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. 2024 Dec 28;14:30808. doi: 10.1038/s41598-024-81111-9

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© The Author(s) 2024

Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.

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Fig. 6 — Comparison of the two models’ estimates. (a) D_KL and (b) SS estimated by the variational Bayes model with various hyperparameter values (𝜎_q). (c) Uncertainty and (d) change point probability (CPP) estimated by the reduced Bayesian model with various hazard rate (HR) values. Both VB and RB models with optimal hyperparameter values predicted similar changes in D_KL /uncertainty. This pattern gradually reversed as σ_q increased (more inaccurate) in the VB model. By contrast, the estimated uncertainty remained higher across trials with an increasing HR. VB model: variational Bayes model; RB model: reduced Bayesian model.