Numerous applications in biology, statistics, science, and engineering require generating samples from high-dimensional probability distributions. In recent years, the Hamiltonian Monte Carlo (HMC) method has emerged as a state-of-the-art Markov chain Monte Carlo technique, exploiting the shape of such high-dimensional target distributions to efficiently generate samples. Despite its impressive empirical success and increasing popularity, its wide-scale adoption remains limited due to the high computational cost of gradient calculation. Moreover, applying this method is impossible when the gradient of the posterior cannot be computed (for example, with black-box simulators). To overcome these challenges, we propose a novel two-stage Hamiltonian Monte Carlo algorithm with a surrogate model. In this multi-fidelity algorithm, the acceptance probability is computed in the first stage via a standard HMC proposal using an inexpensive differentiable surrogate model, and if the proposal is accepted, the posterior is evaluated in the second stage using the high-fidelity (HF) numerical solver. Splitting the standard HMC algorithm into these two stages allows for approximating the gradient of the posterior efficiently, while producing accurate posterior samples by using HF numerical solvers in the second stage. We demonstrate the effectiveness of this algorithm for a range of problems, including linear and nonlinear Bayesian inverse problems with in-silico data and experimental data. The proposed algorithm is shown to seamlessly integrate with various low-fidelity and HF models, priors, and datasets. Remarkably, our proposed method outperforms the traditional HMC algorithm in both computational and statistical efficiency by several orders of magnitude, all while retaining or improving the accuracy in computed posterior statistics.

Investment portfolios, central to finance, balance potential returns and risks. This paper introduces a hybrid approach combining Markowitz's portfolio theory with reinforcement learning, utilizing knowledge distillation for training agents. In particular, our proposed method, called KDD (Knowledge Distillation DDPG), consist of two training stages: supervised and reinforcement learning stages. The trained agents optimize portfolio assembly. A comparative analysis against standard financial models and AI frameworks, using metrics like returns, the Sharpe ratio, and nine evaluation indices, reveals our model's superiority. It notably achieves the highest yield and Sharpe ratio of 2.03, ensuring top profitability with the lowest risk in comparable return scenarios.

Partially observable Markov decision processes (POMDPs) rely on the key assumption that probability distributions are precisely known. Robust POMDPs (RPOMDPs) alleviate this concern by defining imprecise probabilities, referred to as uncertainty sets. While robust MDPs have been studied extensively, work on RPOMDPs is limited and primarily focuses on algorithmic solution methods. We expand the theoretical understanding of RPOMDPs by showing that 1) different assumptions on the uncertainty sets affect optimal policies and values; 2) RPOMDPs have a partially observable stochastic game (POSG) semantic; and 3) the same RPOMDP with different assumptions leads to semantically different POSGs and, thus, different policies and values. These novel semantics for RPOMDPS give access to results for the widely studied POSG model; concretely, we show the existence of a Nash equilibrium. Finally, we classify the existing RPOMDP literature using our semantics, clarifying under which uncertainty assumptions these existing works operate.

We revisit the problem of the existence of the maximum likelihood estimate for multi-class logistic regression. We show that one method of ensuring its existence is by assigning positive probability to every class in the sample dataset. The notion of data separability is not needed, which is in contrast to the classical set up of multi-class logistic regression in which each data sample belongs to one class. We also provide a general and constructive estimate of the convergence rate to the maximum likelihood estimate when gradient descent is used as the optimizer. Our estimate involves bounding the condition number of the Hessian of the maximum likelihood function. The approaches used in this article rely on a simple operator-theoretic framework.

Instrumental variables (IVs) are a popular and powerful tool for estimating causal effects in the presence of unobserved confounding. However, classical approaches rely on strong assumptions such as the $\textit{exclusion criterion}$, which states that instrumental effects must be entirely mediated by treatments. This assumption often fails in practice. When IV methods are improperly applied to data that do not meet the exclusion criterion, estimated causal effects may be badly biased. In this work, we propose a novel solution that provides $\textit{partial}$ identification in linear systems given a set of $\textit{leaky instruments}$, which are allowed to violate the exclusion criterion to some limited degree. We derive a convex optimization objective that provides provably sharp bounds on the average treatment effect under some common forms of information leakage, and implement inference procedures to quantify the uncertainty of resulting estimates. We demonstrate our method in a set of experiments with simulated data, where it performs favorably against the state of the art. An accompanying $\texttt{R}$ package, $\texttt{leakyIV}$, is available from $\texttt{CRAN}$.

