In anytime-valid sequential inference, it is known that any admissible
inference procedure must be based on test martingales and their composite
generalization, called e-processes, which are nonnegative processes whose
expectation at any arbitrary stopping time is upper-bounded by one. An
e-process quantifies the accumulated evidence against a composite null
hypothesis over a sequence of outcomes. This paper studies methods for
combining e-processes that are computed using different information sets, i.e.,
filtrations, for a null hypothesis. Even though e-processes constructed on the
same filtration can be combined effortlessly (e.g., by averaging), e-processes
constructed on different filtrations cannot be combined as easily because their
validity in a coarser filtration does not translate to validity in a finer
filtration. We discuss three concrete examples of such e-processes in the
literature: exchangeability tests, independence tests, and tests for evaluating
and comparing forecasts with lags. Our main result establishes that these
e-processes can be lifted into any finer filtration using adjusters, which are
functions that allow betting on the running maximum of the accumulated wealth
(thereby insuring against the loss of evidence). We also develop randomized
adjusters that can improve the power of the resulting sequential inference
procedure.

本文研究了不同信息集合构建的 e 过程如何结合，通过引入调整因子，该过程能够转移到更精细的 filtration 中，同时开发了随机调整因子以提高顺序推理过程的效力。

结合滤波结果的证据合并

Combining Evidence Across Filtrations

Information networks are becoming increasingly popular to capture complex
relationships across various disciplines, such as social networks, citation
networks, and biological networks. The primary challenge in this domain is
measuring similarity or distance between networks based on topology. However,
classical graph-theoretic measures are usually local and mainly based on
differences between either node or edge measurements or correlations without
considering the topology of networks such as the connected components or holes.
In recent years, mathematical tools and deep learning based methods have become
popular to extract the topological features of networks. Persistent homology
(PH) is a mathematical tool in computational topology that measures the
topological features of data that persist across multiple scales with
applications ranging from biological networks to social networks. In this
paper, we provide a conceptual review of key advancements in this area of using
PH on complex network science. We give a brief mathematical background on PH,
review different methods (i.e. filtrations) to define PH on networks and
highlight different algorithms and applications where PH is used in solving
network mining problems. In doing so, we develop a unified framework to
describe these recent approaches and emphasize major conceptual distinctions.
We conclude with directions for future work. We focus our review on recent
approaches that get significant attention in the mathematics and data mining
communities working on network data. We believe our summary of the analysis of
PH on networks will provide important insights to researchers in applied
network science.

介绍了信息网络中不同的网络测量学方法，重点关注了一种数学工具 —— 持久同调在计算拓扑学中的应用，综述了持久同调应用在网络挖掘问题解决中的不同算法和应用，并强调了最新方法的重要性和潜力。