The DOI matches its Crossref record; 102 of the 103 references it cites were found in a registry or an index.
Bayesian Methods for Neural Networks and Related Models
Abstract
Models such as feed-forward neural networks and certain other structures investigated in the computer science literature are not amenable to closed-form Bayesian analysis. The paper reviews the various approaches taken to overcome this difficulty, involving the use of Gaussian approximations, Markov chain Monte Carlo simulation routines and a class of non-Gaussian but “deterministic” approximations called variational approximations.
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“Bayesian Neural Networks (BNNs) [68,71,72] are stochastic ANNs trained using Bayesian inference.”
From Explainable AI in medical imaging: An overview for clinical practitioners – Beyond saliency-based XAI approaches (2023) · cites this paper as [72]
Citations
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Bugbee, B. D., Breidt, F. J., & van der Woerd, M. J. (2015) · Journal of Computational and Graphical Statistics · doi:10.1080/10618600.2014.983642 · Cited by 3
Scalable Stochastic Parametric Verification with Stochastic Variational Smoothed Model Checking
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Platania, F., Appio, F. P., Hernandez, C. T., Ouadghiri, I. E., & Peillex, J. (2023) · Annals of Operations Research · doi:10.1007/s10479-023-05754-z · Cited by 3
“For a comprehensive study and comparison between the Levenberg–Marquardt and Bayesian regularization algorithms, we recommend Kayri (2016), Gianola et al. (2011), Hagan and Menhaj (1994), Burden and Winkler (2008), Marwalla (2007), Titterington (2004), among others.”
Uncertainty-Aware SAR ATR: Defending Against Adversarial Attacks via Bayesian Neural Networks
Ye, T., Kannan, R., Prasanna, V. K., & Busart, C. (2024) · IEEE Radar Conference (RadarConf) · doi:10.1109/radarconf2458775.2024.10548693 · Cited by 3
“Bayesian Neural Networks (BNNs) [9], [10] are a type of neural network models that leverage the principles of Bayesian statistics.”
BackgroundMethodShirvani, A., Nili‐Ahmadabadi, M., & Ha, M. Y. (2024) · Journal of Mechanical Science and Technology · doi:10.1007/s12206-024-2104-7 · Cited by 3
“MDL indicates a less complex model for the neural network architecture corresponds to higher probability in the Bayesian view [38-41] and a higher performance and generalizability [42].”
Guerra, G. A., Ruiz, P. M., Anamiati, G., & Landberg, L. (2024) · EPJ Photovoltaics · doi:10.1051/epjpv/2024028 · Cited by 3
Ge, X., Given, M. J., & Stewart, B. G. (2022) · IEEE International Conference on High Voltage Engineering and Application (ICHVE) · doi:10.1109/ichve53725.2022.9961792 · Cited by 3
“Among regularization methods, Bayesian regularization not only obtains lower mean squared errors [5], it also linearly combines the sum of squares of weights (E w ) and squared errors to minimize the target function (2) by using (3) so that the trained network has good generalization qualities[6]:”
BackgroundSome Aspects of Latent Structure Analysis
Titterington, D. M. (2006) · Lecture notes in computer science · doi:10.1007/11752790_4 · Cited by 3
“For a review of variational and other approaches to Bayesian analysis in models of this general type see [ 41 ].”
MethodMethods in functional data analysis and functional genomics
Backenroth, D. (2018) · doi:10.7916/d81r82fm · Cited by 3
“3), are computationally efficient and typically yield accurate point estimates for model parameters, although they provide only an approximation to the complete posterior distribution and inference may suffer as a result [Ormerod and Wand, 2012; Jordan, 2004; Jordan et al., 1999; Titterington, 2004].”
BackgroundConcentration Inequalities and Optimal Number of Layers for Stochastic Deep Neural Networks
Caprio, M., & Mukherjee, S. (2023) · IEEE Access · doi:10.1109/access.2023.3268034 · Cited by 3
“A well known example of SNN are Bayesian neural networks (BNNs), that are ANNs trained using a Bayesian approach [10], [12], [22], [25], [42], [44], [45].”
