  The current work is motivated by the need for robust statistical methods for
precision medicine; as such, we address the need for statistical methods that
provide actionable inference for a single unit at any point in time. We aim to
learn an optimal, unknown choice of the controlled components of the design in
order to optimize the expected outcome; with that, we adapt the randomization
mechanism for future time-point experiments based on the data collected on the
individual over time. Our results demonstrate that one can learn the optimal
rule based on a single sample, and thereby adjust the design at any point t
with valid inference for the mean target parameter. This work provides several
contributions to the field of statistical precision medicine. First, we define
a general class of averages of conditional causal parameters defined by the
current context for the single unit time-series data. We define a nonparametric
model for the probability distribution of the time-series under few
assumptions, and aim to fully utilize the sequential randomization in the
estimation procedure via the double robust structure of the efficient influence
curve of the proposed target parameter. We present multiple
exploration-exploitation strategies for assigning treatment, and methods for
estimating the optimal rule. Lastly, we present the study of the data-adaptive
inference on the mean under the optimal treatment rule, where the target
parameter adapts over time in response to the observed context of the
individual. Our target parameter is pathwise differentiable with an efficient
influence function that is doubly robust - which makes it easier to estimate
than previously proposed variations. We characterize the limit distribution of
our estimator under a Donsker condition expressed in terms of a notion of
bracketing entropy adapted to martingale settings.
