A Bayesian parameter inference problem is conducted to estimate the explosive yield of the first atomic explosion at Trinity in New Mexico. The first of its kind, the study advances understanding of fireball dynamics and provides an improved method for the determination of explosive yield. Using fireball radius-time data taken from archival film footage of the explosion and a physical model for the expansion characteristics of the resulting fireball, a yield estimate is made. Bayesian results from the Markov chain indicate that the estimated parameters are consistent with previous calculation except for the critical parameter that modifies the independent time variable. This unique result finds that this parameter deviates in a statistically significant way from previous predictions. Use of the Bayesian parameter estimates computed is found to greatly improve the ability of the fireball model to predict the observed data. In addition, parameter correlations are computed from the Markov chain and discussed. As a result, the method used increases basic understanding of fireball dynamics and provides an improved method for the determination of explosive yields.