In this paper, we explore adaptive inference based on variational Bayes. Although a number of studies have been conducted to analyze contraction properties of variational posteriors, there is still a lack of a general and computationally tractable variational Bayes method that can achieve adaptive optimal contraction of the variational posterior. We propose a novel variational Bayes framework, called adaptive variational Bayes, which can operate on a collection of models with varying dimensions and structures. The proposed framework combines variational posteriors over individual models with certain weights to obtain a variational posterior over the entire model. It turns out that this combined variational posterior minimizes the Kullback-Leibler divergence to the original posterior distribution. We show that the proposed variational posterior achieves optimal contraction rates adaptively under very general conditions and attains model selection consistency when the true model structure exists. We apply the general results obtained for the adaptive variational Bayes to several examples including deep learning models and derive some new and adaptive inference results. Moreover, we consider the use of quasi-likelihood in our framework. We formulate conditions on the quasi-likelihood to ensure the adaptive optimality and discuss specific applications to stochastic block models and nonparametric regression with sub-Gaussian errors.

Discovering causal structures from data is a challenging inference problem of fundamental importance in all areas of science. The appealing scaling properties of neural networks have recently led to a surge of interest in differentiable neural network-based methods for learning causal structures from data. So far differentiable causal discovery has focused on static datasets of observational or interventional origin. In this work, we introduce an active intervention-targeting mechanism which enables a quick identification of the underlying causal structure of the data-generating process. Our method significantly reduces the required number of interactions compared with random intervention targeting and is applicable for both discrete and continuous optimization formulations of learning the underlying directed acyclic graph (DAG) from data. We examine the proposed method across a wide range of settings and demonstrate superior performance on multiple benchmarks from simulated to real-world data. Learning causal structure from data is a challenging but important task that lies at the heart of scientific reasoning and accompanying progress in many disciplines (Sachs et al., 2005; Hill et al., 2016; Lauritzen & Spiegelhalter, 1988; Korb & Nicholson, 2010). While there exists a plethora of methods for the task, computationally and statistically more efficient algorithms are highly desired (Heinze-Deml et al., 2018). As a result, there has been a surge in interest in differentiable structure learning and the combination of deep learning and causal inference (Schölkopf et al., 2021). However, the improvement critically depends on the experiments and interventions available. Despite advances in high-throughput methods for interventional data in specific fields (Dixit et al., 2016), the acquisition of interventional samples in the general settings tends to be costly, technically impossible or even unethical for specific interventions.

Nowadays businesses are focusing on automation. They are trying to automate all manual tasks that consume a lot of human effort and time. Today machine learning algorithms have taken over the process that was considered to be mundane or dangerous. Technology is continuously churning businesses making them efficient, smarter, and capable. As technology has become accessible, new innovations in business processes have emerged. The technology revolution was triggered by the democratization of computing tools and techniques which are now easily available.

This paper proposes a multi-label classification algorithm capable of continual learning by applying an Adaptive Resonance Theory (ART)-based clustering algorithm and the Bayesian approach for label probability computation. The ART-based clustering algorithm adaptively and continually generates prototype nodes corresponding to given data, and the generated nodes are used as classifiers. The label probability computation independently counts the number of label appearances for each class and calculates the Bayesian probabilities. Thus, the label probability computation can cope with an increase in the number of labels. Experimental results with synthetic and real-world multi-label datasets show that the proposed algorithm has competitive classification performance to other well-known algorithms while realizing continual learning.

The Free Energy Principle (FEP) postulates that biological agents perceive and interact with their environment in order to minimize a Variational Free Energy (VFE) with respect to a generative model of their environment. The inference of a policy (future control sequence) according to the FEP is known as Active Inference (AIF). The AIF literature describes multiple VFE objectives for policy planning that lead to epistemic (information-seeking) behavior. However, most objectives have limited modeling flexibility. This paper approaches epistemic behavior from a constrained Bethe Free Energy (CBFE) perspective. Crucially, variational optimization of the CBFE can be expressed in terms of message passing on free-form generative models. The key intuition behind the CBFE is that we impose a point-mass constraint on predicted outcomes, which explicitly encodes the assumption that the agent will make observations in the future. We interpret the CBFE objective in terms of its constituent behavioral drives. We then illustrate resulting behavior of the CBFE by planning and interacting with a simulated T-maze environment. Simulations for the T-maze task illustrate how the CBFE agent exhibits an epistemic drive, and actively plans ahead to account for the impact of predicted outcomes. Compared to an EFE agent, the CBFE agent incurs expected reward in significantly more environmental scenarios. We conclude that CBFE optimization by message passing suggests a general mechanism for epistemic-aware AIF in free-form generative models.

