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The adaptive belief propagation (ABP) algorithm was recently proposed by Jiang and Narayanan for the soft decoding of Reed-Solomon (RS) codes. In this paper, simplified versions of this algorithm are investigated for the turbo decoding of product codes. The complexity of the turbo-oriented adaptive belief propagation (TAB) algorithm is significantly reduced by moving the matrix adaptation step outside of the belief propagation iteration loop. A reduced-complexity version of the TAB algorithm that offers a trade-off between performance and complexity is also proposed. Simulation results for the turbo decoding of product codes show that belief propagation based on adaptive parity check matrices is a practical alternative to the currently very popular Chase-Pyndiah algorithm.