trajectory can be computed by exhaustive enumeration. The wish to stress that the main aim of the simulations on synthetic dataset is to compare effectiveness of different inference techniques; we postpone the actual test whether the model is a good one to our simulations on real data.

We have tested the MCMC methods, namely Gibbs sampling (Gibbs), simulated annnomer to the control and for 10 and 50 sweeps. For each onset y k , the optimal proposal p ( γ k | : ) is computed always on a fixed set, Γ = {0,1/4,2/4 . . . 3}. Figure 6 shows a typical run of MCMC.

Similarly, we have implemented the SMC for N = {1,5,10,50,100} particles. The selection schema was random drawing from the optimal proposal p ( γ k | ) computed using one or two time slices.  Only in the special case of greedy filtering (GF), i.e., when N = 1, we have selected the switch with maximum probability. An example run is shown in Figure 5.

We observe that on average SMC results are superior to MCMC (Figure 7).  We observe that, increasing the number of sweeps for MCMC does not improve the solution significantly. On the other hand, increasing the number of particles seems to improve the quality of the SMC solution monotonically. Moreover, the results suggest that sampling from two time slices jointly (with the exception of SA ) does not have a big effect.  GF outperforms a particle filter with 5 particles that draws randomly from the proposal. That suggests that for PF with a small number of particles N, it may be desirable to use a hybrid selection schema that selects the particle with maximum weight automatically and randomly selects the remaining N - 1.

We compare inference methods in terms of execution time and the quality of solutions (as measured by edit distance). As Figure 8 suggests, using a two slice proposal is not justified. Moreover it seems that for comparable computational effort, SMC tends to outperform all MC methods.

## 4.2  Real Data:  Beatles

We evaluate the performance of the model on polyphonic piano performances. 12 pianists were invited to play two Beatles songs, Michelle and Yesterday.  Both pieces have a relatively simple rhythmic structure with ample opportunity to add expressiveness by fluctuating the tempo. The original score is shown in Figure 9(a). The subjects had different musical education and background: four professional jazz players, four professional classical performers and four amateur classical pianists. Each arrangement had to be played in three tempo conditions, three repetitions per tempo condition. The tempo conditions were normal, slow and fast tempo, all in a musically realistic range and all according to the judgment of the performer. Further details are reported in (Cemgil et al., 2001).

## 4.2.1 Preprocessing

The original performances contained several errors, such as missing notes or additional notes that were not on the original score. Such errors are eliminated by using a matching technique (Heijink, Desain, & Honing, 2000) based on dynamical programming.  However, visual inspection of the resulting dataset suggested still several matching errors that we interpret as outliers. To remove these outliers, we have extended the quantization model with a two state switching observation model, i.e., the discrete space consists of ( γ k, i k ). In this simple