Regression models are used in a wide range of applications providing a powerful scientific tool for researchers from different fields. Linear, or simple parametric, models are often not sufficient to describe complex relationships between input variables and a response. Such relationships can be better described through  flexible approaches such as neural networks, but this results in less interpretable models and potential overfitting. Alternatively, specific parametric nonlinear functions can be used, but the specification of such functions is in general complicated. In this paper, we introduce a  flexible approach for the construction and selection of highly  flexible nonlinear parametric regression models. Nonlinear features are generated hierarchically, similarly to deep learning, but have additional  flexibility on the possible types of features to be considered. This  flexibility, combined with variable selection, allows us to find a small set of important features and thereby more interpretable models. Within the space of possible functions, a Bayesian approach, introducing priors for functions based on their complexity, is considered. A genetically modified mode jumping Markov chain Monte Carlo algorithm is adopted to perform Bayesian inference and estimate posterior probabilities for model averaging. In various applications, we illustrate how our approach is used to obtain meaningful nonlinear models. Additionally, we compare its predictive performance with several machine learning algorithms.  

A natural notion of rationality/consistency for aggregating models is that, for all (possibly aggregated) models $A$ and $B$, if the output of model $A$ is $f(A)$ and if the output model $B$ is $f(B)$, then the output of the model obtained by aggregating $A$ and $B$ must be a weighted average of $f(A)$ and $f(B)$. Similarly, a natural notion of rationality for aggregating preferences of ensembles of experts is that, for all (possibly aggregated) experts $A$ and $B$, and all possible choices $x$ and $y$, if both $A$ and $B$ prefer $x$ over $y$, then the expert obtained by aggregating $A$ and $B$ must also prefer $x$ over $y$. Rational aggregation is an important element of uncertainty quantification, and it lies behind many seemingly different results in economic theory: spanning social choice, belief formation, and individual decision making. Three examples of rational aggregation rules are as follows. (1) Give each individual model (expert) a weight (a score) and use weighted averaging to aggregate individual or finite ensembles of models (experts). (2) Order/rank individual model (expert) and let the aggregation of a finite ensemble of individual models (experts) be the highest-ranked individual model (expert) in that ensemble. (3) Give each individual model (expert) a weight, introduce a weak order/ranking over the set of models/experts, aggregate $A$ and $B$ as the weighted average of the highest-ranked models (experts) in $A$ or $B$. Note that (1) and (2) are particular cases of (3). In this paper, we show that all rational aggregation rules are of the form (3). This result unifies aggregation procedures across different economic environments. Following the main representation, we show applications and extensions of our representation in various separated economics topics such as belief formation, choice theory, and social welfare economics.

Approximate Bayesian Computation (ABC) enables statistical inference in complex models whose likelihoods are difficult to calculate but easy to simulate from. ABC constructs a kernel-type approximation to the posterior distribution through an accept/reject mechanism which compares summary statistics of real and simulated data. To obviate the need for summary statistics, we directly compare empirical distributions with a Kullback-Leibler (KL) divergence estimator obtained via classification. In particular, we blend flexible machine learning classifiers within ABC to automate fake/real data comparisons. We consider the traditional accept/reject kernel as well as an exponential weighting scheme which does not require the ABC acceptance threshold. Our theoretical results show that the rate at which our ABC posterior distributions concentrate around the true parameter depends on the estimation error of the classifier. We derive limiting posterior shape results and find that, with a properly scaled exponential kernel, asymptotic normality holds. We demonstrate the usefulness of our approach on simulated examples as well as real data in the context of stock volatility estimation.

Active inference is a state-of-the-art framework for modelling the brain that explains a wide range of mechanisms such as habit formation, dopaminergic discharge and curiosity. However, recent implementations suffer from an exponential (space and time) complexity class when computing the prior over all the possible policies up to the time horizon. Fountas et al. (2020) used Monte Carlo tree search to address this problem, leading to very good results in two different tasks. Additionally, Champion et al. (2021a) proposed a tree search approach based on structure learning. This was enabled by the development of a variational message passing approach to active inference (Champion et al., 2021b), which enables compositional construction of Bayesian networks for active inference. However, this message passing tree search approach, which we call branching-time active inference (BTAI), has never been tested empirically. In this paper, we present an experimental study of the approach (Champion et al., 2021a) in the context of a maze solving agent. In this context, we show that both improved prior preferences and deeper search help mitigate the vulnerability to local minima. Then, we compare BTAI to standard active inference (AI) on a graph navigation task. We show that for small graphs, both BTAI and AI successfully solve the task. For larger graphs, AI exhibits an exponential (space) complexity class, making the approach intractable. However, BTAI explores the space of policies more efficiently, successfully scaling to larger graphs.

Over the last 10 to 15 years, active inference has helped to explain various brain mechanisms from habit formation to dopaminergic discharge and even modelling curiosity. However, the current implementations suffer from an exponential (space and time) complexity class when computing the prior over all the possible policies up to the time-horizon. Fountas et al (2020) used Monte Carlo tree search to address this problem, leading to impressive results in two different tasks. In this paper, we present an alternative framework that aims to unify tree search and active inference by casting planning as a structure learning problem. Two tree search algorithms are then presented. The first propagates the expected free energy forward in time (i.e., towards the leaves), while the second propagates it backward (i.e., towards the root). Then, we demonstrate that forward and backward propagations are related to active inference and sophisticated inference, respectively, thereby clarifying the differences between those two planning strategies.

