Despite the huge success of deep neural networks (NNs), finding good mechanisms for quantifying their prediction uncertainty is still an open problem. It was recently shown, that using an ensemble of NNs trained with a proper scoring rule leads to results competitive to those of Bayesian NNs. This ensemble method can be understood as finite mixture model with uniform mixing weights. We build on this mixture model approach and increase its flexibility by replacing the fixed mixing weights by an adaptive, input-dependent distribution (specifying the probability of each component) represented by an NN, and by considering uncountably many mixture components. The resulting model can be seen as the continuous counterpart to mixture density networks and is therefore referred to as compound density networks. We empirically show that the proposed model results in better uncertainty estimates and is more robust to adversarial examples than previous approaches.
In this work the authors propose an extension of mixture density networks to the continuous domain, named compound density networks. Specifically the paper builds on top of the idea of the ensemble neural networks (NNs) and introduces a stochastic neural network for handling the mixing components. The mixing distribution is also parameterised by a neural network. The authors claim that the proposed model can result in better uncertainty estimates and the experiments attempt to demonstrate the benefits of the approach, especially in cases of having to deal with adversarial attacks.
The paper in general is well written and easy to follow. I have some concerns regarding the presentation of the main objective and the lack of justification in certain parts of the methodology. Let me elaborate. First of all, I don’t understand how the main equation of the compound density network in Equation (3) is different from the general case of a Bayesian neural network? Can the authors please comment on that?
I also find weird the way that the authors arrive to their final objective in Equation (5). They start from Equation (4) which is incorrectly denoted as the log-marginal distribution while it is the same conditional distribution introduced in Equation (3) with the extra summation for all the available data points. Then they continue to Equation (5) which they present as the combination of the true likelihood with a KL regularisation term. However, what the authors implicitly did was to perform variational inference for maximising their likelihood by introducing a variational distribution q(\theta) = p(\theta | g(x_n; \psi). Is there a reason why the authors do not introduce their objective by following the variational framework?
Furthermore, in the beginning of Section 3.1 the authors present their idea on probabilistic hypernetoworks which “maps x to a distribution over parameters instead of specific value \theta.” How is this different from the case that we were considering so far? If we had a point estimate for \theta we would not require to take an expectation in Equation (3) in the first place.
My biggest concern in the methodology, however, has to do with the selection of the matrix variate normal prior for the weights and the imposition of diagonal covariances (diag(a) and diag(b)). The Kronecker product between two diagonal matrices results in another diagonal matrix, i.e., diagonal covariance, which implies that the weights within a layer are given by an independent multivariate Gaussian. What is the purpose then for introducing the matrix variate Gaussian? I would expect that you would like to impose additional structure to the weights. I expect the authors to comment on that.
Regarding the experimental evaluation of the model rather confusing. The authors have proposed a model that due to the mixing is better suited for predictions with heteroscedastic noise and can better quantify the aleatoric uncertainty. However, the selected experiments on the cubic regression toy data (Section 5.1) and the out-of-distribution classification (Section 5.2) are clear examples of system’s noise, i.e. epistemic uncertainty. The generative process of the toy data clearly states that there is no heteroscedastic noise to handle. The same applies for the notMNIST data which belong to a completely different data set compared to MNIST and thus out of sample prediction cannot benefit from the mixing; i.e., variations have to be explained by system’s noise. So overall I have the feeling that the authors have not succeeded to evaluate the model’s power with these two experiments and we cannot draw any strong conclusions regarding the benefit of the proposed mixing approach.
To continue with the experimental evaluation, I found the plots with the predictive uncertainty in Figure 3 a bit confusing. The plot by itself, as I understood, quantifies the model’s uncertainty in in- and out-of sample prediction. While I agree with the authors that it is generally desirable for a model to be more confident when predicting in MNIST (since it has already seen samples of it) compared to when predicting in notMNIST (completely different data), these plots tells us nothing regarding the predictive power of the model. There is no value in being very confident if you are wrong and vice-versa, so unless there is an accompanying plot/table reporting the accuracy I see not much value from this plot alone.
Finally, it is unclear how the authors have picked the best \lambda parameter for their approach? On page 5 they state that they “pick the value that results in a good trade-off between high uncertainty estimates and high prediction accuracy.” Does this mean that you get to observe the performance in the test in order to select the appropriate value for \lambda? If this is the case this is completely undesirable and is considered a bad practice.