  We study the design of a goal-oriented sampling and scheduling strategy
through a channel with highly variable two-way random delay, which can exhibit
memory (e.g., Delay and Disruption Tolerant Networks). The objective of the
communication is to optimize the performance of remote inference, where an
inference algorithm (e.g., a trained neural network) on the receiver side
predicts a time-varying target signal using the data samples transmitted by a
sensor. Previous formulations to this problem either assumed a channel with IID
transmission delay, neglecting feedback delay, or considered the monotonic
relation that the performance only gets worse as the input information ages. We
show how, with delayed feedback, one can effectively exploit the knowledge
about delay memory through an index-based threshold policy. This policy
minimizes the expected time-average inference error that can be monotone or
non-monotone in age. The index function is expressed in terms of the Age of
Information (AoI) on the receiver side and a parameter regarding the
distribution of subsequent transmission delay, both of which can readily be
tracked.
