Restricting exchangeable nonparametric distributions
Distributions over matrices with exchangeable rows and infinitely many columns are useful in constructing nonparametric latent variable models. However, the distribution implied by such models over the number of features exhibited by each data point may be poorly-suited for many modeling tasks. In this paper, we propose a class of exchangeable nonparametric priors obtained by restricting the domain of existing models. Such models allow us to specify the distribution over the number of features per data point, and can achieve better performance on data sets where the number of features is not well-modeled by the original distribution.
This is an interesting paper describing mechanism to 
      introduce constraints of the priors 1) I have two concerns with 
      how the discussion is formulated: 1.1) The introduction of the paper 
      talks about preserving exchangeability, and a substantial effort is 
      devoted to showing that restrictions on the predictive distributions do 
      not necessarily lead to exchangeable rows. However, given that the authors 
      are working with priors on array-valued processes, they should have framed 
      the discussion in terms of separate exchangeability (Aldous, 1981), which 
      is what we really desired from the model. I think that all the proofs 
      works and the models described are indeed separately exchangeable (rather 
      than unrestrictedly exchangeable), but the current presentation is not 
      really adequate. 1.2) An implicit assumption is that the 
      distribution on $\theta$ admits a density (denoted by $\nu$) with respect 
      to some dominating measure. Besides not stating this clearly, the authors 
      refer to $\nu$ as a measure in various parts of the manuscript. 
      Furthermore, the authors refer to $\theta$ as the "directing measure". 
      $\theta$ is a collection of probabilities, but they do not add up to 1, so 
      talking about a measure here seems incorrect. I think that precision in 
      the use of standard terminology is important. 1.3) In equations 
      (3) and (4), I believe that the authors should also multiply the final 
      expressions by the appropriate indicator (as the support is now 
      constrained) 2) Beyond being a mathematically interesting problem, 
      the motivation for introducing the constraints is not very strong. The 
      best motivation might come from the discussion on HMM models, but even 
      that is relatively week. I guess that the authors could argue that 
      obtaining better results in the experiment in Section 5.2 is motivation 
      enough, but having some intuition on why the proposed model outperforms 
      would have been useful. 3) There is no discussion about how the 
      size of the truncation is to be selected. Now that constraints have been 
      introduced, standard results will not be applicable (particularly if f(S) 
      in equation (5) is heavy tailed). 4) Section 3.2 emphasizes that 
      directly restricting the predictive distributions does not lead to an 
      exchangeable model. The example is nice, but this is a well known property 
      and I am not sure that it needed this much space.