Savage (1972) lays down the foundation of Bayesian decision theory, but asserts that it is not applicable in big worlds where the environment is complex. Using the theory of finite automaton to model belief formation, this paper studies the characteristics of optimal learning behavior in small and big worlds, where the complexity of the environment is low and high, respectively, relative to the cognitive ability of the decision maker. Confirming Savage's claim, optimal learning behavior is closed to Bayesian in small worlds but significantly different in big worlds. In addition, I show that in big worlds, the optimal learning behavior could exhibit a wide range of well-documented non-Bayesian learning behavior, including the use of heuristic, correlation neglect, persistent over-confidence, inattentive learning, and other behaviors of model simplification or misspecification. These results establish a clear and testable relationship between the prominence of non-Bayesian learning behavior, complexity and cognitive ability.

This paper studies the source (event) localization problem in decentralized wireless sensor networks (WSNs) under the fault model without knowing the sensor parameters. Event localizations have many applications such as localizing intruders, Wifi hotspots and users, and faults in power systems. Previous studies assume the true knowledge (or good estimates) of sensor parameters (e.g., fault model probability or Region of Influence (ROI) of the source) for source localization. However, we propose two methods to estimate the source location in this paper under the fault model: hitting set approach and feature selection method, which only utilize the noisy data set at the fusion center for estimation of the source location without knowing the sensor parameters. The proposed methods have been shown to localize the source effectively. We also study the lower bound on the sample complexity requirement for hitting set method. These methods have also been extended for multiple sources localizations. In addition, we modify the proposed feature selection approach to use maximum likelihood. Finally, extensive simulations are carried out for different settings (i.e., the number of sensor nodes and sample complexity) to validate our proposed methods in comparison to centroid, maximum likelihood, FTML, SNAP estimators.

In recent years, bankruptcy forecasting has gained lot of attention from researchers as well as practitioners in the field of financial risk management. For bankruptcy prediction, various approaches proposed in the past and currently in practice relies on accounting ratios and using statistical modeling or machine learning methods. These models have had varying degrees of successes. Models such as Linear Discriminant Analysis or Artificial Neural Network employ discriminative classification techniques. They lack explicit provision to include prior expert knowledge. In this paper, we propose another route of generative modeling using Expert Bayesian framework. The biggest advantage of the proposed framework is an explicit inclusion of expert judgment in the modeling process. Also the proposed methodology provides a way to quantify uncertainty in prediction. As a result the model built using Bayesian framework is highly flexible, interpretable and intuitive in nature. The proposed approach is well suited for highly regulated or safety critical applications such as in finance or in medical diagnosis. In such cases accuracy in the prediction is not the only concern for decision makers. Decision makers and other stakeholders are also interested in uncertainty in the prediction as well as interpretability of the model. We empirically demonstrate these benefits of proposed framework on real world dataset using Stan, a probabilistic programming language. We found that the proposed model is either comparable or superior to the other existing methods. Also resulting model has much less False Positive Rate compared to many existing state of the art methods. The corresponding R code for the experiments is available at Github repository.

Gaussian Process (GPs) models are a rich distribution over functions with inductive biases controlled by a kernel function. Learning occurs through the optimisation of kernel hyperparameters using the marginal likelihood as the objective. This classical approach known as Type-II maximum likelihood (ML-II) yields point estimates of the hyperparameters, and continues to be the default method for training GPs. However, this approach risks underestimating predictive uncertainty and is prone to overfitting especially when there are many hyperparameters. Furthermore, gradient based optimisation makes ML-II point estimates highly susceptible to the presence of local minima. This work presents an alternative learning procedure where the hyperparameters of the kernel function are marginalised using Nested Sampling (NS), a technique that is well suited to sample from complex, multi-modal distributions. We focus on regression tasks with the spectral mixture (SM) class of kernels and find that a principled approach to quantifying model uncertainty leads to substantial gains in predictive performance across a range of synthetic and benchmark data sets. In this context, nested sampling is also found to offer a speed advantage over Hamiltonian Monte Carlo (HMC), widely considered to be the gold-standard in MCMC based inference.

Clinical decision making about treatments and interventions based on personal characteristics leads to effective health improvement. Machine learning (ML) has been the central concern of the diagnosis support and disease prediction based on comprehensive patient information. Because the black-box problem in ML is serious for medical applications, explainable artificial intelligence (XAI) techniques to explain the reasons for ML models predictions have been focused. A remaining important issue in clinical situations is discovery of concrete and realistic treatment processes. This paper proposes an innovative framework to plan concrete treatment processes based on an ML model. A key point of our proposed framework is to evaluate an "actionability" of the treatment process using a stochastic surrogate model constructed through hierarchical Bayesian modeling. The actionability is an essential concept for suggesting a realistic treatment process, which leads to clinical applications for personal health improvement. This paper also presents two experiments to evaluate our framework. We first demonstrate the feasibility of our framework from the viewpoint of the methodology using a synthetic dataset. Subsequently, our framework is applied to an actual health checkup dataset, which comprises 3,132 participants, considering an application to improve systolic blood pressure values at a personal level. We confirmed that the computed treatment processes are actionable and consistent with clinical knowledge for lowering blood pressure. These results demonstrate that our framework can contribute to decision making in the medical field. Our framework can be expected to provide clinicians deeper insights by proposing concrete and actionable treatment process based on the ML model.

