**Observable Operator Model (OOM)**

Observable Operator Models (OOMs) were introduced by Jaeger as a generalization of hidden Markov models (HMMs). The theory of OOMs makes use of both probabilistic and linear algebraic tools, which has an important advantage: using the tools of linear algebra a very simple and efficient learning algorithm can be developed for OOMs. This seems to be better than the known algorithms for HMMs.

A widely used class of models for stochastic systems is hidden Markov models. Systems that can be modeled by hidden Markov models are a proper subclass of linearly dependent processes, a class of stochastic systems known from mathematical investigations carried out over the past four decades. This article provides a novel, simple characterization of linearly dependent processes, called observable operator models . The mathematical properties of observable operator models lead to a constructive learning algorithm for the identification of linearly dependent processes. The core of the algorithm has a time complexity of O(N + nm3), where N is the size of training data, n is the number of distinguishable outcomes of observations, and m is model state-space dimension.

A short introduction to observable operator models of discrete stochastic processes

Observable Operator Models for Discrete Stochastic Time Series

A Consistent Method for Learning OOMs from Asymptotically Stationary Time Series Data Containing Missing Values

➘ “Partially Observable Markov Decision Process” … **Propheticus**

Due to recent technological developments, Machine Learning (ML), a subfield of Artificial Intelligence (AI), has been successfully used to process and extract knowledge from a variety of complex problems. However, a thorough ML approach is complex and highly dependent on the problem at hand. Additionally, implementing the logic required to execute the experiments is no small nor trivial deed, consequentially increasing the probability of faulty code which can compromise the results. Propheticus is a data-driven framework which results of the need for a tool that abstracts some of the inherent complexity of ML, whilst being easy to understand and use, as well as to adapt and expand to assist the user’s specific needs. Propheticus systematizes and enforces various complex concepts of an ML experiment workflow, taking into account the nature of both the problem and the data. It contains functionalities to execute all the different tasks, from data preprocessing, to results analysis and comparison. Notwithstanding, it can be fairly easily adapted to different problems due to its flexible architecture, and customized as needed to address the user’s needs. … **Missing Value PC (MVPC)**

Missing data are ubiquitous in many domains such as healthcare. Depending on how they are missing, the (conditional) independence relations in the observed data may be different from those for the complete data generated by the underlying causal process and, as a consequence, simply applying existing causal discovery methods to the observed data may lead to wrong conclusions. It is then essential to extend existing causal discovery approaches to find true underlying causal structure from such incomplete data. In this paper, we aim at solving this problem for data that are missing with different mechanisms, including missing completely at random (MCAR), missing at random (MAR), and missing not at random (MNAR). With missingness mechanisms represented by missingness Graph (m-Graph), we analyze conditions under which addition correction is needed to derive conditional independence/dependence relations in the complete data. Based on our analysis, we propose missing value PC (MVPC), which combines additional corrections with traditional causal discovery algorithm, in particular, PC. Our proposed MVPC is shown in theory to give asymptotically correct results even using data that are MAR and MNAR. Experiment results illustrate that the proposed algorithm can correct the conditional independence for values MCAR, MAR and rather general cases of values MNAR both with synthetic data as well as real-life healthcare application. … **Probabilistic Robustness**

Neural networks are becoming increasingly prevalent in software, and it is therefore important to be able to verify their behavior. Because verifying the correctness of neural networks is extremely challenging, it is common to focus on the verification of other properties of these systems. One important property, in particular, is robustness. Most existing definitions of robustness, however, focus on the worst-case scenario where the inputs are adversarial. Such notions of robustness are too strong, and unlikely to be satisfied by-and verifiable for-practical neural networks. Observing that real-world inputs to neural networks are drawn from non-adversarial probability distributions, we propose a novel notion of robustness: probabilistic robustness, which requires the neural network to be robust with at least $(1 – \epsilon)$ probability with respect to the input distribution. This probabilistic approach is practical and provides a principled way of estimating the robustness of a neural network. We also present an algorithm, based on abstract interpretation and importance sampling, for checking whether a neural network is probabilistically robust. Our algorithm uses abstract interpretation to approximate the behavior of a neural network and compute an overapproximation of the input regions that violate robustness. It then uses importance sampling to counter the effect of such overapproximation and compute an accurate estimate of the probability that the neural network violates the robustness property. …