In statistical decision theory, a model is said to be Pareto optimal (or admissible) if no other model carries less risk for at least one state of nature while presenting no more risk for others. How can you rationally aggregate/combine a finite set of Pareto optimal models while preserving Pareto efficiency? This question is nontrivial because weighted model averaging does not, in general, preserve Pareto efficiency. This paper presents an answer in four logical steps: (1) A rational aggregation rule should preserve Pareto efficiency (2) Due to the complete class theorem, Pareto optimal models must be Bayesian, i.e., they minimize a risk where the true state of nature is averaged with respect to some prior. Therefore each Pareto optimal model can be associated with a prior, and Pareto efficiency can be maintained by aggregating Pareto optimal models through their priors. (3) A prior can be interpreted as a preference ranking over models: prior $\pi$ prefers model A over model B if the average risk of A is lower than the average risk of B. (4) A rational/consistent aggregation rule should preserve this preference ranking: If both priors $\pi$ and $\pi'$ prefer model A over model B, then the prior obtained by aggregating $\pi$ and $\pi'$ must also prefer A over B. Under these four steps, we show that all rational/consistent aggregation rules are as follows: Give each individual Pareto optimal model a weight, introduce a weak order/ranking over the set of Pareto optimal models, aggregate a finite set of models S as the model associated with the prior obtained as the weighted average of the priors of the highest-ranked models in S. This result shows that all rational/consistent aggregation rules must follow a generalization of hierarchical Bayesian modeling. Following our main result, we present applications to Kernel smoothing, time-depreciating models, and voting mechanisms.

Using a model of the environment and a value function, an agent can construct many estimates of a state's value, by unrolling the model for different lengths and bootstrapping with its value function. Our key insight is that one can treat this set of value estimates as a type of ensemble, which we call an \emph{implicit value ensemble} (IVE). Consequently, the discrepancy between these estimates can be used as a proxy for the agent's epistemic uncertainty; we term this signal \emph{model-value inconsistency} or \emph{self-inconsistency} for short. Unlike prior work which estimates uncertainty by training an ensemble of many models and/or value functions, this approach requires only the single model and value function which are already being learned in most model-based reinforcement learning algorithms. We provide empirical evidence in both tabular and function approximation settings from pixels that self-inconsistency is useful (i) as a signal for exploration, (ii) for acting safely under distribution shifts, and (iii) for robustifying value-based planning with a model.

Multiple techniques for producing calibrated predictive probabilities using deep neural networks in supervised learning settings have emerged that leverage approaches to ensemble diverse solutions discovered during cyclic training or training from multiple random starting points (deep ensembles). However, only a limited amount of work has investigated the utility of exploring the local region around each diverse solution (posterior mode). Using three well-known deep architectures on the CIFAR-10 dataset, we evaluate several simple methods for exploring local regions of the weight space with respect to Brier score, accuracy, and expected calibration error. We consider both Bayesian inference techniques (variational inference and Hamiltonian Monte Carlo applied to the softmax output layer) as well as utilizing the stochastic gradient descent trajectory near optima. While adding separate modes to the ensemble uniformly improves performance, we show that the simple mode exploration methods considered here produce little to no improvement over ensembles without mode exploration.

Local volatility is a versatile option pricing model due to its state dependent diffusion coefficient. Calibration is, however, non-trivial as it involves both proposing a hypothesis model of the latent function and a method for fitting it to data. In this paper we present novel Bayesian inference with Gaussian process priors. We obtain a rich representation of the local volatility function with a probabilistic notion of uncertainty attached to the calibrate. We propose an inference algorithm and apply our approach to S&P 500 market data.

In this work we explore a new framework for approximate Bayesian inference in large datasets based on stochastic control. We advocate stochastic control as a finite time and low variance alternative to popular steady-state methods such as stochastic gradient Langevin dynamics (SGLD). Furthermore, we discuss and adapt the existing theoretical guarantees of this framework and establish connections to already existing VI routines in SDE-based models.

Neural Networks play a growing role in many science disciplines, including physics. Variational Autoencoders (VAEs) are neural networks that are able to represent the essential information of a high dimensional data set in a low dimensional latent space, which have a probabilistic interpretation. In particular the so-called encoder network, the first part of the VAE, which maps its input onto a position in latent space, additionally provides uncertainty information in terms of a variance around this position. In this work, an extension to the Autoencoder architecture is introduced, the FisherNet. In this architecture, the latent space uncertainty is not generated using an additional information channel in the encoder, but derived from the decoder, by means of the Fisher information metric. This architecture has advantages from a theoretical point of view as it provides a direct uncertainty quantification derived from the model, and also accounts for uncertainty cross-correlations. We can show experimentally that the FisherNet produces more accurate data reconstructions than a comparable VAE and its learning performance also apparently scales better with the number of latent space dimensions.

