In this work, trellis based source coding using the MAP criterion is described. The motivation for this criterion is given by rate-distortion theory as well as by the duality between trellis source encoding and decoding of convolutional channel codes. We explain the idea of using tailbiting sequences to combat the negative e?ects of a ?xed initial state or an explicit storage of the initial state. The tailbiting BCJR algorithm is described and its convergence properties are highlighted. In order to ensure tailbiting sequences from the symbol-by-symbol output, the path extension method is introduced. Simulation results reveal that a certain blocklength is required for optimal encoding performance. The comparison to the Viterbi algorithm indicates a superior performance of the TB-BCJR algorithm for short blocks. In addition to that, a way for encoding multiple correlated sources is presented. The approach is based on using vector labels instead of scalar labels on the state transitions. We present simulation results for encoding of two correlated Gaussian sources using the TB-BCJR algorithm.