A fundamental question in neuroscience is how the brain creates an internal model of the world to guide actions using sequences of ambiguous sensory information. This is naturally formulated as a reinforcement learning problem under partial observations, where an agent must estimate relevant latent variables in the world from its evidence, anticipate possible future states, and choose actions that optimize total expected reward. This problem can be solved by control theory, which allows us to find the optimal actions for a given system dynamics and objective function. However, animals often appear to behave suboptimally. Why? We hypothesize that animals have their own flawed internal model of the world, and choose actions with the highest expected subjective reward according to that flawed model. We describe this behavior as rational but not optimal. The problem of Inverse Rational Control (IRC) aims to identify which internal model would best explain an agent's actions. Our contribution here generalizes past work on Inverse Rational Control which solved this problem for discrete control in partially observable Markov decision processes. Here we accommodate continuous nonlinear dynamics and continuous actions, and impute sensory observations corrupted by unknown noise that is private to the animal. We first build an optimal Bayesian agent that learns an optimal policy generalized over the entire model space of dynamics and subjective rewards using deep reinforcement learning. Crucially, this allows us to compute a likelihood over models for experimentally observable action trajectories acquired from a suboptimal agent. We then find the model parameters that maximize the likelihood using gradient ascent.

We investigate partial identification of causal relative and attributable risk---the ratio of two counterfactual proportions and the difference between them---in case-control and case-population studies. The odds ratio is shown to be a sharp upper bound on causal relative risk under the monotone treatment response and monotone treatment selection assumptions, without resorting to strong ignorability, nor to the rare-disease assumption. Sharp bounds on causal attributable risk are also obtained under the same assumptions. Paying special attention to the (conditional) odds ratio, we propose a semiparametrically efficient estimator of the aggregated (log) odds ratio. Further, we develop easy-to-implement causal inference procedures for relative and attributable risk. Finally, we showcase our methodology by applying it to two unique datasets in the literature. We find that attending private school may have little effect on entering a very selective university in Pakistan and that dropping out of school could substantially increase relative and attributable risk of joining a criminal gang in Brazil.

We present a computational model of mathematical reasoning according to which mathematics is a fundamentally stochastic process. That is, on our model, whether or not a given formula is deemed a theorem in some axiomatic system is not a matter of certainty, but is instead governed by a probability distribution. We then show that this framework gives a compelling account of several aspects of mathematical practice. These include: 1) the way in which mathematicians generate research programs, 2) the applicability of Bayesian models of mathematical heuristics, 3) the role of abductive reasoning in mathematics, 4) the way in which multiple proofs of a proposition can strengthen our degree of belief in that proposition, and 5) the nature of the hypothesis that there are multiple formal systems that are isomorphic to physically possible universes. Thus, by embracing a model of mathematics as not perfectly predictable, we generate a new and fruitful perspective on the epistemology and practice of mathematics.

We consider learning a sparse pairwise Markov Random Field (MRF) with continuous-valued variables from i.i.d samples. We adapt the algorithm of Vuffray et al. (2019) to this setting and provide finite-sample analysis revealing sample complexity scaling logarithmically with the number of variables, as in the discrete and Gaussian settings. Our approach is applicable to a large class of pairwise MRFs with continuous variables and also has desirable asymptotic properties, including consistency and normality under mild conditions. Further, we establish that the population version of the optimization criterion employed in Vuffray et al. (2019) can be interpreted as local maximum likelihood estimation (MLE). As part of our analysis, we introduce a robust variation of sparse linear regression a` la Lasso, which may be of interest in its own right.

Human annotations are an important source of information in the development of natural language understanding approaches. As under the pressure of productivity annotators can assign different labels to a given text, the quality of produced annotations frequently varies. This is especially the case if decisions are difficult, with high cognitive load, requires awareness of broader context, or careful consideration of background knowledge. To alleviate the problem, we propose two semi-supervised methods to guide the annotation process: a Bayesian deep learning model and a Bayesian ensemble method. Using a Bayesian deep learning method, we can discover annotations that cannot be trusted and might require reannotation. A recently proposed Bayesian ensemble method helps us to combine the annotators' labels with predictions of trained models. According to the results obtained from three hate speech detection experiments, the proposed Bayesian methods can improve the annotations and prediction performance of BERT models.