# The Domain Mismatch Problem in the Broadcast Speaker Attribution Task
## 4 Methodology
### PLDA Tree-Based Clustering Block  
The considered clustering block relies on the idea originally proposed in [36]. It uses a model of the PLDA family to estimate the set of \(N\) labels \(\Theta=[\theta_{1},\dots,\theta_{N}]\) that best explain the set of \(N\) embeddings \(\mathbf{\Phi}=[\mathbf{\phi}_{1},\dots,\mathbf{\phi}_{N}]\). Thus, the label \(\theta_{j}\) indicates the cluster to which the embedding \(\mathbf{\phi}_{j}\) belongs. The proposed approach works based on the Maximum a Posteriori (MAP) estimation of the labels:  
\[\Theta_{clust}=\operatorname*{arg\,max}_{\Theta}P(\Theta|\mathbf{\Phi})= \operatorname*{arg\,max}_{\Theta}\frac{P(\mathbf{\Phi},\Theta)}{P(\mathbf{\Phi})} \tag{2}\]  
Working in terms of the latter expression, the only term depending on the labels is \(P(\mathbf{\Phi},\Theta)\), which can be decomposed according to the product rule of probability as:  
\[P(\mathbf{\Phi},\Theta)=\prod_{j=2}^{N}P\Big{(}\mathbf{\phi}_{j^{\prime}}\theta_{j}| \mathbf{\phi}_{1}^{j-1},\theta_{1}^{j-1}\Big{)}P(\mathbf{\phi}_{1},\theta_{1}) \tag{3}\]  
respectively, being \(\mathbf{\phi}_{1}^{j-1}\) and \(\theta_{1}^{j-1}\) subsets from \(\mathbf{\Phi}\) and \(\Theta\) with the first \(j-1\) elements \(\Big{(}\mathbf{\phi}_{1}^{j-1}=\Big{[}\mathbf{\phi}_{1},\dots,\mathbf{\phi}_{j-1}\Big{]}\) and \(\theta_{1}^{j-1}=\big{[}\theta_{1},\dots,\theta_{j-1}\big{]}\Big{)}\). Inspired by [34], we can make the term \(P\Big{(}\mathbf{\phi}_{j},\theta_{j}|\mathbf{\phi}_{1}^{j-1},\theta_{1}^{j-1}\Big{)}\) more tractable by decomposing it into a conditional distribution depending on the embeddings and a prior distribution for the labels:  
\[P\Big{(}\mathbf{\phi}_{j},\theta_{j}|\mathbf{\phi}_{1}^{j-1},\theta_{1}^{j-1}\Big{)}= P\Big{(}\mathbf{\phi}_{j}|\theta_{j},\mathbf{\phi}_{1}^{j-1},\theta_{1}^{j-1}\Big{)}P \Big{(}\theta_{j}|\theta_{1}^{j-1}\Big{)} \tag{4}\]  
This decomposition isolates a term \(P\Big{(}\mathbf{\phi}_{j}|\theta_{j},\mathbf{\phi}_{1}^{j-1},\theta_{1}^{j-1}\Big{)}\) in which the embedding \(j\) depends on its \(j\)th label as well as previous embeddings and labels. As the design choice, we define it as a mixture of simpler distributions controlled by the means of the variable \(\theta_{j}\). We define this variable as a one-hot sample with \(I\) values (\(\theta_{j}=\{\theta_{1j},\dots,\theta_{ij},..\theta_{lj}\}\)), where I is the number of candidate clusters. Hence, the \(i\)th component of \(\theta_{j}\) takes the value of one exclusively if the embedding \(j\) belongs to cluster \(i\). Additionally, we also impose the restriction that the embedding \(j\), when belonging to the cluster \(i\), should be exclusively explained by those embeddings already assigned to this cluster. This subset of previous embeddings assigned to cluster \(i\) at time \(j\) is denoted as \(\mathbf{\Phi}_{ij}\). Under these conditions:  
\[P\Big{(}\mathbf{\phi}_{j}|\theta_{j},\mathbf{\phi}_{1}^{j-1},\theta_{1}^{j-1}\Big{)}= \prod_{i=1}^{I}P\Big{(}\mathbf{\phi}_{j}|\mathbf{\Phi}_{ij}\Big{)}^{\theta_{ij}} \tag{5}\]  
As we described before, we want this model to be based on the PLDA family. This family of models makes use of a latent variable \(\mathbf{y}_{i}\) that is shared by all elements from the same cluster. Consequently, we can redefine \(P\Big{(}\mathbf{\phi}_{j}|\mathbf{\Phi}_{ij}\Big{)}\) as:  
\[P\Big{(}\mathbf{\phi}_{j}|\mathbf{\Phi}_{ij}\Big{)}=\int P\Big{(}\mathbf{\phi}_{j}|\mathbf{ y}_{i}\Big{)}P\big{(}\mathbf{y}_{i}|\mathbf{\Phi}_{ij}\Big{)}d\mathbf{y}_{i} \tag{6}\]  
By going backwards, we have obtained two familiar distributions when dealing with the PLDA family: \(P\Big{(}\mathbf{\phi}_{j}|\mathbf{y}_{i}\Big{)}\) and \(P\big{(}\mathbf{y}_{i}|\mathbf{\Phi}_{ij}\big{)}\). In our formulations, we consider our model to be a Fully Bayesian PLDA (FBPLDA) [50] with a single latent variable.