MethodA variational Bayes approach to a semiparametric regression using Gaussian process priors
Ong, V. M.-H., Mensah, D. K., Nott, D. J., Jo, S., Park, B., & Choi, T. (2017) · Electronic Journal of Statistics · doi:10.1214/17-ejs1324 · Cited by 3
“For Bayesian inference with the posterior density p(δ|y), which is often mathematically intractable, variational approximation methods (e.g. Jordan (2004), Titterington (2004), and Ormerod and Wand (2010)) can be employed.”
BackgroundMethodInfluentialSingh, C., Sawle, Y., Kumar, N., Jha, U., & Arunkumar, L. (2023) · doi:10.1002/9781394167524.ch3 · Cited by 3
Adaptive Generation of Training Data for ML Reduced Model Creation
Cianciosa, M. R., Archibald, R. K., Elwasif, W. R., Gainaru, A., Park, J. M., & Whitfield, R. E. (2022) · IEEE International Conference on Big Data (BigData) · doi:10.1109/bigdata55660.2022.10020884 · Cited by 2
Ho, A., Zanisi, L., de Leeuw, B., Galvan, V., Rodriguez-Fernandez, P., & Howard, N. T. (2025) · Physics of Plasmas · doi:10.1063/5.0276455 · Cited by 2
Cairoli, F., & Bortolussi, L. (2025) · International Journal of Systems Science · doi:10.1080/00207721.2025.2490618 · Cited by 2
“The core idea of Bayesian neural networks (BNNs) (Bishop, 2006; Goan & Fookes,2020 ; Lampinen & Vehtari, 2001; Titterington, 2004), is to place a probability distribution over the weights w of a neural network f w : (cid:5) → [0, 1], transforming the latter into a probabilistic model.”
Layek, K., Samanta, S., Sadhu, A. K., Maity, S. P., & Barui, A. (2018) · Elsevier eBooks · doi:10.1016/b978-0-12-813087-2.00013-0 · Cited by 2
Méthodes bayésiennes variationnelles : concepts et applications en neuroimagerie
Keribin, C. (2010) · HAL · Cited by 2
Implementation of Hopfield Neural Network for its Capacity with Finger Print Images
Chandra, R., Kumar, S., & Goswami, P. (2016) · International Journal of Computer Applications · doi:10.5120/ijca2016909625 · Cited by 2
Disentangling the Gauss-Newton Method and Approximate Inference for Neural NetworksPreprint
Immer, A. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2007.11994 · Cited by 4
“r3 FromNeuralNetworktoGaussian ProcesswithLaplaceandGauss-Newton The Laplace approximation is often used as a baseline for Bayesian neural networks and in some cases achieves state of the art results [43, 44, 49]. For neural networks, the diagonal or a Kronecker-factored generalized Gauss-Newton approximation to the H…”
MethodFinancial Bridges and Network CommunitiesPreprint
Casarin, R., Costola, M., & Yenerdag, E. (2018) · SSRN Electronic Journal · doi:10.2139/ssrn.3178053 · Cited by 2
Aye, S. A. (2014) · UpSpace Institutional Repository (University of Pretoria) · Cited by 2
Lall, P., Deshpande, S., & Nguyen, L. T. (2017) · IEEE Transactions on Components Packaging and Manufacturing Technology · doi:10.1109/tcpmt.2017.2672498 · Cited by 1
Morphological Study of the Infrared Sensory Pits of Pit Viper, Python and Boa Snakes
Weyand, B., Israelowitz, M., Reuter, M., Bohlmann, S., Wagner, R. A., Rizvi, S. W. H., Gille, C., Reimers, K., Vogt, P. M., & von Schroeder, H. P. (2021) · Series in bioengineering · doi:10.1007/978-3-319-53214-1_7 · Cited by 1
Biomimetics Strategies to Overcoming Noise
Rizvi, S. W. H., Weyand, B., Israelowitz, M., Gille, C., Reuter, M., Bohlmann, S., Reimers, K., Vogt, P. M., & von Schroeder, H. P. (2021) · Series in bioengineering · doi:10.1007/978-3-319-53214-1_10 · Cited by 1
A variational inference for the Lévy adaptive regression with multiple kernels
Lee, Y., Jo, S., & Lee, J. (2022) · Computational Statistics · doi:10.1007/s00180-022-01200-z · Cited by 1
“…distribution by iterating the following update equations (see, e.g., Bishop 2006; Blei et al. 2017): and Although the factorization we use could not capture correlations between parameters (see, e.g., Titterington 2004; Wang and Titterington 2005), it makes inference procedure tractable and fast.”
