Partial information decompositions (PIDs), which quantify information interactions between three or more variables in terms of uniqueness, redundancy and synergy, are gaining traction in many application domains. However, our understanding of the operational interpretations of PIDs is still incomplete for many popular PID definitions. In this paper, we discuss the operational interpretations of unique information through the lens of two well-known PID definitions. We reexamine an interpretation from statistical decision theory showing how unique information upper bounds the risk in a decision problem. We then explore a new connection between the two PIDs, which allows us to develop an informal but appealing interpretation, and generalize the PID definitions using a common Lagrangian formulation. Finally, we provide a new PID definition that is able to capture the information that is unique. We also show that it has a straightforward interpretation and examine its properties.The full version of this paper is available online [1].

Bivariate partial information decompositions (PIDs) characterize how the information in a "message" random variable is decomposed between two "constituent" random variables in terms of unique, redundant and synergistic information components. These components are a function of the joint distribution of the three variables, and are typically defined using an optimization over the space of all possible joint distributions. This makes it computationally challenging to compute PIDs in practice and restricts their use to low-dimensional random vectors. To ease this burden, we consider the case of jointly Gaussian random vectors in this paper. This case was previously examined by Barrett [1], who showed that certain operationally well-motivated PIDs reduce to a closed form expression for scalar messages. Here, we show that Barrett’s result does not extend to vector messages in general, and characterize the set of multivariate Gaussian distributions that reduce to closed-form. Then, for all other multivariate Gaussian distributions, we propose a convex optimization framework for approximately computing a specific PID definition based on the statistical concept of deficiency. Using simplifying assumptions specific to the Gaussian case, we provide an efficient algorithm to approximately compute the bivariate PID for multivariate Gaussian variables with tens or even hundreds of dimensions. We also theoretically and empirically justify the goodness of this approximation.

PURPOSE: Anesthesiologists simultaneously manage several aspects of patient care during general anesthesia. Automating administration of hypnotic agents could enable more precise control of a patient's level of unconsciousness and enable anesthesiologists to focus on the most critical aspects of patient care. Reinforcement learning (RL) algorithms can be used to fit a mapping from patient state to a medication regimen. These algorithms can learn complex control policies that, when paired with modern techniques for promoting model interpretability, offer a promising approach for developing a clinically viable system for automated anesthestic drug delivery. METHODS: We expand on our prior work applying deep RL to automated anesthetic dosing by now using a continuous-action model based on the actor-critic RL paradigm. The proposed RL agent is composed of a policy network that maps observed anesthetic states to a continuous probability density over propofol-infusion rates and a value network that estimates the favorability of observed states. We train and test three versions of the RL agent using varied reward functions. The agent is trained using simulated pharmacokinetic/pharmacodynamic models with randomized parameters to ensure robustness to patient variability. The model is tested on simulations and retrospectively on nine general anesthesia cases collected in the operating room. We utilize Shapley additive explanations to gain an understanding of the factors with the greatest influence over the agent's decision-making. RESULTS: The deep RL agent significantly outperformed a proportional-integral-derivative controller (median episode median absolute performance error 1.9% ± 1.8 and 3.1% ± 1.1). The model that was rewarded for minimizing total doses performed the best across simulated patient demographics (median episode median performance error 1.1% ± 0.5). When run on real-world clinical datasets, the agent recommended doses that were consistent with those administered by the anesthesiologist. CONCLUSIONS: The proposed approach marks the first fully continuous deep RL algorithm for automating anesthestic drug dosing. The reward function used by the RL training algorithm can be flexibly designed for desirable practices (e.g. use less anesthetic) and bolstered performances. Through careful analysis of the learned policies, techniques for interpreting dosing decisions, and testing on clinical data, we confirm that the agent's anesthetic dosing is consistent with our understanding of best-practices in anesthesia care.

We present a sample path dependent measure of causal influence between time series. The proposed causal measure is a random sequence, a realization of which enables identification of specific patterns that give rise to high levels of causal influence. We show that these patterns cannot be identified by existing measures such as directed information (DI). We demonstrate how sequential prediction theory may be leveraged to estimate the proposed causal measure and introduce a notion of regret for assessing the performance of such estimators. We prove a finite sample bound on this regret that is determined by the worst case regret of the sequential predictors used in the estimator. Justification for the proposed measure is provided through a series of examples, simulations, and application to stock market data. Within the context of estimating DI, we show that, because joint Markovicity of a pair of processes does not imply the marginal Markovicity of individual processes, commonly used plug-in estimators of DI will be biased for a large subset of jointly Markov processes. We introduce a notion of DI with “stale history”, which can be combined with a plug-in estimator to upper and lower bound the DI when marginal Markovicity does not hold.

When estimating the directed information between two jointly stationary Markov processes, it is typically assumed that the recipient of the directed information is itself Markov of the same order as the joint process. While this assumption is often made explicit in the presentation of such estimators, a characterization of when we can expect the assumption to hold is lacking. Using the concept of d-separation from Bayesian networks, we present sufficient conditions for which this assumption holds. We further show that the set of parameters for which these conditions are not also necessary has Lebesgue measure zero. Given the strictness of these conditions, we introduce a notion of partial directed information, which can be used to bound the bias of directed information estimates when the directed information recipient is not itself Markov. Lastly we estimate this bound on simulations in a variety of settings to assess the extent to which the bias should be cause for concern.

We present a sample path dependent measure of causal influence between two time series. The proposed measure is a random variable whose expected sum is the directed information. A realization of the proposed measure may be used to identify the specific patterns in the data that yield a greater flow of information from one process to another, even in stationary processes. We demonstrate how sequential prediction theory may be leveraged to obtain accurate estimates of the causal measure at each point in time and introduce a notion of regret for assessing the performance of estimators of the measure. We prove a finite sample bound on this regret that is determined by the regret of the sequential predictors used in obtaining the estimate. We estimate the causal measure for a simulated collection of binary Markov processes using a Bayesian updating approach. Finally, given that the measure is a function of time, we demonstrate how estimators of the causal measure may be extended to effectively capture causality in time-varying scenarios.