<p>We present a hybrid Bayesian inference and Levenberg-Marquardt (LM) framework for solving the inverse problem of input heat estimation that uses solid materials as fuel. This framework aims to estimate the boundary heat flow rate based solely on drum pressure and level measurements, using a physics-based model of the boiler dynamics. In the industry, it is crucial to know this variable, as input energy directly affects engineering projects and process operations. In this approach, the heat flow rate samples in the Bayesian inference were initially generated using the Random Walk (RW) algorithm and refined at each iteration by the Levenberg-Marquardt scheme. The heat flow rate was represented through the Karhunen-Loève (KL) expansion, utilizing a Gaussian covariance whose coefficients were estimated iteratively by minimizing a misfit function. This method effectively reduces the high dimensionality of the problem, allowing for an accurate approximation of the heat flow rate using a relatively small number of realizations. The numerical results demonstrate that the proposed framework successfully reduces the computational costs typically associated with traditional Markov Chain Monte Carlo (MCMC) method while maintaining high solution accuracy. This study also reveals that the hybrid RW-LM approach is more efficient than both the RW and LM methods when used in isolation, especially in scenarios with low spatial correlation and high variability, where traditional methods often fail. Furthermore, the misfit function used here as an objective measure ensures that the coefficients of the KL expansion associated with the heat flow rate efficiently converge to their true values. This hybrid RW-LM method effectively balances computational efficiency and precision in solving the inverse problem. It offers a viable alternative to standard methods and holds promise for broader applications in process control and predictive modeling.</p>