We present a novel task that measures how people generalize objects' causal powers based on observing a single (Experiment 1) or a few (Experiment 2) causal interactions between object pairs. We propose a computational modeling framework that can synthesize human-like generalization patterns in our task setting, and sheds light on how people may navigate the compositional space of possible causal functions and categories efficiently. Our modeling framework combines a causal function generator that makes use of agent and recipient objects' features and relations, and a Bayesian non-parametric inference process to govern the degree of similarity-based generalization. Our model has a natural "resource-rational" variant that outperforms a naive Bayesian account in describing participants, in particular reproducing a generalization-order effect and causal asymmetry observed in our behavioral experiments. We argue that this modeling framework provides a computationally plausible mechanism for real world causal generalization.

In this paper, we propose a deterministic variational inference approach and generate low-discrepancy points by minimizing the kernel discrepancy, also known as the Maximum Mean Discrepancy or MMD. Based on the general energetic variational inference framework by Wang et. al. (2021), minimizing the kernel discrepancy is transformed to solving a dynamic ODE system via the explicit Euler scheme. We name the resulting algorithm EVI-MMD and demonstrate it through examples in which the target distribution is fully specified, partially specified up to the normalizing constant, and empirically known in the form of training data. Its performances are satisfactory compared to alternative methods in the applications of distribution approximation, numerical integration, and generative learning. The EVI-MMD algorithm overcomes the bottleneck of the existing MMD-descent algorithms, which are mostly applicable to two-sample problems. Algorithms with more sophisticated structures and potential advantages can be developed under the EVI framework.

A few years ago, at the Princeton Club in Manhattan, I chanced on a memorable chat with the Harvard psychologist Steven Pinker. His spouse, the philosopher Rebecca Goldstein, with whom he was tagging along, had been invited onto a panel to discuss the conflict between religion and science and Einstein's so-called "God letter," which was being auctioned at Christie's. Pinker had recently published Enlightenment Now: The Case for Reason, Science, Humanism, and Progress. I was eager to pepper him with questions, mainly on religion, rationality, and evolutionary psychology. I remember I wanted Pinker's take on something Harvey Whitehouse, one of the founders of the cognitive science of religion, told me in an interview--that my own little enlightenment, of becoming an atheist in college, was probably mostly a product of merely changing my social milieu. I wasn't so much moved by rational arguments against the ethics and existence of God but by being distanced from my old life and meeting new, non-religious friends. I recall Pinker almost pouncing on that argument, defending reason's power to change our minds. He noted that people especially high in "intellectance," a personality trait now more commonly called "openness to experience," tend to be more curious, intelligent, and willing to entertain new ideas. I still think that Pinker's way of seeing things made more sense of my experience in those heady days. I really was, for the first time, trying my best to think things through, and it was exhilarating. We talked until the event staff shelved the wine, and parted ways at a chilly midtown intersection.

False alerts due to misconfigured/ compromised IDS in ICS networks can lead to severe economic and operational damage. To solve this problem, research has focused on leveraging deep learning techniques that help reduce false alerts. However, a shortcoming is that these works often require or implicitly assume the physical and cyber sensors to be trustworthy. Implicit trust of data is a major problem with using artificial intelligence or machine learning for CPS security, because during critical attack detection time they are more at risk, with greater likelihood and impact, of also being compromised. To address this shortcoming, the problem is reframed on how to make good decisions given uncertainty. Then, the decision is detection, and the uncertainty includes whether the data used for ML-based IDS is compromised. Thus, this work presents an approach for reducing false alerts in CPS power systems by dealing uncertainty without the knowledge of prior distribution of alerts. Specifically, an evidence theoretic based approach leveraging Dempster Shafer combination rules are proposed for reducing false alerts. A multi-hypothesis mass function model is designed that leverages probability scores obtained from various supervised-learning classifiers. Using this model, a location-cum-domain based fusion framework is proposed and evaluated with different combination rules, that fuse multiple evidence from inter-domain and intra-domain sensors. The approach is demonstrated in a cyber-physical power system testbed with Man-In-The-Middle attack emulation in a large-scale synthetic electric grid. For evaluating the performance, plausibility, belief, pignistic, etc. metrics as decision functions are considered. To improve the performance, a multi-objective based genetic algorithm is proposed for feature selection considering the decision metrics as the fitness function.

Generative Flow Networks (GFlowNets) have been introduced as a method to sample a diverse set of candidates in an active learning context, with a training objective that makes them approximately sample in proportion to a given reward function. In this paper, we show a number of additional theoretical properties of GFlowNets. They can be used to estimate joint probability distributions and the corresponding marginal distributions where some variables are unspecified and, of particular interest, can represent distributions over composite objects like sets and graphs. GFlowNets amortize the work typically done by computationally expensive MCMC methods in a single but trained generative pass. They could also be used to estimate partition functions and free energies, conditional probabilities of supersets (supergraphs) given a subset (subgraph), as well as marginal distributions over all supersets (supergraphs) of a given set (graph). We introduce variations enabling the estimation of entropy and mutual information, sampling from a Pareto frontier, connections to reward-maximizing policies, and extensions to stochastic environments, continuous actions and modular energy functions.