  Off-policy reinforcement learning enables near-optimal policy from suboptimal
experience, thereby provisions opportunity for artificial intelligence
applications in healthcare. Previous works have mainly framed patient-clinician
interactions as Markov decision processes, while true physiological states are
not necessarily fully observable from clinical data. We capture this situation
with partially observable Markov decision process, in which an agent optimises
its actions in a belief represented as a distribution of patient states
inferred from individual history trajectories. A Gaussian mixture model is
fitted for the observed data. Moreover, we take into account the fact that
nuance in pharmaceutical dosage could presumably result in significantly
different effect by modelling a continuous policy through a Gaussian
approximator directly in the policy space, i.e. the actor. To address the
challenge of infinite number of possible belief states which renders exact
value iteration intractable, we evaluate and plan for only every encountered
belief, through heuristic search tree by tightly maintaining lower and upper
bounds of the true value of belief. We further resort to function
approximations to update value bounds estimation, i.e. the critic, so that the
tree search can be improved through more compact bounds at the fringe nodes
that will be back-propagated to the root. Both actor and critic parameters are
learned via gradient-based approaches. Our proposed policy trained from real
intensive care unit data is capable of dictating dosing on vasopressors and
intravenous fluids for sepsis patients that lead to the best patient outcomes.
