For the smooth functioning of Continuous Casting (CC) machinery, accurately estimating the heat flux at the interface between the CC mold and molten steel is crucial. To tackle this challenge, we've developed a novel approach by formulating a stochastic inverse heat transfer problem (IHTP). This method aims to deduce the transient heat flux, which is considered an unknown Neumann boundary condition. We employ an innovative Data Assimilation (DA) technique known as Ensemble-based Simultaneous Input and State Filtering (EnSISF), which is further enhanced by integrating Radial Basis Functions (RBFs) for improved computational efficiency. This approach allows for the simultaneous prediction of temperature distribution and estimation of heat flux based on sequential temperature data from thermocouples within the mold. Our research marks a significant advancement in achieving real-time probabilistic boundary condition estimation. It effectively deals with the challenges posed by noisy measurements and model inaccuracies, leading to efficient real-time monitoring and control. This is essential for avoiding caster shutdowns and ensuring the continuous operation of CC machinery.