for future methodological research, enhancements, and extensions.

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Notes

  1. In geostatistical literature, the terms “prediction” and “estimation” are often used interchangeably. In the following, we distinguish between the two terms by following the statistical vernacular. In it, prediction is the inference of random variable values, whereas estimation designates the inference tasks where unknown parameters are involved.

  2. In geostatistical literature, the term “homogeneity” is commonly used to refer to what the statistical vernacular calls “spatial stationarity”.

  3. There are, of course, strong arguments in support of using observed data, too. For example, data-based results via well-designed experimental designs can be used to investigate potential flaws in unestablished theoretical rules and principles.

  4. As of May 2017, available at http://seksgui.org.

  5. As of May 2017, available at http://www.unc.edu/depts/case/BMEGUI.

  6. As of May 2017, available at http://140.112.63.249/STARBME/index.html.

  7. In leave-one-out cross-validation, observations are removed from a dataset one at a time. The attribute is predicted at each vacated location from the remaining data, and the result is compared to the observed value. In general, it is recommended to keep validation data as a separate set from training observations, so that cross-validation produces a nearly unbiased estimate of the true error expected on an independent test set (e.g., Efron 1983; Varma and Simon 2006).

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Acknowledgements

The work was supported by the National Natural Science Foundation of China (No. 529105-N11701ZJ). Alexander Kolovos received no funding.

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  1. Institute of Islands and Coastal Ecosystems, Ocean College, Zhejiang University, Zhoushan, China

    Junyu He

  2. SpaceTimeWorks, San Diego, CA, USA

    Alexander Kolovos

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Correspondence to Alexander Kolovos.

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He, J., Kolovos, A. Bayesian maximum entropy approach and its applications: a review. Stoch Environ Res Risk Assess 32, 859–877 (2018). https://doi.org/10.1007/s00477-017-1419-7

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