Chase-Pyndiah decoding is widely used for decoding product codes. However, this method is suboptimal and requires scaling the soft information exchanged during the iterative processing. In this paper, we propose a framework for obtaining the scaling coefficients based on maximizing the generalized mutual information. Our approach yields gains up to 0.11 dB for product codes with two-error correcting extended BCH component codes over the binary-input additive white Gaussian noise channel compared to the original Chase-Pyndiah decoder with heuristically obtained coefficients. We also introduce an extrinsic version of the Chase-Pyndiah decoder and associate product codes with a turbo-like code ensemble to derive a Monte Carlo-based density evolution analysis. The resulting iterative decoding thresholds accurately predict the onset of the waterfall region.