Testing For Typicality with Respect to an Ensemble of Learned Distributions
Good methods of performing anomaly detection on high-dimensional data sets are needed, since algorithms which are trained on data are only expected to perform well on data that is similar to the training data. There are theoretical results on the ability to detect if a population of data is likely to come from a known base distribution, which is known as the goodness-of-fit problem, but those results require knowing a model of the base distribution. The ability to correctly reject anomalous data hinges on the accuracy of the model of the base distribution. For high dimensional data, learning an accurate-enough model of the base distribution such that anomaly detection works reliably is very challenging, as many researchers have noted in recent years. Existing methods for the goodness-of-fit problem do not ac- count for the fact that a model of the base distribution is learned. To address that gap, we offer a theoretically motivated approach to account for the density learning procedure. In particular, we propose training an ensemble of density models, considering data to be anomalous if the data is anomalous with respect to any member of the ensemble. We provide a theoretical justification for this approach, proving first that a test on typicality is a valid approach to the goodness-of-fit problem, and then proving that for a correctly constructed ensemble of models, the intersection of typical sets of the models lies in the interior of the typical set of the base distribution. We present our method in the context of an example on synthetic data in which the effects we consider can easily be seen.
Summary: This paper analyzes and extends a recently proposed goodness-of-fit test based on typicality [Nalisnick et al., ArXiv 2019]. Firstly, the authors give bounds on the type-II error of this test, showing it can be characterized as a function of KLD[q || p_true] where p is the true data generating process and q is an alternative generative process. The paper then shifts to the main contribution: an in-depth study of a Gaussian mixture simulation along with accompanying theoretical results. The simulation shows that maximum likelihood estimation (MLE)---due to it optimizing KLD[p_true || p_model]---does not penalize the model for placing probability in places not occupied by p_true. This means that while samples from p_true should fall within the model’s typical set, the model typical set may be broader than p_true’s. Table 1 makes this clear by showing that only 30-40% of samples from the model fall within the typical set of p_true. Yet >93% of samples from p_true fall within the models’ typical sets. The paper then makes the observation that the models do not have high overlap in their typical sets, and thus p_true’s typical set could be well approximated by the intersection of the various models’ typical sets. Applying this procedure to the Gaussian mixture simulation, the authors observe that ~95% of samples drawn from the intersection of the ensemble fall within p_true’s typical set. Moreover, ~97% of samples from p_true are in the ensemble (intersection) typical set. The paper closes by proving that the diversity of the ensemble controls the overlap in their typical sets, and hence increasing diversity should only improve the approximation of p_true’s typical set. ____ Pros: This paper contributes some interesting ideas to a recent topic of interest in the community---namely, that deep generative models assign high likelihood to out-of-distribution (OOD) data [Nalisnick et al., ICLR 2019] and how should we address this problem if we are to use them for anomaly detection, model validation [Bishop, 1994], etc. This paper makes some careful distinctions between the true data process, the model, and the alternative distribution, which I have not seen done often in this literature. And while the mass-covering effect of MLE on the resulting model fit is well known, this paper is the first with which I am aware that translates that fact into a practical recommendation (i.e. their intersection method). Furthermore, this connection to ensembling may provide important theoretical grounding to other ensemble-based methods for OOD detection [Choi et al., ArXiv 2019]. ____ Cons: The primary deficiency in the paper is experimental. While the text does make some compelling arguments in the Gaussian mixture simulations, some validation on real data must be provided. Ideally experiments on CIFAR-10 vs SVHN (OOD) and FashionMNIST vs MNIST (OOD) should be reported as these data set pairings have become the benchmark cases in this line of literature. Besides the lack of experiments on real data, I find the paper’s material to be a bit disjointed and ununified. For instance, Theorem 1 is never discussed again after it is presented in Section 2.1. I thought for sure the presence of the KLD-term would be referenced again to relate the ensembling methodology back to the bound on the type-II error. For another example, normalizing flows are discussed in Section 2.3 and the change-of-variables formula given in Equation 5. However, normalizing flows are never mentioned again except in passing in the Related Work section. ____ Final Evaluation: While I find the paper to contain interesting ideas, it is too unfinished for me to recommend acceptance at this time. Experiments on real data must be included and the overall coherence of the draft improved.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Testing For Typicality with Respect to an Ensemble of Learned Distributions
