16 · A. Plangprasopchok and K. Lerman

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In this work, we choose another solution by extending our finite model to have "countably" infinite numbers of topics and interests. By "countably" infinite number of components, we mean that such numbers are flexible and can vary according to the number of observations. Intuitively, there is a higher chance that more topics and interests will be found in a data set that has more resources and users. Such unbounded number of components can be dealt with within a Bayesian framework, as mentioned in the previous works
[Neal 2000; Rasmussen 2000; Teh et al. 2006]. This approach helps bypass the problem of selecting values for these parameters.

Following [Neal 2000], we set both NZ and NX to approach ∞. This will give the model the ability to select not only previously used topic/interest components but also to instantiate "unused" components when required. However, the model that we derived in the previous section cannot be extended directly under this framework due to the use of symmetric Dirichlet priors. As pointed out by [Teh et al. 2006], when the number of components grows, using the symmetric Dirichlet prior results in a very low - even zero probability - chance that a mixture component is shared across groups of data. That is, in our context, there is a higher chance that a certain topic is only used within one resource rather than utilized by many of them. Considering Eq. (1), if we set NZ to approach ∞,
we can obtain posterior probability of z as follows ACM Journal Name, Vol. x, No. y, zz 2010.