Nonparametric Bayesian Discrete Latent Variable Models for ...

4.5.4 Conclusions
4.6 Discussion
EBA is a choice model which has correspondences to several models in economics and
psychology. The model assumes the choice probabilities to result from the non-shared
features of the options. The usefulness of the EBA model has been hampered by the lack
of a method that can accurately infer the unknown features of the alternatives. We have
suggested to use an infinite latent feature matrix to represent the unknown features. We
showed empirically that the infinite latent feature model (iEBA) can capture the latent
structure in the choice data as well as the handcrafted model (tEBA). For data for
which we have less prior information it might not be possible to handcraft a reasonable
feature matrix.
Different feature matrices can result in the same choice probabilities and therefore are
not distinguished by the model. For example, features that are shared by all options
do not affect the likelihood. Furthermore, only the ratio of the weights affects the
likelihood. What seems to be a non-identifiability problem is not an issue for sampling
since we are interested in inferring the choice probabilities, not the ”true” features.
On a more conceptual side, the non-identifiability of the features makes the samples
from the posterior hard to interpret. For instance, for the celebrities data one might
have hoped to find feature matrices that correspond to a tree or at least find matrices
with some other directly interpretable structure. However, although the experiment by
Rumelhart and Greeno (1971) was designed with the tree structure in mind Figure 4.16,
we do not know the true latent features that lead to the choices of the subjects. Nevertheless,
we can use the posterior to predict future choices from past data, assess the
similarity of the options and cluster or rank them.
The EBA model does not take into account neither the time ordering of choice outcomes
nor the identities of the choice makers. For some choice scenarios this information
is irrelevant or not available. However, a model that can make use of the additional information
when available would potentially be more powerful in representing the structure.
This is a direction that the EBA model can be improved.
Another possible improvement of the EBA model would be to take into account the
shared features as well when calculating the choice probabilities. One possibility is to
add to eq. (4.63) the effect of the shared features with a scaling factor s ∈ [0, 1]. The
EBA model would be recovered for s = 0, whereas the BTL model would be obtained
when the similar features are treated the same as the differences, that is, with s = 1.
Using a small but nonzero scale value may improve recovering the latent structure of the
alternatives especially when there are pairs of alternatives that have not been compared.
4.6 Discussion
The Indian buffet process has been recently introduced by Griffiths and Ghahramani
(2005), and rapidly attracted interest due to its conceptual simplicity and promising
flexibility for defining nonparametric latent feature models. Its relation to the DP and
related distributions has inspired models and inference algorithms from the DP literature
to be adjusted to the IBP models. The different approaches with which the equivalent
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