We consider the problem of estimating a temperature-dependent thermal conductivity model (curve) from temperature measurements. We apply a Bayesian estimation approach that takes into account measurement errors and limited prior information of system properties. The approach intertwines system simulation and Markov chain Monte Carlo (MCMC) sampling. We investigate the impact of assuming different model classes - cubic polynomials and piecewise linear functions - their parametrization, and different types of prior information - ranging from uninformative to informative. Piecewise linear functions require more parameters (conductivity values) to be estimated than the four parameters (coefficients or conductivity values) needed for cubic polynomials. The former model class is more flexible, but the latter requires less MCMC samples. While parametrizing polynomials with coefficients may feel more natural, it turns out that parametrizing them using conductivity values is far more natural for the specification of prior information. Robust estimation is possible for all model classes and parametrizations, as long as the prior information is accurate or not too informative. Gaussian Markov random field priors are especially well-suited for piecewise linear functions.

翻译：我们考虑从温度测量数据中估计温度依赖的热导率模型（曲线）的问题。采用贝叶斯估计方法，该方法考虑了测量误差和系统属性的有限先验信息。该方法将系统仿真与马尔可夫链蒙特卡洛（MCMC）采样相结合。我们研究了假设不同模型类别（三次多项式与分段线性函数）及其参数化方式的影响，以及不同先验信息类型（从无信息到有信息）的影响。分段线性函数需要估计的参数（热导率值）多于三次多项式所需的四个参数（系数或热导率值）。前者模型类别更灵活，但后者所需MCMC样本量更少。虽然用系数参数化多项式可能更直观，但研究发现采用热导率值进行参数化对先验信息的设定更为自然。只要先验信息准确或不太强，所有模型类别和参数化方式均可实现稳健估计。高斯马尔可夫随机场先验特别适用于分段线性函数。