MethodGlobal Surrogate Modeling by Neural Network-Based Model Uncertainty
Leifsson, L., Nagawkar, J. R., Barnet, L., Bryden, K. M., Koziel, S. M., & Pietrenko‐Dabrowska, A. (2022) · Lecture notes in computer science · doi:10.1007/978-3-031-08757-8_35 · Cited by 1
Okut, H., Gianola, D., WEIGEL, K. A., & Rosa, G. J. M. (2022) · Livestock Studies · doi:10.46897/livestockstudies.1159627 · Cited by 1
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Sun, M. (2025) · International Conference on Algorithm, Artificial Intelligence and Computer Vision (AAICV) · doi:10.1109/aaicv66571.2025.00060 · Cited by 1
“Titterington (2004) pointed out that Bayesian methods include the use of Gaussian approximation, MCMC simulation and other techniques, which can play an important role in uncertainty propagation [9].”
Prognose makroökonomischer Zeitreihen: Ein Vergleich linearer Modelle mit neuronalen Netzen
Koller, W. (2014) · Peter Lang D eBooks · doi:10.3726/978-3-653-03344-1 · Cited by 1
Target Detection on Hyperspectral Images Using MCMC and VI Trained Bayesian Neural Networks
Ries, D., Adams, J., & Zollweg, J. (2022) · IEEE Aerospace Conference (AERO) · doi:10.1109/aero53065.2022.9843344 · Cited by 1
“[18] provides a review of BNN and their common estimation methods at the time, MCMC, Gaussian approximation, and early variational inference (VI), from a statistical perspective.”
MethodEfficient Uncertainty Quantification Using Sequential Sampling-Based Neural Networks
Koratikere, P., Leifsson, L., Koziel, S. M., & Pietrenko‐Dabrowska, A. (2023) · Lecture notes in computer science · doi:10.1007/978-3-031-36024-4_41 · Cited by 1
Martín, A. G. (2010) · doi:10.14201/gredos.76447 · Cited by 1
Zhang, S. (2017) · ScholarWorks - WMU (Western Michigan University) · Cited by 1
Artificial neural networks for the computation of the inverse kinematics of redundant manipulators
Habibkhah, S. (2020) · oURspace (University of Regina) · doi:10.82465/4074 · Cited by 1
Infante, S., Cedeño, F., & Candel, J. M. O. (2010) · Redalyc (Universidad Autónoma del Estado de México)
Bayesian non-linear matching of pairwise microarray gene expressionsPreprint
Nieto, C. Á., & Diazaraque, J. M. M. (2008) · RePEc: Research Papers in Economics
Predicting flavonoid UGT regioselectivity with graphical residue models and machine learning
Jackson, A. R. (2009) · Digital Commons - East Tennessee State University (East Tennessee State University)
Parameter estimation of general regression neural network using Bayesian approach
Choir, A. S., Prasetyo, R. B., Ulama, B. S. S., Iriawan, N., Fitriasari, K., & Dokhi, M. (2016) · AIP conference proceedings · doi:10.1063/1.4940858
Shi, J. Q., & Choi, T. (2011) · Chapman & Hall/CRC eBooks · doi:10.1201/b11038-11
Financial Bridges and Network CommunitiesPreprint
Casarin, R., Costola, M., & Yenerdag, E. (2017) · SSRN Electronic Journal · doi:10.2139/ssrn.2951716
Onunka, C. (2017) · International Conference on Mechatronics and Machine Vision in Practice (M2VIP) · doi:10.1109/m2vip.2017.8211487
“Thus, there is cross-fertilization in the implementation of the methodologies in the EEG signal analysis [4].”