Good methods of performing anomaly detection on high-dimensional data sets are needed, since algorithms which are trained on data are only expected to perform well on data that is similar to the training data. There are theoretical results on the ability to detect if a population of data is likely to come from a known base distribution, which is known as the goodness-of-fit problem, but those results require knowing a model of the base distribution. The ability to correctly reject anomalous data hinges on the accuracy of the model of the base distribution. For high dimensional data, learning an accurate-enough model of the base distribution such that anomaly detection works reliably is very challenging, as many researchers have noted in recent years. Existing methods for the goodness-of-fit problem do not ac- count for the fact that a model of the base distribution is learned. To address that gap, we offer a theoretically motivated approach to account for the density learning procedure. In particular, we propose training an ensemble of density models, considering data to be anomalous if the data is anomalous with respect to any member of the ensemble. We provide a theoretical justification for this approach, proving first that a test on typicality is a valid approach to the goodness-of-fit problem, and then proving that for a correctly constructed ensemble of models, the intersection of typical sets of the models lies in the interior of the typical set of the base distribution. We present our method in the context of an example on synthetic data in which the effects we consider can easily be seen.
user
<think>Okay, to do a decent peer review, I should start with summarizing what the work is about. This includes novelty, contributions, clarity, and impact. I might include suggestions to issues that I find critical to the usefulness of the work. Starting with novelty, there are different types of novelty. The first aspect of novelty is scope and scale. Is the work covering a broader perspective than previous works? The second aspect is methodology. Is the proposed methodology novel compared to previous methods? Do they provide any significant advantage over previous methodologies? The third aspect is results. Are the results different compared previous work? Does this lead to any novel theories or an extension of current theories? Next is clarity, I will be reading the work in depth and analyze if any sentences can be potentially ambiguous or misleading. If any sentences are ambiguous, I will state them and give a suggestion on how to rephrase such sentence or how to make more clear to the audience. Lastly, impact is important. I will attempt to predict how such work will impact the scientific community. This might include the applications of such work and how the work might transform the way we study a certain topic. Finally, I will might only mention issues that are critical. This is to keep the review as concise and helpful as possible.</think>



Summary: This paper analyzes and extends a recently proposed goodness-of-fit test based on typicality [Nalisnick et al., ArXiv 2019]. Firstly, the authors give bounds on the type-II error of this test, showing it can be characterized as a function of KLD[q || p_true] where p is the true data generating process and q is an alternative generative process. The paper then shifts to the main contribution: an in-depth study of a Gaussian mixture simulation along with accompanying theoretical results. The simulation shows that maximum likelihood estimation (MLE)---due to it optimizing KLD[p_true || p_model]---does not penalize the model for placing probability in places not occupied by p_true. This means that while samples from p_true should fall within the model’s typical set, the model typical set may be broader than p_true’s. Table 1 makes this clear by showing that only 30-40% of samples from the model fall within the typical set of p_true. Yet >93% of samples from p_true fall within the models’ typical sets. The paper then makes the observation that the models do not have high overlap in their typical sets, and thus p_true’s typical set could be well approximated by the intersection of the various models’ typical sets. Applying this procedure to the Gaussian mixture simulation, the authors observe that ~95% of samples drawn from the intersection of the ensemble fall within p_true’s typical set. Moreover, ~97% of samples from p_true are in the ensemble (intersection) typical set. The paper closes by proving that the diversity of the ensemble controls the overlap in their typical sets, and hence increasing diversity should only improve the approximation of p_true’s typical set. ____ Pros: This paper contributes some interesting ideas to a recent topic of interest in the community---namely, that deep generative models assign high likelihood to out-of-distribution (OOD) data [Nalisnick et al., ICLR 2019] and how should we address this problem if we are to use them for anomaly detection, model validation [Bishop, 1994], etc. This paper makes some careful distinctions between the true data process, the model, and the alternative distribution, which I have not seen done often in this literature. And while the mass-covering effect of MLE on the resulting model fit is well known, this paper is the first with which I am aware that translates that fact into a practical recommendation (i.e. their intersection method). Furthermore, this connection to ensembling may provide important theoretical grounding to other ensemble-based methods for OOD detection [Choi et al., ArXiv 2019]. ____ Cons: The primary deficiency in the paper is experimental. While the text does make some compelling arguments in the Gaussian mixture simulations, some validation on real data must be provided. Ideally experiments on CIFAR-10 vs SVHN (OOD) and FashionMNIST vs MNIST (OOD) should be reported as these data set pairings have become the benchmark cases in this line of literature. Besides the lack of experiments on real data, I find the paper’s material to be a bit disjointed and ununified. For instance, Theorem 1 is never discussed again after it is presented in Section 2.1. I thought for sure the presence of the KLD-term would be referenced again to relate the ensembling methodology back to the bound on the type-II error. For another example, normalizing flows are discussed in Section 2.3 and the change-of-variables formula given in Equation 5. However, normalizing flows are never mentioned again except in passing in the Related Work section. ____ Final Evaluation: While I find the paper to contain interesting ideas, it is too unfinished for me to recommend acceptance at this time. Experiments on real data must be included and the overall coherence of the draft improved.
