The Source Separation model will be easier to explain if the mechanics of a “cocktail party” are ﬁrst described. The cocktail party is an easily understood example of where the Source Separation model can be applied. There are many other applications where the Source Separation model is appropriate. At a cocktail party there are partygoers or speakers holding conversations while at the same time there are microphones recording or observing the speakers also called underlying sources. The cocktail party is illustrated in adapted Figure 1.1. Partygoers, speakers, and sources will be used interchangeably as will be recorded and observed. The cocktail party will be returned to often when describing new concepts or material. At the cocktail party there are typically several small groups of speakers
holding conversations. In each group, typically only one person is speaking at a time. Consider the closest two speakers in Figure 1.1. In this group, person one (left) speaks, then person two (right) speaks, then person one again, and so on. The speakers are obviously negatively correlated. In the Bayesian Source Separation model of this text, the speakers are allowed to be correlated and not constrained to be independent. At a cocktail party, there are p microphones that record or observe m
partygoers or speakers at n time increments. This notation is consistent with traditional Multivariate Statistics. The observed conversations consist of mixtures of true unobservable conversations. A given microphone is not placed to a given speakers’ mouth and is not shielded from the other speakers. The microphones do not observe the speakers’ conversation in isolation. The recorded conversations are mixed. The problem is to unmix or recover the original conversations from the recorded mixed conversations. Consider the following example. There is a party with m = 4 speakers
and p = 3 microphones as seen in Figure 1.2. At time increment i, where i = 1, . . . ,n, the conversation emitted from speaker 1 is si1, speaker 2 is si2, speaker 3 is si3, and speaker 4 is si4. Then, the recorded conversation at microphone 1 is xi1, at microphone 2 is xi2, and at microphone 3 is xi3. There is an unknown function f as illustrated in Figure 1.3 called the mixing
function which takes the emitted source signals and mixes them to produce
The cocktail party.