  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.