assistant
Summary: This paper analyzes and extends a recently proposed goodness-of-fit test based on typicality [Nalisnick et al., ArXiv 2019]. Firstly, the authors give bounds on the type-II error of this test, showing it can be characterized as a function of KLD[q || p_true] where p is the true data generating process and q is an alternative generative process. The paper then shifts to the main contribution: an in-depth study of a Gaussian mixture simulation along with accompanying theoretical results. The simulation shows that maximum likelihood estimation (MLE)---due to it optimizing KLD[p_true || p_model]---does not penalize the model for placing probability in places not occupied by p_true. This means that while samples from p_true should fall within the model’s typical set, the model typical set may be broader than p_true’s. Table 1 makes this clear by showing that only 30-40% of samples from the model fall within the typical set of p_true. Yet >93% of samples from p_true fall within the models’ typical sets. The paper then makes the observation that the models do not have high overlap in their typical sets, and thus p_true’s typical set could be well approximated by the intersection of the various models’ typical sets. Applying this procedure to the Gaussian mixture simulation, the authors observe that ~95% of samples drawn from the intersection of the ensemble fall within p_true’s typical set. Moreover, ~97% of samples from p_true are in the ensemble (intersection) typical set. The paper closes by proving that the diversity of the ensemble controls the overlap in their typical sets, and hence increasing diversity should only improve the approximation of p_true’s typical set. ____ Pros: This paper contributes some interesting ideas to a recent topic of interest in the community---namely, that deep generative models assign high likelihood to out-of-distribution (OOD) data [Nalisnick et al., ICLR 2019] and how should we address this problem if we are to use them for anomaly detection, model validation [Bishop, 1994], etc. This paper makes some careful distinctions between the true data process, the model, and the alternative distribution, which I have not seen done often in this literature. And while the mass-covering effect of MLE on the resulting model fit is well known, this paper is the first with which I am aware that translates that fact into a practical recommendation (i.e. their intersection method). Furthermore, this connection to ensembling may provide important theoretical grounding to other ensemble-based methods for OOD detection [Choi et al., ArXiv 2019]. ____ Cons: The primary deficiency in the paper is experimental. While the text does make some compelling arguments in the Gaussian mixture simulations, some validation on real data must be provided. Ideally experiments on CIFAR-10 vs SVHN (OOD) and FashionMNIST vs MNIST (OOD) should be reported as these data set pairings have become the benchmark cases in this line of literature. Besides the lack of experiments on real data, I find the paper’s material to be a bit disjointed and ununified. For instance, Theorem 1 is never discussed again after it is presented in Section 2.1. I thought for sure the presence of the KLD-term would be referenced again to relate the ensembling methodology back to the bound on the type-II error. For another example, normalizing flows are discussed in Section 2.3 and the change-of-variables formula given in Equation 5. However, normalizing flows are never mentioned again except in passing in the Related Work section. ____ Final Evaluation: While I find the paper to contain interesting ideas, it is too unfinished for me to recommend acceptance at this time. Experiments on real data must be included and the overall coherence of the draft improved.