Combining Evidence Across Filtrations

1. Problem Statement

The paper addresses a fundamental challenge in Sequential Anytime-Valid Inference (SAVI): how to combine evidence processes (specifically e-processes) that are constructed under different filtrations (information structures) for the same null hypothesis.

• Context: In SAVI, an e-process is a non-negative adapted process that satisfies the optional stopping property ($E_P[e_\tau] \leq 1$) for any stopping time $\tau$. This allows researchers to stop data collection at any time based on the data itself without inflating Type I error.
• The Conflict: While e-processes defined in the same filtration can be combined effortlessly (e.g., by averaging), e-processes defined in different filtrations cannot.
  • If filtration $\mathcal{G}$ is a sub-filtration (coarser) of $\mathcal{F}$, an e-process valid in $\mathcal{G}$ is not necessarily valid in $\mathcal{F}$.
  • Motivating Example: In testing the randomness of financial time series, one might use a Universal Inference (UI) e-process (valid in the full data filtration $\mathcal{F}$) and a Conformal Test Martingale (CTM) (valid only in a coarser filtration $\mathcal{G}$ generated by conformal p-values). Simply averaging these two processes fails because the CTM can exceed expectation 1 at stopping times defined in $\mathcal{F}$ (e.g., stopping based on raw data patterns) even if it is valid in $\mathcal{G}$.
• Goal: Develop a general method to "lift" e-processes from a coarser filtration to a finer one so they can be validly combined.

2. Methodology

The authors propose a framework based on adjusters and lifting lemmas.

2.1. Key Concepts

• Filtrations: $\mathcal{G} \subseteq \mathcal{F}$, where $\mathcal{G}$ contains less information than $\mathcal{F}$.
• P-processes vs. E-processes:
  • P-processes (sequential p-values) satisfy $P(p_\tau \leq \alpha) \leq \alpha$.
  • E-processes satisfy $E_P[e_\tau] \leq 1$.
• The Asymmetry: The paper establishes that p-processes can be lifted from $\mathcal{G}$ to $\mathcal{F}$ "for free" (without modification), whereas e-processes generally cannot.

2.2. The "Adjust-Then-Combine" Procedure

The core methodology involves two steps:

1. Lifting via Adjusters: To lift an e-process $e'$ from a coarser filtration $\mathcal{G}$ to a finer filtration $\mathcal{F}$, one must apply an adjuster function $A$ to the running maximum of the process ($e^*_t = \sup_{i \leq t} e_i$).
   • Definition of an Adjuster: A function $A: [1, \infty] \to [0, \infty)$ is an adjuster if $\int_1^\infty \frac{A(e)}{e^2} de \leq 1$.
   • The Lifted Process: The new process is defined as $e^{adj}_t = A(e^*_t)$.
   • Theorem 2 (e-lifting): If $e$ is an e-process in $\mathcal{G}$, then $A(e^*)$ is an e-process in any finer filtration $\mathcal{F}$.
2. Combination: Once the lifted process is an e-process in $\mathcal{F}$, it can be combined with other e-processes in $\mathcal{F}$ (like the UI e-process) via simple averaging.
   • Corollary 2: $\bar{e}_t = \gamma e_t + (1-\gamma)A(e'^*_t)$ is a valid e-process in $\mathcal{F}$.

2.3. Theoretical Characterization

The authors prove a characterization theorem (Theorem 3) showing that the use of adjusters is not just sufficient but necessary for lifting arbitrary e-processes. If a function maps the running maximum of any e-process in a sub-filtration to a valid e-process in a finer filtration, that function must be an adjuster. This formalizes the "cost" of lifting: one must sacrifice some evidence (via the integral constraint of the adjuster) to regain validity.

3. Key Contributions

1. Identification of the Filtration Problem: The paper rigorously identifies that combining e-processes across different information sets is invalid without correction, a subtle issue often overlooked in sequential testing literature.
2. The e-Lifting Theorem: It provides a constructive proof that any e-process in a sub-filtration can be transformed into a valid e-process in a finer filtration using an adjuster applied to its running maximum.
3. Necessity of Adjusters: The characterization theorem proves that adjusters are the unique class of functions capable of performing this lift for arbitrary e-processes.
4. Distinction between P and E lifting: It clarifies that while p-processes enjoy "free" lifting (Theorem 1), e-processes require a "tax" (the adjuster) to achieve the same property.
5. Practical "Adjust-Then-Combine" Recipe: A generalizable algorithm for combining diverse sequential tests (e.g., Markovian vs. Changepoint detectors) that were previously incompatible.

4. Results and Experiments

The authors validate their theory through simulations and real-world financial data.

• Simulation (Randomness Testing):
  • Setup: Testing binary data for IID randomness using a UI e-process (powerful against Markovian alternatives) and a Conformal Test Martingale (powerful against changepoints).
  • Finding: The unadjusted average of the two processes fails validity checks (mean > 1) at data-dependent stopping times. The adjusted version (using the mixture adjuster $A_{mix}$) maintains validity (mean $\leq 1$).
  • Power: The combined, adjusted process successfully detects both Markovian dependencies and changepoints, leveraging the strengths of both components. While the adjuster incurs a logarithmic "cost" (reducing the growth rate slightly compared to the raw average), the combined process remains powerful.
• Real-World Application (Financial Data):
  • Task: Sequentially testing the randomness of high-volatility days in S&P 500, Amazon, Nike, and USD-CNY data (2019–2023).
  • Result: The combined e-process detected distinct periods of non-randomness. For example, during the 2020 pandemic, the UI component drove the signal (Markovian clustering), while during 2022 inflation spikes, the conformal component (changepoint) drove the signal. The combined process captured both deviations, whereas individual processes missed one or the other.
• Additional Use Cases:
  • Independence Testing: Combining pairwise betting e-processes and rank-based e-processes.
  • Forecast Evaluation: Comparing $h$-step-ahead forecasters where evidence is collected in staggered sub-filtrations.

5. Significance and Implications

• Theoretical Impact: The paper resolves a gap in the theory of SAVI. It proves that powerful e-processes always exist in the data filtration if they exist in a coarser filtration (Corollary 3). This contrasts with test martingales, where powerful martingales may not exist in the full filtration.
• Practical Impact: It enables the construction of robust, anytime-valid tests for complex hypotheses (like randomness or independence) by allowing researchers to mix and match specialized testing strategies (e.g., betting strategies, conformal methods, universal inference) that were previously incompatible due to filtration constraints.
• Cost of Validity: The paper quantifies the trade-off: achieving validity in a finer filtration requires a logarithmic loss in statistical power (evidence accumulation). However, this cost is necessary to prevent "data peeking" from invalidating the test.
• Future Directions: The authors suggest that while adjusters are necessary for arbitrary e-processes, specific problem structures might allow for more powerful, non-adjuster lifts, and explore randomized lifting strategies as a potential avenue for improvement.

In summary, this paper provides the mathematical machinery to unify disparate sequential testing methods, ensuring that the resulting combined evidence remains valid regardless of when or how the experiment is stopped, thereby enhancing the reliability of sequential inference in complex, real-world scenarios.