We propose a Bayesian framework [1] to estimate and quantify the uncertainties of a temperature-dependent thermal conductivity. We consider a numerical unidimensional heat conduction problem, which involves a rod heated at one edge and cooled at the other. The thermal conductivity depends on the temperature, which characterizes the non-linearity of the problem [2]. We use transient temperature measurements at fixed positions on the rod to estimate and quantify the uncertainties of the conductivity function. The Bayesian framework is conducted based on space and time discretization of the problem. To model the temperature-dependent thermal conductivity, in line with the isogeometric analysis paradigm [3], we use cubic splines due to their smooth behavior. This aligns with the physical expectations of the conductivity. Splines also provide flexibility for capturing non-linear relationships while ensuring numerical stability [3]. For the Bayesian framework, the prior distribution is modeled in terms of the values of the conductivity at the interpolation points used in the spline representation, while the likelihood function is modeled in terms of the simulated measurement noise. The prior and the likelihood are combined into a posterior, and we obtain its samples with an adaptive Metropolis-Hastings algorithm [4], which is a Markov Chain Monte Carlo (MCMC) [5] method. Results demonstrate the effectiveness of the Bayesian spline-based framework in providing reliable estimation and uncertainty quantification of the temperature-dependent thermal conductivity. REFERENCES [1] A. Gelman, et al, Bayesian data analysis. Chapman and Hall/CRC, 1995. [2] M. N. Özışık, Heat conduction. John Wiley & Sons, 1993. [3] J. A. Cottrell, T. J. R. Hughes, Y. Bazilevs, Isogeometric analysis: toward integration of CAD and FEA. John Wiley & Sons, 2009. [4] M. Vihola, Robust adaptive Metropolis algorithm with coerced acceptance rate. Statistics and computing 22 (2012): 997-1008. [5] S. Brooks, et al., Handbook of Markov chain Monte Carlo. CRC press, 2011.