General Table Completion using a Bayesian Nonparametric Model
Even though heterogeneous databases can be found in a broad variety of applications, there exists a lack of tools for estimating missing data in such databases. In this paper, we provide an efficient and robust table completion tool, based on a Bayesian nonparametric latent feature model. In particular, we propose a general observation model for the Indian buffet process (IBP) adapted to mixed continuous (real-valued and positive real-valued) and discrete (categorical, ordinal and count) observations. Then, we propose an inference algorithm that scales linearly with the number of observations. Finally, our experiments over five real databases show that the proposed approach provides more robust and accurate estimates than the standard IBP and the Bayesian probabilistic matrix factorization with Gaussian observations.
In the manuscript "General Table Completion using a Bayesian Nonparametric Model" the authors propose a method for database completion using the Indian buffet process (IBP) which is a Bayesian nonparametric prior on binary matrices with a fixed number of rows and an unbounded number of columns (which correspond to features of the data associated with the rows). The authors suppose that the columns of the database fall into the following 5 classes: reals, nonnegative reals, discrete, ordinal or integer. For each of these classes, the authors provide a latent IBP model linking the entries of the database with latent features for each row. They find the latent features and likelihood parameters using standard MCMC methods for the IBP and apply their model to predicting missing entries in 5 diverse datasets ranging in size from 20k to 2m entries. This model is important for the NIPS community because it constitutes a very general application of Bayesian nonparametrics. Recently in the machine learning community there has been lots of interest in automatic statistics (see Lloyd et al. "Automatic construction and natural-language description of nonparametric regression models" 2014).In each of the 5  classes, the latent features interact with the likelihood through a Gaussian random variable which is passed through a link function. The covariance matrix of the Gaussian random variable is learned, as are the parameters of the link functions. As the authors point out, this approach is not incredibly innovative (the IBP has been used to model each of these classes separately before, as can be seen by the citations in the manuscript). The novel aspect of the authors' work is a uniform framework in which all of these classes are combined in order to do inference on an arbitrary database. Further, in the experiments I think that the authors give quite good results defending the idea that adding the link functions (i.e., treating the classes differently instead of just versions of Gaussian random variables that just happen to be observed at positive or at integral values) really does add more predictive accuracy.  Another innovative aspect of the manuscript was in the Gibbs updates, which build upon the IBP sampler from Chu and Ghahramani "Gaussian processes for ordinal regression" (2005) which marginalizes over the latent feature assignments of the data items. The authors extend this method to also marginalize over the weighting factors of the features, improving the efficiency of the algorithm.Overall, I thought that this manuscript was good and that it should be accepted. The datasets they applied their model to are new and their results do show that learning link functions for IBP based methods improves upon standard Gaussian models. I did, however think that they should have compared with more methods. For example, they could have compared with methods based on Dirichlet process mixtures such as CrossCat (the likelihoods and link functions used in this paper could also be used to augment the CrossCat model (Shafto et al. 2011) in a way such that it becomes appropriate for the same sort of mixed-class matrices).This manuscript still left some questions about inference based on the IBP model open: the IBP has symmetries arising from simultaneous transformations of the latent feature weights and also the feature assignments. For example, if the sign of all the entries in the feature weight matrix change, and if the feature matrix is simultaneously inverted (so 0s go to 1s and 1s go to 0s) then the likelihood of the data remains unchanged. Because modes in MCMC are attractive, this leads to slower mixing when the MCMC state is `the same distance' between two equivalent modes, it is not pulled strongly to either of them. Unfortunately, the inference method used by the authors did not address this.In the introduction of the manuscript, the authors mention that Bayesian methods can be used to detect errors in databases (i.e., entries where there has been some corruption or mistake in data entry). But I found that they did not actually explore this in the paper. I do think that this is an excellent direction for Bayesian nonparametrics so I was disappointed that it wasn't picked up upon again in the manuscript after the introduction. They could also look at detection of the class of a column, which might not always be obvious by looking at the entries of the column alone. For example, in a database it might turn out that the discrete classes of a column actually have an ordinal relation. Also, a bunch of -9.0000's in a column of real numbers might suggest that those entries are actually missing. Or, a column with both marked missing entries and entries with 0.0000, might actually have the 0.0000 entries through a default, missing value also.I felt like the experiments conducted by the authors demonstracted clearly that using link functions over improves the accuracy. However, I did wish that the authors explored the inference on the experiments more. For example, why does the SIBP (which is an IBP with purely Gaussian assumptions) method outperform the author's method for the `internet usage' dataset? Also, by examining the columns in Figure 3(c) for example, it is clear that there is some structure in the order in which the columns are registered in the database. Why does the BPMF method perform poorly on the first and last columns of the Nesarc dataset?