We introduce ADAPQUEST, a software tool written in Java for the development of adaptive questionnaires based on Bayesian networks. Adaptiveness is intended here as the dynamical choice of the question sequence on the basis of an evolving model of the skill level of the test taker. Bayesian networks offer a flexible and highly interpretable framework to describe such testing process, especially when coping with multiple skills. ADAPQUEST embeds dedicated elicitation strategies to simplify the elicitation of the questionnaire parameters. An application of this tool for the diagnosis of mental disorders is also discussed together with some implementation details.

In this work, we examine the problem of online decision making under uncertainty, which we formulate as planning in the belief space. Maintaining beliefs (i.e., distributions) over high-dimensional states (e.g., entire trajectories) was not only shown to significantly improve accuracy, but also allows planning with information-theoretic objectives, as required for the tasks of active SLAM and information gathering. Nonetheless, planning under this "smoothing" paradigm holds a high computational complexity, which makes it challenging for online solution. Thus, we suggest the following idea: before planning, perform a standalone state variable reordering procedure on the initial belief, and "push forwards" all the predicted loop closing variables. Since the initial variable order determines which subset of them would be affected by incoming updates, such reordering allows us to minimize the total number of affected variables, and reduce the computational complexity of candidate evaluation during planning. We call this approach PIVOT: Predictive Incremental Variable Ordering Tactic. Applying this tactic can also improve the state inference efficiency; if we maintain the PIVOT order after the planning session, then we should similarly reduce the cost of loop closures, when they actually occur. To demonstrate its effectiveness, we applied PIVOT in a realistic active SLAM simulation, where we managed to significantly reduce the computation time of both the planning and inference sessions. The approach is applicable to general distributions, and induces no loss in accuracy.

Markov chain Monte Carlo (MCMC) provides asymptotically consistent estimates of intractable posterior expectations as the number of iterations tends to infinity. However, in large data applications, MCMC can be computationally expensive per iteration. This has catalyzed interest in sampling methods such as approximate MCMC, which trade off asymptotic consistency for improved computational speed. In this article, we propose estimators based on couplings of Markov chains to assess the quality of such asymptotically biased sampling methods. The estimators give empirical upper bounds of the Wassertein distance between the limiting distribution of the asymptotically biased sampling method and the original target distribution of interest. We establish theoretical guarantees for our upper bounds and show that our estimators can remain effective in high dimensions. We apply our quality measures to stochastic gradient MCMC, variational Bayes, and Laplace approximations for tall data and to approximate MCMC for Bayesian logistic regression in 4500 dimensions and Bayesian linear regression in 50000 dimensions.

This paper studies the regret minimization problem for finite horizon, episodic reinforcement learning (RL) with infinitely large state and action spaces. Empirically, RL has achieved success in diverse domains, even when the problem size (measured in the number of states and actions) explodes [35, 44, 28]. The key to developing sample-efficient algorithms is to leverage function approximation, enabling us to generalize across different state-action pairs. Much theoretical progress has been made towards understanding function approximation in RL. Existing theory typically requires strong linearity assumptions on transition dynamics [e.g., 55, 26, 10, 36] or action-value functions [e.g., 30, 57] of the Markov Decision Process (MDP). However, most real world problems are nonlinear, and our theoretical understanding of these settings remains limited. Thus, we ask the question: Can we design provably efficient RL algorithms in nonlinear environments? Recently, Chowdhury et al. [13] introduced a nonlinear setting where the state-transition measures are finitely parameterized exponential family models, and they proposed to estimate model parameters via maximum likelihood estimation (MLE). The exponential family is a well-studied and powerful statistical framework, so it is a natural model class to consider beyond linear models.

A central concept in active inference is that the internal states of a physical system parametrise probability measures over states of the external world. These can be seen as an agent's beliefs, expressed as a Bayesian prior or posterior. Here we begin the development of a general theory that would tell us when it is appropriate to interpret states as representing beliefs in this way. We focus on the case in which a system can be interpreted as performing either Bayesian filtering or Bayesian inference. We provide formal definitions of what it means for such an interpretation to exist, using techniques from category theory.