MethodZárate-Solano, H. M., & Zapata-Sanabria, D. R. (2018) · doi:10.18235/0001264
Titterington, M. (2014) · Wiley eBooks · doi:10.1002/9781118445112.stat00448
(2004) · Wiley series in probability and statistics · doi:10.1002/0470090456.refs
Titterington, M. (2005) · Wiley eBooks · doi:10.1002/0471667196.ess1112.pub3
Nicodemus, E. R., Ambarkar, V. M., Ray, S., & Schipperijn, F. W. (2022) · SAE International Journal of Engines · doi:10.4271/03-16-03-0017
ÖZGÜR, B., & Yolcu, U. (2022) · Turkish Journal of Forecasting · doi:10.34110/forecasting.1223653
Research on global illumination exploration in dynamic scenes
Ji, X. H., Yang, H., & Han, C. (2023) · International Conference on Computer Vision and Pattern Analysis (ICCPA) · doi:10.1117/12.2684526
“Reviewing the neural network-based Bayesian approach proposed by Titterington [13] , much research progress has been made in this area.”
MethodUncertainty-informed SAR image classification with Bayesian neural networks
Tian, Y., Kannan, R., Prasanna, V. K., Wang, X., & Busart, C. (2024) · Automatic target recognition. Conference · doi:10.1117/12.3013963
“13,14 BNNs are neural networks where model parameters are probabilistic instead of deterministic.”
Concentration of Measure and Global Optimization of Bayesian Multilayer Perceptron. Part I
Temyanov, B. K., & Nigmatullin, R. R. (2024) · Lobachevskii Journal of Mathematics · doi:10.1134/s1995080224600651
Salim, M. S. (2024) · International Journal of Sensors Wireless Communications and Control · doi:10.2174/0122103279346234240910104932
Artificial neural network weights-based new formula for solar power plant energy prediction
Salim, M. S., Sabri, N., & Dheyab, A. A. R. (2024) · Edelweiss Applied Science and Technology · doi:10.55214/25768484.v8i6.3998
Hao, J., & Pan, Y. (2025) · Frontiers in Agronomy · doi:10.3389/fagro.2025.1684447
Herrmann, L., Jokeit, M., Weeger, O., & Kollmannsberger, S. (2025) · Springer eBooks · doi:10.1007/978-3-031-89529-6_3
Mustafa, R., Khan, F. U., Farooq, A., & Rani, A. (2025) · Chromatographia · doi:10.1007/s10337-025-04463-y
“Bayesian regularization adaptively adapts regularization parameters, unlike manual tuning in conventional methods, making certain that the model relics both generalizable and accurate [17–20].”
Advancing cosmological field-level inference with physics-informed Bayesian neural networks
Ding, S. (2025) · theses.fr (ABES) · doi:10.70675/b3b47cd9z9b36z4663zb245z1dce12a4a9fa
Approximate Inference with Exponential Tilting Densities: Theory and Applications
Shi, X., Wang, X., Wong, A. C. M., & Lin, W. (2026) · Statistical Science · doi:10.1214/24-sts971
Clustering and Registration of Functional Data with Applications in Time Course Genomics Data
Zhang, Y. (2014) · eScholarship (California Digital Library)
Metadata
- Authors
- Titterington, D. M.
- Corresponding author
- Titterington, D. M.
- Venue
- Statistical Science
- Publisher
- Institute of Mathematical Statistics
- Published
- 2004-02-01
- Volume
- 19
- Issue
- 1
- Pages
- 128-139
- ISSN
- 0883-4237 (ISSN-L)
- 2168-8745
- OpenAlex
- W2019449965
- Cited by
- 162
- References
- 103
- Citations by year
- 2026: 2 · 2025: 9 · 2024: 20 · 2023: 16 · 2022: 17 · 2021: 9 · 2020: 9 · 2019: 7 · 2018: 6 · 2017: 7 · 2016: 10 · 2015: 9 · 2014: 8 · 2013: 3 · 2012: 1
- FWCI
- 3.05
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Statistical Science (publishedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Markov Chains and Monte Carlo Methods
- Physical Sciences › Computer Science › Artificial Intelligence › Gaussian Processes and Bayesian Inference
- Physical Sciences › Computer Science › Artificial Intelligence › Neural Networks and Applications
- Keywords
- Bayesian neural networks, feedforward neural networks, Gaussian approximation, Markov chain Monte Carlo, variational approximation, computational intractability
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