Transformation-based methods have been an attractive approach in non-parametric inference for problems such as unconditional and conditional density estimation due to their unique hierarchical structure that models the data as flexible transformation of a set of common latent variables. More recently, transformation-based models have been used in variational inference (VI) to construct flexible implicit families of variational distributions. However, their use in both non-parametric inference and variational inference lacks theoretical justification. We provide theoretical justification for the use of non-linear latent variable models (NL-LVMs) in non-parametric inference by showing that the support of the transformation induced prior in the space of densities is sufficiently large in the $L_1$ sense. We also show that, when a Gaussian process (GP) prior is placed on the transformation function, the posterior concentrates at the optimal rate up to a logarithmic factor. Adopting the flexibility demonstrated in the non-parametric setting, we use the NL-LVM to construct an implicit family of variational distributions, deemed GP-IVI. We delineate sufficient conditions under which GP-IVI achieves optimal risk bounds and approximates the true posterior in the sense of the Kullback-Leibler divergence. To the best of our knowledge, this is the first work on providing theoretical guarantees for implicit variational inference.

A structural causal model is made of endogenous (manifest) and exogenous (latent) variables. In a recent paper, it has been shown that endogenous observations induce linear constraints on the probabilities of the exogenous variables. This allows to exactly map a causal model into a \emph{credal network}. Causal inferences, such as interventions and counterfactuals, can consequently be obtained by standard credal network algorithms. These natively return sharp values in the identifiable case, while intervals corresponding to the exact bounds are produced for unidentifiable queries. In this paper we present an approximate characterization of the constraints on the exogenous probabilities. This is based on a specialization of the EM algorithm to the treatment of the missing values in the exogenous observations. Multiple EM runs can be consequently used to describe the causal model as a set of Bayesian networks and, hence, a credal network to be queried for the bounding of unidentifiable queries. Preliminary empirical tests show how this approach might provide good inner bounds with relatively few runs. This is a promising direction for causal analysis in models whose topology prevents a straightforward specification of the credal mapping.

Bayesian change-point detection, together with latent variable models, allows to perform segmentation over high-dimensional time-series. We assume that change-points lie on a lower-dimensional manifold where we aim to infer subsets of discrete latent variables. For this model, full inference is computationally unfeasible and pseudo-observations based on point-estimates are used instead. However, if estimation is not certain enough, change-point detection gets affected. To circumvent this problem, we propose a multinomial sampling methodology that improves the detection rate and reduces the delay while keeping complexity stable and inference analytically tractable. Our experiments show results that outperform the baseline method and we also provide an example oriented to a human behavior study.

Deep generative networks trained via maximum likelihood on a natural image dataset like CIFAR10 often assign high likelihoods to images from datasets with different objects (e.g., SVHN). We refine previous investigations of this failure at anomaly detection for invertible generative networks and provide a clear explanation of it as a combination of model bias and domain prior: Convolutional networks learn similar low-level feature distributions when trained on any natural image dataset and these low-level features dominate the likelihood. Hence, when the discriminative features between inliers and outliers are on a high-level, e.g., object shapes, anomaly detection becomes particularly challenging. To remove the negative impact of model bias and domain prior on detecting high-level differences, we propose two methods, first, using the log likelihood ratios of two identical models, one trained on the in-distribution data (e.g., CIFAR10) and the other one on a more general distribution of images (e.g., 80 Million Tiny Images). We also derive a novel outlier loss for the in-distribution network on samples from the more general distribution to further improve the performance. Secondly, using a multi-scale model like Glow, we show that low-level features are mainly captured at early scales. Therefore, using only the likelihood contribution of the final scale performs remarkably well for detecting high-level feature differences of the out-of-distribution and the in-distribution. This method is especially useful if one does not have access to a suitable general distribution. Overall, our methods achieve strong anomaly detection performance in the unsupervised setting, and only slightly underperform state-of-the-art classifier-based methods in the supervised setting. Code can be found at https://github.com/boschresearch/hierarchical_anomaly_detection.

This paper is a tutorial and literature review on sampling algorithms. We have two main types of sampling in statistics. The first type is survey sampling which draws samples from a set or population. The second type is sampling from probability distribution where we have a probability density or mass function. In this paper, we cover both types of sampling. First, we review some required background on mean squared error, variance, bias, maximum likelihood estimation, Bernoulli, Binomial, and Hypergeometric distributions, the Horvitz-Thompson estimator, and the Markov property. Then, we explain the theory of simple random sampling, bootstrapping, stratified sampling, and cluster sampling. We also briefly introduce multistage sampling, network sampling, and snowball sampling. Afterwards, we switch to sampling from distribution. We explain sampling from cumulative distribution function, Monte Carlo approximation, simple Monte Carlo methods, and Markov Chain Monte Carlo (MCMC) methods. For simple Monte Carlo methods, whose iterations are independent, we cover importance sampling and rejection sampling. For MCMC methods, we cover Metropolis algorithm, Metropolis-Hastings algorithm, Gibbs sampling, and slice sampling. Then, we explain the random walk behaviour of Monte Carlo methods and more efficient Monte Carlo methods, including Hamiltonian (or hybrid) Monte Carlo, Adler's overrelaxation, and ordered overrelaxation. Finally, we summarize the characteristics, pros, and cons of sampling methods compared to each other. This paper can be useful for different fields of statistics, machine learning, reinforcement learning, and computational physics.