Predictive modeling of criminal recidivism, or whether people will re-offend in the future, has a long and contentious history. Modern causal inference methods allow us to move beyond prediction and target the "treatment effect" of a specific intervention on an outcome in an observational dataset. In this paper, we look specifically at the effect of incarceration (prison time) on recidivism, using a well-known dataset from North Carolina. Two popular causal methods for addressing confounding bias are explained and demonstrated: directed acyclic graph (DAG) adjustment and double machine learning (DML), including a sensitivity analysis for unobserved confounders. We find that incarceration has a detrimental effect on recidivism, i.e., longer prison sentences make it more likely that individuals will re-offend after release, although this conclusion should not be generalized beyond the scope of our data. We hope that this case study can inform future applications of causal inference to criminal justice analysis.

The high dimensional nature of genomics data complicates feature selection, in particular in low sample size studies - not uncommon in clinical prediction settings. It is widely recognized that complementary data on the features, `co-data', may improve results. Examples are prior feature groups or p-values from a related study. Such co-data are ubiquitous in genomics settings due to the availability of public repositories. Yet, the uptake of learning methods that structurally use such co-data is limited. We review guided adaptive shrinkage methods: a class of regression-based learners that use co-data to adapt the shrinkage parameters, crucial for the performance of those learners. We discuss technical aspects, but also the applicability in terms of types of co-data that can be handled. This class of methods is contrasted with several others. In particular, group-adaptive shrinkage is compared with the better-known sparse group-lasso by evaluating feature selection. Finally, we demonstrate the versatility of the guided shrinkage methodology by showing how to `do-it-yourself': we integrate implementations of a co-data learner and the spike-and-slab prior for the purpose of improving feature selection in genetics studies.

Many aspects of human learning have been proposed as a process of constructing mental programs: from acquiring symbolic number representations to intuitive theories about the world. In parallel, there is a long-tradition of using information processing to model human cognition through Rate Distortion Theory (RDT). Yet, it is still poorly understood how to apply RDT when mental representations take the form of programs. In this work, we adapt RDT by proposing a three way trade-off among rate (description length), distortion (error), and computational costs (search budget). We use simulations on a melody task to study the implications of this trade-off, and show that constructing a shared program library across tasks provides global benefits. However, this comes at the cost of sensitivity to curricula, which is also characteristic of human learners. Finally, we use methods from partial information decomposition to generate training curricula that induce more effective libraries and better generalization.

Curriculum Analytics (CA) studies curriculum structure and student data to ensure the quality of educational programs. One desirable property of courses within curricula is that they are not unexpectedly more difficult for students of different backgrounds. While prior work points to likely variations in course difficulty across student groups, robust methodologies for capturing such variations are scarce, and existing approaches do not adequately decouple course-specific difficulty from students' general performance levels. The present study introduces Differential Course Functioning (DCF) as an Item Response Theory (IRT)-based CA methodology. DCF controls for student performance levels and examines whether significant differences exist in how distinct student groups succeed in a given course. Leveraging data from over 20,000 students at a large public university, we demonstrate DCF's ability to detect inequities in undergraduate course difficulty across student groups described by grade achievement. We compare major pairs with high co-enrollment and transfer students to their non-transfer peers. For the former, our findings suggest a link between DCF effect sizes and the alignment of course content to student home department motivating interventions targeted towards improving course preparedness. For the latter, results suggest minor variations in course-specific difficulty between transfer and non-transfer students. While this is desirable, it also suggests that interventions targeted toward mitigating grade achievement gaps in transfer students should encompass comprehensive support beyond enhancing preparedness for individual courses. By providing more nuanced and equitable assessments of academic performance and difficulties experienced by diverse student populations, DCF could support policymakers, course articulation officers, and student advisors.

We propose a post-processing algorithm for fair classification that mitigates model bias under a unified family of group fairness criteria covering statistical parity, equal opportunity, and equalized odds, applicable to multi-class problems and both attribute-aware and attribute-blind settings. It achieves fairness by re-calibrating the output score of the given base model with a "fairness cost" -- a linear combination of the (predicted) group memberships. Our algorithm is based on a representation result showing that the optimal fair classifier can be expressed as a linear post-processing of the loss function and the group predictor, derived via using these as sufficient statistics to reformulate the fair classification problem as a linear program. The parameters of the post-processor are estimated by solving the empirical LP. Experiments on benchmark datasets show the efficiency and effectiveness of our algorithm at reducing disparity compared to existing algorithms, including in-processing, especially on larger problems.