Chris@19: 
Chris@19: I agree with you - having a look at a model that does not support 
Chris@19: convolution would help. You'll find attached 'hnmf.m', which is 
Chris@19: essentially the same model without convolution. So the simplified model 
Chris@19: is: P(w,t) = P(t) \sum_{s,p} P(w|p,s)P(p|t)P(s|p,t)
Chris@19: 
Chris@19: Also, a more recent (and much more efficient) version of the CMJ system 
Chris@19: converts the model from a convolutive to a linear one, but still keeping 
Chris@19: the shift-invariance support. That is achieved by having a pre-extracted 
Chris@19: 4-D dictionary that also supports templates that are pre-shifted across 
Chris@19: log-frequency (so that the system would not need to compute the 
Chris@19: convolutions during the EM step). I have uploaded the source code on 
Chris@19: SoundSoftware [i], and you can find the related paper in [ii]. This 
Chris@19: system has the exact same performance with the CMJ one, but is much 
Chris@19: easier to understand/implement, and is over 50 times faster. 
Chris@19: 
Chris@19: [i] https://code.soundsoftware.ac.uk/projects/amt_mssiplca_fast
Chris@19: [ii] http://www.ecmlpkdd2013.org/wp-content/uploads/2013/09/MLMU_benetos.pdf
Chris@19: 
Chris@19: > In eqn 12,
Chris@19: >    Pt(p) =
Chris@19: >        sum[w,f,s] ( P(p,f,s|w,t) Vw,t ) /
Chris@19: >        sum[p,w,f,s] ( P(p,f,s|w,t) Vw,t )
Chris@19: >
Chris@19: > P(p,f,s|w,t) is the result of the E-step (and a time-frequency
Chris@19: > distribution), and Vw,t is the input spectrogram (also a
Chris@19: > time-frequency distribution), right?
Chris@19: 
Chris@19: Right! Basically, P(p,f,s|w,t) is a 5-dimensional matrix, essentially 
Chris@19: the model without the sums (the sums convert P(p,f,s|w,t) into a 2-D 
Chris@19: matrix P(w,t)).
Chris@19: 
Chris@19: > So I read this as something like: update the pitch probability
Chris@19: > distribution for time t so that its value for a pitch p is the ratio
Chris@19: > of the sum of the expression P(p,f,s|w,t) Vw,t for *that* pitch
Chris@19: > variable to the sum of the same expression across *all* pitch
Chris@19: > variables.
Chris@19: 
Chris@19: The equation you put essentially takes the 5-dimensional quantity 
Chris@19: P(p,f,s|w,t) Vw,t and marginalises it to P(p,t), i.e. it sums over all 
Chris@19: other dimensions. All these 'unknown' parameters, e.g. P(s|p,t), are 
Chris@19: generated from this 5-dimensional posterior distribution.
Chris@19: 
Chris@19: > But what does it mean to refer to P(p,f,s|w,t) for a single pitch
Chris@19: > variable, given that P(p,f,s|w,t) is just a time-frequency
Chris@19: > distribution? There doesn't seem to be any dependence on p in it. I
Chris@19: > think this is where I'm missing the (hopefully obvious) fundamental
Chris@19: > thing.
Chris@19: 
Chris@19: P(p,f,s|w,t) is not a time-frequency distribution; it is a 5-dimensional 
Chris@19: posterior distribution of the 3 unknown model parameters given time and 
Chris@19: frequency.
Chris@19: 
Chris@19: The basic concept of EM is that you have a latent variable in your 
Chris@19: model, e.g. p; in the E-step, you compute the posterior given the 
Chris@19: known/input data (e.g. P(p|w,t)). For the M-step, you compute the 
Chris@19: complete likelihood given the original input, P(p|w,t)Vwt; and you 
Chris@19: marginalise over the variables that you don't care about, e.g. if you 
Chris@19: want to find P(p|t), you compute \sum_w P(p|w,t)V_wt; finally, you 
Chris@19: normalise that result according to your model, so if your model has a 
Chris@19: P(p|t) component, you normalise so that P(p) for a given timeframe sums 
Chris@19: to one (this is the denominator in the equation you showed).
Chris@19: