Abstract
The purpose of this chapter is to introduce a different way of implementing Bayes theorem using a distributed, parallel algorithm first introduced by Specht (1990), which he named the probabilistic neural network (PNN). We have discussed in Chap. 6 the difficulties in constructing Bayesian networks using both the data-driven and expert-driven approaches. A significant advantage of the Bayesian PNN (BPNN) is that the node edge architecture is theoretically predetermined by the Parzen (1962)-Cacoullos (1966) theoretical formulation.
Specifically, this chapter covers the following topics.
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It develops the mathematical formulation for the PNN.
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It demonstrates that the normal PNN can be configured as an optimal Bayesian classifier (BPNN).
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It shows how Parzen’s theorem maps into Cacoullos’s theorem.
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It provides an illustrative toy example, showing a BPNN analysis for two classes and nine samples (four benign and five malignant) for a twofold cross-validation analysis.
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It shows how to develop the optimal standard or variance sigma value for the Gaussian density function (a significant problem) and discusses PNN training methods.
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It provides a BPNN application to the Alzheimer’s speech data.
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Abbreviations
- AUC:
-
Area under the ROC curve
- BPNN:
-
Bayesian probabilistic neural network
- GA:
-
Genetic algorithm
- GRNN:
-
Generalized regression neural network
- pdf:
-
probability density function
- PNN:
-
Probabilistic neural network
- ROC:
-
Receiver operator characteristic
- SVM:
-
Support vector machine
References
Cacoullos T (1966) Estimation of a multivariate density. Ann Inst Stat Math 18(2):179–189
Georgiou VL, Maloflsi SN, Alavizus L, Vrahatis MN (2006) Evolutionary Bayesian probabilistic neural networks. In: International Conference on Numerical Analysis and Applied Mathematics (ICNAAM 2006), pp 393–396
Land WH Jr, Margolis D, Kallergi M, Heine JJ (2010) A kernel approach for ensemble decision combinations with two-view mammography applications. Int J Funct Inform Personal Med 3(2):157–182
Land WH, Masters T, Lo JY (2000) Performance evaluation using the GRNN Oracle and a new evolutionary programming/adaptive boosting hybrid for breast cancer benign/malignant diagnostic aids, ANNIE
Masters T (1995) Advanced algorithms for neural networks: a C++ source book. Wiley, New York
Parzen E (1962) On estimation of a probability density function and mode. Ann Math Stat 33(3):1065–1076
Specht D (1990) Probabilistic neural networks. Neural Netw 3:109–118
Xu R, Wumsch DC II (2009) Clustering. Wiley, New York. ISBN 978-0-470-27680-8
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Land, W.H., Schaffer, J.D. (2020). Bayesian Probabilistic Neural Network (BPNN). In: The Art and Science of Machine Intelligence. Springer, Cham. https://doi.org/10.1007/978-3-030-18496-4_7
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DOI: https://doi.org/10.1007/978-3-030-18496-4_7
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