• D. A. Klyushin Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
  • O. S. Maistrenko Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
Keywords: explained artificial intelligence, non-parametric statistics, postulates of machine learning, deep learning, convolutive neural network


The paper proposes a non-parametrical approach to explainable artificial intelligence based on the compactness postulate, which states that objects of one class in the feature space are, as a rule, located closer to each other than to objects of other classes. Objects are considered similar if they are located close to each other in the feature space. Meanwhile, the properties of objects in real life are often random values. Such objects are not described by a vector of features, but by a random sample or several samples of features, and the postulate of compactness should be replaced by the postulate of statistical homogeneity. Objects are considered statistically homogeneous if their features obey the same distributions. The paper describes a non-parametric measure of homogeneity and an illustration of its use in medical applications, in particular for the diagnosis of breast cancer within the framework of similarity-based explainable artificial intelligence.For comparison, the results of diagnostics of the same data set using deep learning of an artificial neural network are given. We formulate new statistical postulates of machine learning and propose to consider a machine learning algorithm as explanatory and interpretable if it satisfies these postulates.


Adadi A., Berrada M. Peeking Inside the Black-Box: A Survey on Explainable Artificial Intelligence (XAI). IEEE Access 2018. Vol. 6. P. 52138–52160.

Alabdulhadi M., Coolen-Maturi T., Coolen F. Nonparametric predictive inference for comparison of two diagnostic tests. Communications in Statistics – Theory and Methods. 2021. Vol. 50. P. 4470–4486.

Amann J. et al. To explain or not to explain? – Artificial intelligence explainability in clinical decision support systems. PLOS Digital Health. 2022. Vol. 1(2). P. e0000016.

Andreichuk A. V., Boroday N. V., Golubeva K. M., Klyushin D. A. Artificial Intelligence System for Breast Cancer Screening Based on Malignancy-Associated

Changes in Buccal Epithelium. In: Enabling AI Applications in Data Science. Part of the Studies in Computational Intelligence book series (SCI, 2022, volume 911) Springer, 2022, pp. 267–285.

Arrieta A. B. et al. Explainable Artificial Intelligence (XAI): Concepts, taxonomies, opportunities, and challenges toward responsible A. Information Fusion. 2020. Vol. 58. P. 82–115.

Bakalis E. et al. (2022) Universal Markers Unveil Metastatic Cancerous Cross-Sections at Nanoscale. Cancers. 2022. Vol. 14. No. 15. P. 3728.

Barnett V. (1976) The ordering of multivariate data. Journal of the Royal Statistical Society. Series A (General). Vol. 139. No. 3. P.318–355.

Bi W. et al. Artificial intelligence in cancer imaging: Clinical challenges and applications. CA Cancer Journal for Clinicians. 2019. Vol. 69. No. 2). P. 127–157.

Borys K. et al. Explainable AI in medical imaging: An overview for clinical practitioners – Saliency-based XAI approaches. European Journal of Radiology. 2023. Vol. 162. P. 110787.

Cascos I. (2007) Depth function as based of a number of observation of a random vector. Working Paper 07-29, Statistic and Econometric Series 07, 2:1–28.

Chaddad A, Peng J, Xu J, Bouridane A. Survey of Explainable AI Techniques in Healthcare. Sensors. 2023. Vol. 23. No. 2. P. 634.

Chen Z. et al. Artificial intelligence for assisting cancer diagnosis and treatment in the era of precision medicine. Cancer Communications 2021. Vol. 41. No. 11. P. 1100–1115.

Elkington L., Adhikari P., Pradhan P. Fractal Dimension Analysis to Detect the Progress of Cancer Using Transmission Optical Microscopy. Biophysica. 2022. Vol. 2. No. 1. P. 59–69.

Hacking S., Yakirevich E., Wang Y. From Immunohistochemistry to New Digital Ecosystems: A State-of-the-Art Biomarker Review for Precision Breast Cancer Medicine. Cancers. 2022. Vol. 14. No. 14. P. 3469.

Hill B. Posterior distribution of percentiles: Bayes’ theorem for sampling from a population. Journal of American Statistical Association. 1968. Vol. 63. P. 677–691.

Hill B. De Finetti’s theorem, induction, and A(n) or Bayesian nonparametric predictive inference (with discussion). In: D. V. Lindley, J. M. Bernardo, M. H. DeGroot, and A. F. M. Smith (Eds.), Bayesian statistics (1988, Vol. 3, pp. 211–241). Oxford: Oxford University Press.

Klyushin D. A., Petunin Yu. I. A Nonparametric Test for the Equivalence of Populations Based on a Measure of Proximity of Samples. Ukrainian Mathematical Journal. 2003. Vol. 55. No. 2. P. 181–198.

Klyushin D., Golubeva K.. Boroday N., Shervarly D. Breast cancer diagnosis using machine learning and fractal analysis of malignancy-associated changes in buccal epithelium. Chapter in: Artificial Intelligence, Machine Learning, and Data Science Technologies Future Impact and Well-Being for Society 5.0 N. Mohan, R. Singla, P. Kaushal, and S. Kadry (Eds.), CRC Press, 2021, P. 1–18.