This paper discusses the estimation of the generalization gap, the difference between a generalization error and an empirical error, for overparameterized models (e.g., neural networks). We first show that a functional variance, a key concept in defining a widely-applicable information criterion, characterizes the generalization gap even in overparameterized settings where a conventional theory cannot be applied. We also propose a computationally efficient approximation of the function variance, the Langevin approximation of the functional variance (Langevin FV). This method leverages only the $1$st-order gradient of the squared loss function, without referencing the $2$nd-order gradient; this ensures that the computation is efficient and the implementation is consistent with gradient-based optimization algorithms. We demonstrate the Langevin FV numerically by estimating the generalization gaps of overparameterized linear regression and non-linear neural network models.

We introduce Reactive Message Passing (RMP) as a framework for executing schedule-free, robust and scalable message passing-based inference in a factor graph representation of a probabilistic model. RMP is based on the reactive programming style that only describes how nodes in a factor graph react to changes in connected nodes. The absence of a fixed message passing schedule improves robustness, scalability and execution time of the inference procedure. We also present ReactiveMP.jl, which is a Julia package for realizing RMP through minimization of a constrained Bethe free energy. By user-defined specification of local form and factorization constraints on the variational posterior distribution, ReactiveMP.jl executes hybrid message passing algorithms including belief propagation, variational message passing, expectation propagation, and expectation maximisation update rules. Experimental results demonstrate the improved performance of ReactiveMP-based RMP in comparison to other Julia packages for Bayesian inference across a range of probabilistic models. In particular, we show that the RMP framework is able to run Bayesian inference for large-scale probabilistic state space models with hundreds of thousands of random variables on a standard laptop computer.

This specialization program is especially dedicated to statistics. In this program, you will learn basic and intermediate concepts of statistical analysis using the Python programming language. In this program, you will learn the following topics- where data come from, what types of data can be collected, study data design, data management, and how to effectively carry out data exploration and visualization. Along with that, you will work on a variety of assignments that will help you to check your knowledge and ability. This specialization program is a 3-course series. Let's see the details of the courses-

Variational inference is a powerful paradigm for approximate Bayesian inference with a number of appealing properties, including support for model learning and data subsampling. By contrast MCMC methods like Hamiltonian Monte Carlo do not share these properties but remain attractive since, contrary to parametric methods, MCMC is asymptotically unbiased. For these reasons researchers have sought to combine the strengths of both classes of algorithms, with recent approaches coming closer to realizing this vision in practice. However, supporting data subsampling in these hybrid methods can be a challenge, a shortcoming that we address by introducing a surrogate likelihood that can be learned jointly with other variational parameters. We argue theoretically that the resulting algorithm permits the user to make an intuitive trade-off between inference fidelity and computational cost. In an extensive empirical comparison we show that our method performs well in practice and that it is well-suited for black-box inference in probabilistic programming frameworks.

A Bayesian Network is a directed acyclic graph (DAG) on a set of $n$ random variables (identified with the vertices); a Bayesian Network Distribution (BND) is a probability distribution on the rv's that is Markovian on the graph. A finite mixture of such models is the projection on these variables of a BND on the larger graph which has an additional "hidden" (or "latent") random variable $U$, ranging in $\{1,\ldots,k\}$, and a directed edge from $U$ to every other vertex. Models of this type are fundamental to research in Causal Inference, where $U$ models a confounding effect. One extremely special case has been of longstanding interest in the theory literature: the empty graph. Such a distribution is simply a mixture of $k$ product distributions. A longstanding problem has been, given the joint distribution of a mixture of $k$ product distributions, to identify each of the product distributions, and their mixture weights. Our results are: (1) We improve the sample complexity (and runtime) for identifying mixtures of $k$ product distributions from $\exp(O(k^2))$ to $\exp(O(k \log k))$. This is almost best possible in view of a known $\exp(\Omega(k))$ lower bound. (2) We give the first algorithm for the case of non-empty graphs. The complexity for a graph of maximum degree $\Delta$ is $\exp(O(k(\Delta^2 + \log k)))$. (The above complexities are approximate and suppress dependence on secondary parameters.)