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相关热导率模型（曲线）的问题。采用贝叶斯估计方法，该方法能充分考虑测量误差及系统特性的有限先验信息，将系统仿真与马尔可夫链蒙特卡洛（MCMC）采样相结合。我们研究了不同模型类假设（三次多项式与分段线性函数）及其参数化方式，以及不同类型先验信息（从无信息到信息型）的影响。分段线性函数需要估计更多参数（热导率值），而三次多项式仅需四个参数（系数或热导率值）。前者模型类更灵活，但后者所需MCMC样本量更少。尽管用系数参数化多项式可能更直观，但使用热导率值进行参数化对先验信息的描述更为自然。只要先验信息准确或不过度信息型，所有模型类与参数化方式均可实现鲁棒估计。高斯马尔可夫随机场先验尤其适用于分段线性函数。