Koopaie M., Kolahdooz S., Fatahzadeh M., Manifar S. Salivary biomarkers in breast cancer diagnosis: A systematic review and diagnostic meta-analysis. Cancer Medicine. 2022. Vol. 11. No. 13. P. 2644–2661.

Koshevoy G., Mosler K. Zonoid trimming for multivariate distributions. Annals of Statistics. 1997. Vol. 25. P. 1998–2017.

Li J., Du Q., Sun C. An improved box-counting method for image fractal dimension estimation. Pattern Recognition. Vol. 42. No. 11. P. 2460–2469.

Liang W. et al. Cancer cells corrupt normal epithelial cells through miR-let-7c-rich small extracellular vesicle-mediated downregulation of p53/PTEN. Intertational Journal of Oral Science. Vol. 14. No. 36.

Lipton Z. The mythos of model interpretability: In machine learning, the concept of interpretability is both important and slippery. ACM Queue. 2018. Vol. 16. No. 3. P. 31–57.

Liu R. J. On a notion of data depth based on random simplices. Annals of Statistics. 1990. Vol. 18. P. 405–414.

Love P. et al. Explainable Artificial Intelligence (XAI): Precepts, Methods, and Opportunities for Research in Construction. arXiv:2211.06579v2, 2022.

Lyashko S. Klyushin D., Alexeyenko V. Mulrivariate ranking using elliptical peeling. Cybernetic and Systems Analysis. 2013. Vol. 49. No. 4. P. 511–516.

Mosler K., Mozharovskyi P. Choosing among notions of multivariate depth statistics. Statistical Science. Vol. 37. No. 3. P. 348–368.

Mottl V., Seredin O., Krasotkina O. Compactness Hypothesis, Potential Functions, and Rectifying Linear Space in Machine Learning. In: International Conference Commemorating the 40th Anniversary of Emmanuil Braverman’s Decease, Boston, MA, USA, April 28-30, 2017, Invited Talks.

Nazir S., Dickson D., Akram M. Survey of explainable artificial intelligence techniques for biomedical imaging with deep neural networks. Computers in Biology and Medicine. 2023. Vol. 156. P. 106668.

Oja H. Descriptive statistics for multivariate distributions. Statistics and Probability Letters. 1983. Vol. 1. P. 327–332.

Petunin Yu., Rublev B. Pattern recognition using quadratic discriminant functions. Numerical and Applied Mathematics. 1996. Vol. 80. P. 89–104.

Polverini P., Nor F., Nor J. Crosstalk between cancer stem cells and the tumor microenvironment drives progression of premalignant oral epithelium. Frontiers in Oral Health. 2023. Vol. 3. No. 1095842.

Rudin C. Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead. Nature Machine Intelligence. 2019. No. 1. P. 206–215.

Rudin C. et al. Interpretable machine learning: Fundamental principles and 10 grand challenges. Statistical Surveys. 2022. Vol. 16. P. 1-85.

Sanchez J., Martin-Landrove M. Morphological and Fractal Properties of Brain Tumors. Frontiers in Physiology. 2022. Vol. 13. No. 878391.

Sheu R.-K., Pardeshi M. A Survey on Medical Explainable AI (XAI): Recent Progress, Explainability Approach, Human Interaction and Scoring System. Sensors. 2022. Vol. 22. No. 8068.

Subramanian H. et al. Procedures for risk-stratification of lung cancer using buccal nanocytology. Biomedical Optics Express. (2016). Vol. 7. No. 9. P. 3795–3810.

Tukey J. Mathematics and the picturing of data. In: Proceedings of the International Congress of Mathematician (1975, pp. 523–531). Montreal, Canada.

Us-Krasovec M. et. al. Malignancy associated changes in epithelial cells of buccal mucosa: a potential cancer detection test. Analytical and Quantitative Cytology and Histology. 2005. Vol. 27. No. 5. P. 254–262.

Wu C. et al. Cancer-Associated Adipocytes and Breast Cancer: Intertwining in the Tumor Microenvironment and Challenges for Cancer Therapy. Cancers. 2023. Vol. 15. No. 3. P. 726.

Xu C. et al. Modeling and analysis fractal order cancer model with effects of chemotherapy. Chaos, Solitons and Fractals. 2022. Vol. 161. No. 112325.

Yang S., Folke T., Shafto P. A psychological theory of explainability. In: Proceedings of the 39th International Conference on Machine Learning, Baltimore, Maryland, USA, 2022, PMLR 162.

Zhang Y., Weng Y., Lund J. Applications of Explainable Artificial Intelligence in Diagnosis and Surgery. Diagnostics. 2022. Vol. 12. No. 2. P. 237.

Zuo Y., Serfling R. General notions of statistical depth function. Annals of Statistics. Vol. 28. P. 461–482.