What gives the Bellman equation its form? We show that the recursive properties of optimal value functions follow from three conditions: that the dynamics decomposes through sufficient statistics, that the return decomposes recursively, and that the aggregation of uncertainty is compatible with both. When all three conditions hold on a common state, the Bellman equation arises from their mutual consistency; when one fails, tractability can often be recovered by augmenting the state or by deforming return or dynamics. The same conditions are shown to give rise to three dualities: one between probability and return, one between return and aggregation, and one between aggregation and probability. Our framework reveals these dualities as arising from a single construction, unifying methods developed separately across reinforcement learning, control, and decision theory.
Digital Twins rely on surrogate models to mirror physical systems in real time, yet these models can degrade as operating conditions evolve, a phenomenon known as concept drift. Maintaining surrogate fidelity under drift, particularly when models must also capture aleatoric uncer…
Many signal processing systems ultimately exist to {act}. Whenever the state variable that determines the action to be taken by a decision maker, or agent, is uncertain, the way that uncertainty is represented decides how well the agent performs and how much its performance can b…
Machine learning models for medical image analysis typically lack a reliable measure of confidence, limiting their use in ambiguous or atypical cases. Here we show that Monte Carlo dropout, applied to a multi-task chest-radiograph classifier (eight thoracic findings, 137,593 trai…
Streaming inference pipelines increasingly pair lightweight fast models with Large Language Models (LLMs) that provide rich semantic understanding at substantial cost. The central question of when to invoke the LLM has received limited formal treatment. We cast this as a risk-bas…
Large language models (LLMs) exhibit strong reasoning and world-knowledge capabilities, yet often struggle to gather information effectively across the multi-turn interactions required in sequential decision-making settings. We introduce Amortised Sequential Information Gathering…
Reinforcement Learning (RL) is a sequential decision-making framework in which an agent learns optimal policies through interaction with an environment by maximizing cumulative rewards. Among RL methods, Bayesian Reinforcement Learning (BRL) addresses common practical challenges…