Summary: Stochastic inverse problems in heat conduction with consideration of uncertainties in the measured temperature data, temperature sensor locations and thermophysical properties are addressed using a Bayesian statistical inference method. Both parameter estimation and thermal history reconstruction problems, including boundary heat flux and heat source reconstruction, are studied. Probabilistic specification of the unknown variables is deduced from temperature measurements. Hierarchical Bayesian models are adopted to relax the prior assumptions on the unknowns. The use of a hierarchical Bayesian method for automatic selection of the regularization parameter in the function estimation inverse problem is discussed. In addition, the method explores the length scales in the estimation of thermal variables varying in space and time. Markov chain Monte Carlo (MCMC) simulation is conducted to explore the high dimensional posterior state space. The methodologies presented are general and applicable to a number of data-driven engineering inverse problems.