For debugging and verification of computer vision convolutional deep neural networks (CNNs) human inspection of the learned latent representations is imperative. Therefore, state-of-the-art eXplainable Artificial Intelligence (XAI) methods globally associate given natural language semantic concepts with representing vectors or regions in the CNN latent space supporting manual inspection. Yet, this approach comes with two major disadvantages: They are locally inaccurate when reconstructing a concept label and discard information about the distribution of concept instance representations. The latter, though, is of particular interest for debugging, like finding and understanding outliers, learned notions of sub-concepts, and concept confusion. Furthermore, current single-layer approaches neglect that information about a concept may be spread over the CNN depth. To overcome these shortcomings, we introduce the local-to-global Guided Concept Projection Vectors (GCPV) approach: It (1) generates local concept vectors that each precisely reconstruct a concept segmentation label, and then (2) generalizes these to global concept and even sub-concept vectors by means of hiearchical clustering. Our experiments on object detectors demonstrate improved performance compared to the state-of-the-art, the benefit of multi-layer concept vectors, and robustness against low-quality concept segmentation labels. Finally, we demonstrate that GCPVs can be applied to find root causes for confusion of concepts like bus and truck, and reveal interesting concept-level outliers. Thus, GCPVs pose a promising step towards interpretable model debugging and informed data improvement.

We define a copula process which describes the dependencies between arbitrarily many random variables independently of their marginal distributions. As an example, we develop a stochastic volatility model, Gaussian Copula Process Volatility (GCPV), to predict the latent standard deviations of a sequence of random variables. To make predictions we use Bayesian inference, with the Laplace approximation, and with Markov chain Monte Carlo as an alternative. We find our model can outperform GARCH on simulated and financial data. And unlike GARCH, GCPV can easily handle missing data, incorporate covariates other than time, and model a rich class of covariance structures.

We define a copula process which describes the dependencies between arbitrarily many random variables independently of their marginal distributions. As an example, we develop a stochastic volatility model, Gaussian Copula Process Volatility (GCPV), to predict the latent standard deviations of a sequence of random variables. To make predictions we use Bayesian inference, with the Laplace approximation, and with Markov chain Monte Carlo as an alternative. We find our model can outperform GARCH on simulated and financial data. And unlike GARCH, GCPV can easily handle missing data, incorporate covariates other than time, and model a rich class of covariance structures.

We define a copula process which describes the dependencies between arbitrarily many random variables independently of their marginal distributions. As an example, we develop a stochastic volatility model, Gaussian Copula Process Volatility (GCPV), to predict the latent standard deviations of a sequence of random variables. To make predictions we use Bayesian inference, with the Laplace approximation, and with Markov chain Monte Carlo as an alternative. We find our model can outperform GARCH on simulated and financial data. And unlike GARCH, GCPV can easily handle missing data, incorporate covariates other than time, and model a rich class of covariance structures.