Modern GPUs often use near memory or high-bandwidth memory, which may be managed as cache when the application data is too large to fit in the near memory. Unlike CPU caches, the near memory cache has a much larger size. A recent approach is statistical caching, which shows near optimal results when managing large memory for file caching. The prior work is ideal and not practical. This paper outlines two extensions. It first formulates a new caching algorithm called least expected use (LEU) replacement and shows, through examples, that the statistical solution automatically integrates two otherwise disparate policies. Then the paper describes a system design to implement LEU. To position the new design for discussion, the paper draws parallels with two familiar ideas, branch prediction and spectral analysis, and considers a set of opportunities and challenges of achieving statistical caching in near memory. CCS CONCEPTS • Computer systems organization → Heterogeneous (hybrid) systems; • Software and its engineering → Memory management.