## 2 Proposed Method
### Model Integration
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{6}{c}{**Bach Chorales**} \\ \cline{2-7}  & \(F_{ba}\) & \(F_{s}\) & \(F_{a}\) & \(F_{t}\) & \(F_{b}\) \\ \hline [PERSON] [49] & 28.12 (4.38) & 24.23 (10.28) & 22.98 (11.85) & 29.35 (12.43) & 35.92 (10.97) \\ [PERSON] and [PERSON] [50] & 24.83 (5.31) & 30.03 (12.63) & 25.24 (10.92) & 21.09 (9.91) & 22.95 (9.30) \\ [PERSON] et al. [4] & 18.30 (4.87) & 13.43 (7.03) & 15.52 (6.50) & 17.14 (6.77) & 27.10 (8.44) \\ [PERSON] and [PERSON] [51] & 44.05 (4.60) & 40.18 (11.28) & 43.34 (7.38) & 41.54 (7.02) & 50.56 (6.16) \\ [PERSON] and [PERSON] [21] & 20.31 (3.40) & 20.42 (5.36) & 21.27 (4.75) & 14.49 (1.37) & 25.05 (2.12) \\ VOCAL4-MP & 21.84 (9.37) & 12.99 (11.23) & 10.27 (10.13) & 22.72 (6.72) & 41.37 (9.41) \\ VOCAL4-VA & **56.49 (10.48)** & **52.37 (12.92)** & **49.13 (11.22)** & **53.10 (11.71)** & **71.38 (6.06)** \\ \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{6}{c}{**Barbershop Quartets**} \\ \cline{2-7}  & \(F_{ba}\) & \(F_{s}\) & \(F_{a}\) & \(F_{t}\) & \(F_{b}\) \\ \[PERSON] [49] & 20.90 (5.79) & 2.53 (4.82) & 29.02 (13.25) & 7.94 (7.48) & 44.09 (14.26) \\ Salamon and Gomez [50] & 20.38 (6.61) & 11.14 (10.27) & 35.14 (14.04) & 8.44 (8.22) & 26.81 (13.69) \\ [PERSON] et al. [4] & 19.13 (8.52) & 10.20 (8.25) & 17.97 (9.03) & 15.93 (8.85) & 32.41 (12.41) \\ [PERSON] and [PERSON] [51] & 37.19 (8.62) & 30.68 (13.94) & **36.15 (11.70)** & 29.15 (13.90) & 52.78 (10.37) \\ [PERSON] and [PERSON] [21] & 23.98 (4.34) & 24.45 (6.36) & 31.61 (6.79) & 13.55 (2.18) & 26.34 (2.03) \\ VOCAL4-MP & 18.35 (7.56) & 2.40 (5.54) & 10.56 (13.92) & 16.61 (7.31) & 43.85 (3.46) \\ VOCAL4-VA & **49.06 (14.65)** & **41.78 (18.78)** & 34.62 (16.29) & **35.59 (16.93)** & **84.25 (6.58)** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Voice assignment results, where standard deviations are shown in parentheses. The post-processing refinement step described in Section 2.1.1 was also run on the output of all cited methods. For those that do not output any voice assignment information ([PERSON] [49], [PERSON] and [PERSON] [50], [PERSON] et al. [4] and [PERSON] and [PERSON] [51]), the music language model was run once on its output with default settings and \(M=4\). VOCAL4-MP represents our proposed method with the acoustic model only. For VOCAL4-MP and [21], voice assignments are derived from each model’s probabilistic voice assignment estimates (\(P_{t}(v|p)\) for [21] and \(P_{t}(p|v)\) for VOCAL4-MP). VOCAL4-VA refers to our fully-integrated model. a uniform distribution over all pitches). The hyperparameter of Equation (18) acts as a soft mask, reweighing the pitch contribution of each voice based on detected pitches from the previous iteration. Performance depends on a proper alignment between the voice types present in each song and the voice types present in the PLCA dictionary. Therefore, we dynamically assign one of the five voice types present in the dictionary (see Section 2.1.2) to each of the voices extracted by the music language model. During the first integrated EM iteration, the acoustic model's voice probabilities \(P_{t}(p|v)\) are set to a uniform distribution upon input to the music language model. Additionally, we cannot be sure which voice types are present in a given song, so we run the language model with \(M=5\). Here, the acoustic model's detections contain many overtones, and we do not want to simply use \(M=4\), because many of the overtones are actually assigned a slightly greater probability than the correct notes by the acoustic model. Rather, the overtones tend to be higher in pitch than the correct notes and, thus, are almost exclusively assigned to the fifth voice by the HMM. These decisions combined allow the music language model to drive the acoustic model towards the correct decomposition without being influenced by the acoustic model's initially noisy voice type probabilities.