In the analysis of observational data in social sciences and businesses, it is difficult to obtain a "(quasi) single-source dataset" in which the variables of interest are simultaneously observed. Instead, multiple-source datasets are typically acquired for different individuals or units. Various methods have been proposed to investigate the relationship between the variables in each dataset, e.g., matching and latent variable modeling. It is necessary to utilize these datasets as a single-source dataset with missing variables. Existing methods assume that the datasets to be integrated are acquired from the same population or that the sampling depends on covariates. This assumption is referred to as missing at random (MAR) in terms of missingness. However, as will been shown in application studies, it is likely that this assumption does not hold in actual data analysis and the results obtained may be biased. We propose a data fusion method that does not assume that datasets are homogenous. We use a Gaussian process latent variable model for non-MAR missing data. This model assumes that the variables of concern and the probability of being missing depend on latent variables. A simulation study and real-world data analysis show that the proposed method with a missing-data mechanism and the latent Gaussian process yields valid estimates, whereas an existing method provides severely biased estimates. This is the first study in which non-random assignment to datasets is considered and resolved under resonable assumptions in data fusion problem.

As a regression technique in spatial statistics, spatiotemporally varying coefficient model (STVC) is an important tool to discover nonstationary and interpretable response-covariate associations over both space and time. However, it is difficult to apply STVC for large-scale spatiotemporal analysis due to the high computational cost. To address this challenge, we summarize the spatiotemporally varying coefficients using a third-order tensor structure and propose to reformulate the spatiotemporally varying coefficient model as a special low-rank tensor regression problem. The low-rank decomposition can effectively model the global patterns of the large data with substantially reduced number of parameters. To further incorporate the local spatiotemporal dependencies among the samples, we place Gaussian process (GP) priors on the spatial and temporal factor matrices to better encode local spatial and temporal processes on each factor component. We refer to the overall framework as Bayesian Kernelized Tensor Regression (BKTR). For model inference, we develop an efficient Markov chain Monte Carlo (MCMC) algorithm, which uses Gibbs sampling to update factor matrices and slice sampling to update kernel hyperparameters. We conduct extensive experiments on both synthetic and real-world data sets, and our results confirm the superior performance and efficiency of BKTR for model estimation and parameter inference.

In Bayesian learning of Gaussian graphical model structure, it is common to restrict attention to certain classes of graphs and approximate the posterior distribution by repeatedly moving from one graph to another, using MCMC or methods such as stochastic shotgun search (SSS). I give two corrected versions of an algorithm for non-decomposable graphs and discuss random graph distributions, in particular as prior distributions. The main topic of the thesis is Bayesian structure-learning with forests or trees. Restricting attention to these graphs can be justified using theorems on random graphs. I describe how to use the Chow$\unicode{x2013}$Liu algorithm and the Matrix Tree Theorem to find the MAP forest and certain quantities in the posterior distribution on trees. I give adapted versions of MCMC and SSS for approximating the posterior distribution for forests and trees, and systems for storing these graphs so that it is easy to choose moves to neighbouring graphs. Experiments show that SSS with trees does well when the true graph is a tree or sparse graph. SSS with trees or forests does better than SSS with decomposable graphs in certain cases. Graph priors improve detection of hubs but need large ranges of probabilities. MCMC on forests fails to mix well and MCMC on trees is slower than SSS. (For a longer abstract see the thesis.)

The Free-Energy-Principle (FEP) is an influential and controversial theory which postulates a deep and powerful connection between the stochastic thermodynamics of self-organization and learning through variational inference. Specifically, it claims that any self-organizing system which can be statistically separated from its environment, and which maintains itself at a non-equilibrium steady state, can be construed as minimizing an information-theoretic functional -- the variational free energy -- and thus performing variational Bayesian inference to infer the hidden state of its environment. This principle has also been applied extensively in neuroscience, and is beginning to make inroads in machine learning by spurring the construction of novel and powerful algorithms by which action, perception, and learning can all be unified under a single objective. While its expansive and often grandiose claims have spurred significant debates in both philosophy and theoretical neuroscience, the mathematical depth and lack of accessible introductions and tutorials for the core claims of the theory have often precluded a deep understanding within the literature. Here, we aim to provide a mathematically detailed, yet intuitive walk-through of the formulation and central claims of the FEP while also providing a discussion of the assumptions necessary and potential limitations of the theory. Additionally, since the FEP is a still a living theory, subject to internal controversy, change, and revision, we also present a detailed appendix highlighting and condensing current perspectives as well as controversies about the nature, applicability, and the mathematical assumptions and formalisms underlying the FEP.

Description logics are a powerful tool for describing ontological knowledge bases. That is, they give a factual account of the world in terms of individuals, concepts and relations. In the presence of uncertainty, such factual accounts are not feasible, and a subjective or epistemic approach is required. Aleatoric description logic models uncertainty in the world as aleatoric events, by the roll of the dice, where an agent has subjective beliefs about the bias of these dice. This provides a subjective Bayesian description logic, where propositions and relations are assigned probabilities according to what a rational agent would bet, given a configuration of possible individuals and dice. Aleatoric description logic is shown to generalise the description logic ALC, and can be seen to describe a probability space of interpretations of a restriction of ALC where all roles are functions. Several computational problems are considered and model-checking and consistency checking algorithms are presented. Finally, aleatoric description logic is shown to be able to model learning, where agents are able to condition their beliefs on the bias of dice according to observations.