In this work we couple the Metropolis-Hastings algorithm with the volcanic ash transport model TEPHRA2, and present the coupled algorithm as a new method to estimate the Eruption Source Parameters of volcanic eruptions based on mass per unit area or thickness measurements of tephra fall deposits. Basic elements in the algorithm and how to implement it are introduced. Experiments are done with synthetic datasets. These experiments are designed to demonstrate that the algorithm works, and to show how inputs affect its performance. Results are presented as sample posterior distribution estimates for variables of interest. Advantages of the algorithm are that it has the ability to i) incorporate prior knowledge; ii) quantify the uncertainty; and iii) capture correlations between variables of interest in the estimated Eruption Source Parameters. A limitation is that some of the inputs need to be specifed subjectively. How and why such inputs affect the performance of the algorithm and how to specify them properly are explained and listed. Correlation between variables of interest are well-explained by the physics of tephra transport. We point out that in tephra deposit inversion, caution is needed in attempting to estimate Eruption Source Parameters, and wind direction and speed at each elevation level, as this increases the number of variables to be estimated. The algorithm is applied to a mass per unit area dataset of the tephra deposit from the 2011 Kirishima-Shinmoedake eruption. Simulation results from TEPHRA2 using posterior means from the algorithm are consistent with field observations, suggesting that this approach reliably reconstructs Eruption Source Parameters and wind conditions from the deposit.