We introduce an extension of the Pólya tree approach for constructing distributions on the space of probability measures. By using optional stopping and optional choice of splitting variables, the construction gives rise to random measures that are absolutely continuous with piecewise smooth densities on partitions that can adapt to fit the data. The resulting "optional Pólya tree" distribution has large support in total variation topology and yields posterior distributions that are also optional Pólya trees with computable parameter values.

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Brain edema is a severe stroke complication that is associated with prolonged hospitalization and poor outcomes. Swollen tissues in the brain compromise cerebral perfusion and may also result in transtentorial herniation. As a physical and biochemical barrier between the peripheral circulation and the central nervous system (CNS), the blood–brain barrier (BBB) plays a vital role in maintaining the stable microenvironment of the CNS. Under pathological conditions, such as ischemic stroke, the dysfunction of the BBB results in increased paracellular permeability, directly contributing to the extravasation of blood components into the brain and causing cerebral vasogenic edema. Recent studies have led to the discovery of the glymphatic system and meningeal lymphatic vessels, which provide a channel for cerebrospinal fluid (CSF) to enter the brain and drain to nearby lymph nodes and communicate with the peripheral immune system, modulating immune surveillance and brain responses. A deeper understanding of the function of the cerebral lymphatic system calls into question the known mechanisms of cerebral edema after stroke. In this review, we first discuss how BBB disruption after stroke can cause or contribute to cerebral edema from the perspective of molecular and cellular pathophysiology. Finally, we discuss how the cerebral lymphatic system participates in the formation of cerebral edema after stroke and summarize the pathophysiological process of cerebral edema formation after stroke from the two directions of the BBB and cerebral lymphatic system.

Testing and characterizing the difference between two data samples is of fundamental interest in statistics. Existing methods such as Kolmogorov-Smirnov and Cramer-von Mises tests do not scale well as the dimensionality increases and provide no easy way to characterize the difference should it exist. In this work, we propose a theoretical framework for inference that addresses these challenges in the form of a prior for Bayesian nonparametric analysis. The new prior is constructed based on a random-partition-and-assignment procedure similar to the one that defines the standard optional Pólya tree distribution, but has the ability to generate multiple random distributions jointly. These random probability distributions are allowed to "couple," that is, to have the same conditional distribution, on subsets of the sample space. We show that this "coupling optional Pólya tree" prior provides a convenient and effective way for both the testing of two sample difference and the learning of the underlying structure of the difference. In addition, we discuss some practical issues in the computational implementation of this prior and provide several numerical examples to demonstrate its work. Supplementary materials for this article are available online.