In this paper, the issue of designing an iterative-detection-and-decoding (IDD)-aided receiver, relying on the low-complexity probabilistic data association (PDA) method, is addressed for turbo-coded multiple-input-multiple-output (MIMO) systems using general M-ary modulations. We demonstrate that the classic candidate-search-aided bit-based extrinsic log-likelihood ratio (LLR) calculation method is not applicable to the family of PDA-based detectors. Additionally, we reveal that, in contrast to the interpretation in the existing literature, the output symbol probabilities of existing PDA algorithms are not the true a posteriori probabilities (APPs) but, rather, the normalized symbol likelihoods. Therefore, the classic relationship, where the extrinsic LLRs are given by subtracting the a priori LLRs from the a posteriori LLRs, does not hold for the existing PDA-based detectors. Motivated by these revelations, we conceive a new approximate Bayesian-theorem-based logarithmic-domain PDA (AB-Log-PDA) method and unveil the technique of calculating bit-based extrinsic LLRs for the AB-Log-PDA, which facilitates the employment of the AB-Log-PDA in a simplified IDD receiver structure. Additionally, we demonstrate that we may dispense with inner iterations within the AB-Log-PDA in the context of IDD receivers. Our complexity analysis and numerical results recorded for Nakagami-m fading channels demonstrate that the proposed AB-Log-PDA-based IDD scheme is capable of achieving a performance comparable with that of the optimal maximum a posteriori (MAP)-detector-based IDD receiver, while imposing significantly lower computational complexity in the scenarios considered.