Weak supervision (WS) frameworks are a popular way to bypass hand-labeling large datasets for training data-hungry models. These approaches synthesize multiple noisy but cheaply-acquired estimates of labels into a set of high-quality pseudolabels for downstream training. However, the synthesis technique is specific to a particular kind of label, such as binary labels or sequences, and each new label type requires manually designing a new synthesis algorithm. Instead, we propose a universal technique that enables weak supervision over any label type while still offering desirable properties, including practical flexibility, computational efficiency, and theoretical guarantees. We apply this technique to important problems previously not tackled by WS frameworks including learning to rank, regression, and learning in hyperbolic manifolds. Theoretically, our synthesis approach produces a consistent estimator for learning a challenging but important generalization of the exponential family model. Experimentally, we validate our framework and show improvement over baselines in diverse settings including real-world learning-to-rank and regression problems along with learning on hyperbolic manifolds.

Probabilistic context-free grammars (PCFGs) and dynamic Bayesian networks (DBNs) are widely used sequence models with complementary strengths and limitations. While PCFGs allow for nested hierarchical dependencies (tree structures), their latent variables (non-terminal symbols) have to be discrete. In contrast, DBNs allow for continuous latent variables, but the dependencies are strictly sequential (chain structure). Therefore, neither can be applied if the latent variables are assumed to be continuous and also to have a nested hierarchical dependency structure. In this paper, we present Recursive Bayesian Networks (RBNs), which generalise and unify PCFGs and DBNs, combining their strengths and containing both as special cases. RBNs define a joint distribution over tree-structured Bayesian networks with discrete or continuous latent variables. The main challenge lies in performing joint inference over the exponential number of possible structures and the continuous variables. We provide two solutions: 1) For arbitrary RBNs, we generalise inside and outside probabilities from PCFGs to the mixed discrete-continuous case, which allows for maximum posterior estimates of the continuous latent variables via gradient descent, while marginalising over network structures. 2) For Gaussian RBNs, we additionally derive an analytic approximation, allowing for robust parameter optimisation and Bayesian inference. The capacity and diverse applications of RBNs are illustrated on two examples: In a quantitative evaluation on synthetic data, we demonstrate and discuss the advantage of RBNs for segmentation and tree induction from noisy sequences, compared to change point detection and hierarchical clustering. In an application to musical data, we approach the unsolved problem of hierarchical music analysis from the raw note level and compare our results to expert annotations.

Obtaining first-order regret bounds -- regret bounds scaling not as the worst-case but with some measure of the performance of the optimal policy on a given instance -- is a core question in sequential decision-making. While such bounds exist in many settings, they have proven elusive in reinforcement learning with large state spaces. In this work we address this gap, and show that it is possible to obtain regret scaling as $\mathcal{O}(\sqrt{V_1^\star K})$ in reinforcement learning with large state spaces, namely the linear MDP setting. Here $V_1^\star$ is the value of the optimal policy and $K$ is the number of episodes. We demonstrate that existing techniques based on least squares estimation are insufficient to obtain this result, and instead develop a novel robust self-normalized concentration bound based on the robust Catoni mean estimator, which may be of independent interest.

Estimating individualized treatment effects (ITEs) from observational data is crucial for decision-making. In order to obtain unbiased ITE estimates, a common assumption is that all confounders are observed. However, in practice, it is unlikely that we observe these confounders directly. Instead, we often observe noisy measurements of true confounders, which can serve as valid proxies. In this paper, we address the problem of estimating ITE in the longitudinal setting where we observe noisy proxies instead of true confounders. To this end, we develop the Deconfounding Temporal Autoencoder, a novel method that leverages observed noisy proxies to learn a hidden embedding that reflects the true hidden confounders. In particular, the DTA combines a long short-term memory autoencoder with a causal regularization penalty that renders the potential outcomes and treatment assignment conditionally independent given the learned hidden embedding. Once the hidden embedding is learned via DTA, state-of-the-art outcome models can be used to control for it and obtain unbiased estimates of ITE. Using synthetic and real-world medical data, we demonstrate the effectiveness of our DTA by improving over state-of-the-art benchmarks by a substantial margin.