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A false-lock monitor is checked repeatedly and often after seeing earlier warnings. Fixed-time p-values can lose their meaning under that optional stopping. An e-process can accumulate evidence with an anytime-valid type-I guarantee—but only when its conditional supermartingale assumptions match the dependent alarm stream.
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Chapter contents
Choose the route closest to the decision, or read from Chapter 1. The chapters remain open; these grouped routes are a navigable index.
Route 1 · Define the decision
Route 2 · Work the ledgers
Route 3 · Transfer the method
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE DECISION
1. Repeated alarms change the error question
If a monitor unlocks whenever any one of hundreds of checks crosses a fixed-time threshold, its false-unlock probability can greatly exceed the nominal level. The stopping rule is part of the procedure.
The null should describe a correctly identified, stable active manifold. Evidence against it may come from residual drift, inactive-coordinate optimality or feasibility, but every channel is observed through an evolving algorithm.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. An e-process is evidence indexed by time
An e-process is non-negative and calibrated so that, under every distribution in the null, its expected behaviour is controlled at stopping times. A threshold such as 1 divided by alpha converts the process into an anytime-valid test under the stated conditions.
Validity is defined relative to a filtration: the information available before each update. Adaptive step sizes, labels and alarm choices must be measurable within that history, and the conditional betting factor must remain valid.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. A dependence audit for alarm channels
Three monitors may look separate while sharing gradients, iterates and thresholds.
| Channel | Shared inputs | Conditional evidence model | Combination caution |
|---|---|---|---|
| Fast/slow noise ratio | Same accepted steps | Block-aware null | Strong serial dependence |
| Inactive-coordinate residual | Same gradient call | Held-out coordinates | Not independent of label |
| Feasibility check | Separate constraint sample | Conditional e-value | Can provide fresher evidence |
Calling channels independent because they have different names is not enough. Write down the common data path before selecting a combination rule.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Worked anytime-valid unlock ledger
Let alpha be 0.05, so a single valid e-process rejects when evidence reaches 20. Suppose a predeclared mixture averages two channel e-processes with equal weights.
| Check | Noise e-value | Feasibility e-value | Mixture e-value | Action |
|---|---|---|---|---|
| 1 | 1.8 | 0.9 | 1.35 | Stay local |
| 2 | 5.0 | 1.4 | 3.20 | Quarantine acceleration |
| 3 | 18.0 | 6.0 | 12.00 | Keep monitoring |
| 4 | 32.0 | 12.0 | 22.00 | Unlock and reset |
A convex mixture of valid e-values is safe at one time, and a properly constructed mixture e-process remains safe over time. Multiplying dependent channels needs stronger conditional validity and must not be improvised.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Dependence belongs inside the update rule
Serially correlated stochastic gradients, overlapping windows and feedback from rejected steps can invalidate an i.i.d. likelihood ratio. Build increments from a null model or supermartingale bound that accommodates the intended dependence.
When that model is uncertain, use conservative predictable bounds, block-level evidence or a separately sampled gate. Anytime-valid does not mean assumption-free.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Separate warning, quarantine and unlock states
A warning can slow local acceleration without claiming the manifold is wrong. Quarantine gathers fresh global-space evidence. Unlock changes the active structure only after the anytime-valid certificate crosses its boundary.
Reset the e-process only under a predeclared renewal rule. Repeatedly restarting after disappointing evidence can recreate the same multiplicity problem in a new form.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Changed-context transfer: support identification
Transfer the method to sparse optimisation where an inactive coefficient may need to re-enter. An e-process can monitor signed inactive gradients while a separate feasibility stream checks model mismatch.
The transfer check specifies predictable bets, gradient dependence and what survives a restart. It compares false unlocks, missed changes and detection delay over fresh random seeds.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connect the false-lock branch
The dual-window guide separates change alarms from independent gates. The false-lock guide defines the feedback loop and escape state.
Use the adaptive-hysteresis guide to control chattering after a valid unlock.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
An e-value is not the posterior probability that the manifold is wrong. It is a calibrated evidence quantity under a specified null.
Anytime validity handles optional stopping, not arbitrary model misspecification. Dependence, channel selection and resets must be covered by the construction.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Practical evidence checklist
Archive the null, filtration, predictable updates, channel data, dependence assumptions, combination rule, boundary, warning states, reset policy and every stopping time.
Validate type-I error, expected stopping time and power on stable, switching, burst-noise and degenerate cases. Include adversarial dependence patterns, not only independent simulations.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Why not monitor an ordinary p-value repeatedly?
A fixed-time p-value generally does not promise validity after arbitrary optional stopping. Use a sequential method designed for that purpose.
Can dependent e-values be averaged?
Convex averaging is often safer than multiplication, but the whole time-indexed construction still needs e-process validity under the joint null.
Must an unlock happen exactly at 1/alpha?
That is the standard evidence boundary for one level-alpha e-process test. Operational safeguards can require a higher boundary or an additional safety gate.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Howard and colleagues develop time-uniform nonparametric confidence sequences. Ramdas, Ruf, Larsson and Koolen study the martingale structure behind admissible anytime-valid inference. Ramdas and colleagues survey e-processes, confidence sequences and safe anytime-valid inference. The false-unlock ledger above is an original safeguard example.
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