A fundamental problem for LHC measurements Among the sciences, particle physics has the luxury of having a very well established theoretical basis. Quantum field theory provides a framework not only for the Standard Model, but also for theories of physics beyond the standard model (BSM). We almost take for granted the predictive power of our theories, but the way our field formulates searches for new new physics in terms of hypothesis tests and confidence intervals is critically tied to the fact that we have predictive models to test in the first place. Often we seem to equate the predictions of a theory with Feynman diagrams and the matrix element for a hard scattering process, which in turn can be used to predict a fully differential cross-section. Of course, that is not the full story, as one must include parton density functions and quarks and gluons give rise to a parton shower and subsequent hadronization process. Moreover, we observe electronic signatures tied to scintillation, ionization, etc. in our detectors, not the final-state particles directly. Therefore the predictive model for a theory must incorporate the response of the detector to the final state particles. While all of these points are well known to an experimental particle physicist, it has not been customary to describe the full simulation chain explicitly as a probabilistic model for the data.

Motivated by decentralized approaches to machine learning, we propose a collaborative Bayesian learning algorithm taking the form of decentralized Langevin dynamics in a non-convex setting. Our analysis show that the initial KL-divergence between the Markov Chain and the target posterior distribution is exponentially decreasing while the error contributions to the overall KL-divergence from the additive noise is decreasing in polynomial time. We further show that the polynomial-term experiences speed-up with number of agents and provide sufficient conditions on the time-varying step-sizes to guarantee convergence to the desired distribution. The performance of the proposed algorithm is evaluated on a wide variety of machine learning tasks. The empirical results show that the performance of individual agents with locally available data is on par with the centralized setting with considerable improvement in the convergence rate.

Modeling the temporal behavior of data is of primordial importance in many scientific and engineering fields. Baseline methods assume that both the dynamic and observation equations follow linear-Gaussian models. However, there are many real-world processes that cannot be characterized by a single linear behavior. Alternatively, it is possible to consider a piecewise-linear model which, combined with a switching mechanism, is well suited when several modes of behavior are needed. Nevertheless, switching dynamical systems are intractable because of their computational complexity increases exponentially with time. In this paper, we propose a variational approximation of piecewise linear dynamical systems. We provide full details of the derivation of two variational expectation-maximization algorithms, a filter and a smoother. We show that the model parameters can be split into two sets, static and dynamic parameters, and that the former parameters can be estimated off-line together with the number of linear modes, or the number of states of the switching variable. We apply the proposed method to a visual tracking problem, namely head-pose tracking, and we thoroughly compare our algorithm with several state of the art trackers.

A Bayesian network is a probabilistic graphical model that consists of a directed acyclic graph (DAG), where each node is a random variable and attached to each node is a conditional probability distribution (CPD). A Bayesian network can be learned from data using the well-known score-and-search approach, and within this approach a key consideration is how to simultaneously learn the global structure in the form of the underlying DAG and the local structure in the CPDs. Several useful forms of local structure have been identified in the literature but thus far the score-and-search approach has only been extended to handle local structure in form of context-specific independence. In this paper, we show how to extend the score-and-search approach to the important and widely useful case of noisy-OR relations. We provide an effective gradient descent algorithm to score a candidate noisy-OR using the widely used BIC score and we provide pruning rules that allow the search to successfully scale to medium sized networks. Our empirical results provide evidence for the success of our approach to learning Bayesian networks that incorporate noisy-OR relations.

The log-likelihood of a generative model often involves both positive and negative terms. For a temporal multivariate point process, the negative term sums over all the possible event types at each time and also integrates over all the possible times. As a result, maximum likelihood estimation is expensive. We show how to instead apply a version of noise-contrastive estimation---a general parameter estimation method with a less expensive stochastic objective. Our specific instantiation of this general idea works out in an interestingly non-trivial way and has provable guarantees for its optimality, consistency and efficiency. On several synthetic and real-world datasets, our method shows benefits: for the model to achieve the same level of log-likelihood on held-out data, our method needs considerably fewer function evaluations and less wall-clock time.