# A Bayesian Modeling Approach to Situated Design of Personalized Soundscaping Algorithms
## 5 Conclusions
### Source Separation  
A recursive implementation of online informed source separation as described by (3) leads to a generalized Kalman filter, which can be realized by variational message passing. During source separation the inferred model parameters are used for separating the sources. From a graphical perspective, the messages colliding on the edges of \(s_{n}^{k,f}\) result in the marginal beliefs over these latent variables representing the constituents signals in the observed mixture. The mean values of these marginal beliefs are extracted and regarded as separated signals. The choice of the Gaussian mixture model for the individual sources and the corresponding variational approximation has some implications for performing source separation in this model. The variational messages \(\overleftarrow{\
u}\left(s_{n}^{k,f}\right)\) will be Gaussian distributions, because of the variational approximations in the Gaussian mixture node. These local approximations are not always appropriate for source separation and can lead to biased estimates of the posterior selection variables and therefore to biased inference results. To resolve this problem, the source separation problem is approached as a Bayesian model averaging problem [44], which can be generalized for any problem containing multiple mixture models. Alternatively, techniques such as Bayesian model selection or Bayesian model combination [45] can also be used. Here each Gaussian mixture node is expanded into \(D^{k}\) distinct models, in which the Gaussian mixture node is replaced by one of its mixture components. This means that the entire generative model is expanded into \(D_{\text{tot}}=\prod_{k=1}^{K}D^{k}\) distinct models, where \(D^{k}\) represents the number of components used to model the \(k\)th constituent source. In the rest of this section an individual model will be denoted by \(m_{d}\) where \(d\in\mathcal{D}\) encodes a unique combination of mixture components. \(d^{k}\) refers to the original cluster index of the \(k\)th constituent source. The set of all unique combinations is denoted by \(\mathcal{D}\) and has cardinality \(|\mathcal{D}|=D_{\text{tot}}\). With Bayesian model averaging we wish to calculate the posterior distribution \(p(\mathbf{x}\mid\mathbf{y})\) of some latent states \(\mathbf{x}\), which in our case represent the constituent signals, given some observations \(\mathbf{y}\). This posterior distribution can obtained by calculating the posterior distribution for each of the models \(p(\mathbf{x}\mid\mathbf{y},\;m_{d})\) and by \"averaging\" them with the posterior model probability \(p(m_{d}\mid\mathbf{y})\) as  
\[p(\mathbf{x}\mid\mathbf{y})=\sum_{d\in\mathcal{D}}\;p(\mathbf{x}\mid\mathbf{y},\;m_{d})p(m_{d }\mid\mathbf{y}). \tag{10}\]  
In this equation the posterior distribution of the latent states for a given model \(p(\mathbf{x}\mid\mathbf{y},\;m_{d})\) can be obtained by performing probabilistic inference in the model \(m_{d}\). The model posterior \(p(m_{d}\mid\mathbf{y})\) on the other hand can be determined as  
\[p(m_{d}\mid\mathbf{y})=\frac{p(\mathbf{y}\mid m_{d})p(m_{d})}{\sum_{d\in\mathcal{D}}p( \mathbf{y}\mid m_{d})p(m_{d})}, \tag{11}\]  
where \(p(m_{d})\) specifies the prior probability of the model \(m_{d}\) and where \(p(\mathbf{y}\mid m_{d})\) denotes the model evidence of model \(m_{d}\). In the proposed models, all required quantities \(p(\mathbf{x}\mid\mathbf{y},\;m_{d})\), \(p(\mathbf{y}\mid m_{d})\) and \(p(m_{d})\) are not easily computable in their exact form, because of intractability's in the model.