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<p>This study develops a Bayesian framework for analyzing failure times under the Weibull distribution, addressing challenges in prior specification caused by the absence of conjugate priors and multi-dimensional sufficient statistics. We propose an adaptive semi-parametric MCMC algorithm, combining a hierarchical Bayesian model with a noninformative LogNormal hyper-prior and the No-U-Turn Sampler (NUTS) in STAN. The framework is evaluated across 24 prior combinations and 72 simulated datasets. Results show that the proposed approach achieves up to 40% lower weighted relative efficiency (WRE) than classical methods such as maximum likelihood estimation (MLE), method of moments, and ordinary least squares, under both increasing and decreasing hazard rate scenarios. For large samples, priors such as Exponential–HalfCauchy yield WRE values as low as 0.83, demonstrating improved regularization. In a real-data application using prostate cancer survival data (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n=87\)</EquationSource> </InlineEquation>), the algorithm predicts mean residual lifetime with deviations below 0.06% from the integrated model, outperforming MLE with deviations of 2.17%. These results highlight the efficiency and accuracy of the proposed framework, offering a scalable tool for reliability and survival analysis.</p>

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Modeling complex life systems: Bayesian inference for Weibull failure times using adaptive MCMC

  • 期刊论文

  • 发表日期：

    2025年12月9日

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  • Tobias Oketch,
  • Mohammad Sepehrifar

摘要

This study develops a Bayesian framework for analyzing failure times under the Weibull distribution, addressing challenges in prior specification caused by the absence of conjugate priors and multi-dimensional sufficient statistics. We propose an adaptive semi-parametric MCMC algorithm, combining a hierarchical Bayesian model with a noninformative LogNormal hyper-prior and the No-U-Turn Sampler (NUTS) in STAN. The framework is evaluated across 24 prior combinations and 72 simulated datasets. Results show that the proposed approach achieves up to 40% lower weighted relative efficiency (WRE) than classical methods such as maximum likelihood estimation (MLE), method of moments, and ordinary least squares, under both increasing and decreasing hazard rate scenarios. For large samples, priors such as Exponential–HalfCauchy yield WRE values as low as 0.83, demonstrating improved regularization. In a real-data application using prostate cancer survival data ( \(n=87\) ), the algorithm predicts mean residual lifetime with deviations below 0.06% from the integrated model, outperforming MLE with deviations of 2.17%. These results highlight the efficiency and accuracy of the proposed framework, offering a scalable tool for reliability and survival analysis.