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Several alarm channels may watch the same active manifold: residual drift, feasibility, inactive-coordinate pressure and noise change. Multiplying their evidence can be invalid when they share data. A predeclared weighted mixture of valid e-processes can preserve anytime-valid evidence under a common null while distributing sensitivity across different failure modes.
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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. Mixtures hedge across evidence channels
Let each channel provide a non-negative e-process that is valid for the same stable-manifold null relative to the declared information history. Fixed non-negative weights that sum to one form a mixture process by weighted addition.
The mixture protects against choosing the wrong single channel in advance. It does not create information: a very small weight can delay detection for the failure mode that only one channel sees.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. Dependence changes combination, not merely correlation labels
Residual, feasibility and noise monitors often use the same iterates and gradients. Their evidence can be strongly dependent even when their numerical correlation appears small in one simulation.
A convex mixture does not require channel independence when each component is valid under the common null. Products usually require stronger conditional structure and should not be substituted because they grow faster.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. A channel-weight allocation ledger
A false-lock safeguard allocates mixture weight across three prevalidated e-processes.
| Channel | Weight | Target failure mode | Evidence cost |
|---|---|---|---|
| Inactive residual | 0.50 | Missed active coordinate | One gradient projection |
| Feasibility drift | 0.30 | Wrong local constraint set | Fresh constraint sample |
| Noise shift | 0.20 | Threshold calibration failure | Dual-window update |
The allocation reflects consequence and expected sensitivity, not a belief that the channels are independent. Keep the rationale and any lower weight floor in the audit record.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Worked mixture-evidence path
With level alpha 0.05, the standard crossing boundary is 20. The mixture is 0.50E1 + 0.30E2 + 0.20E3 at each check.
| Check | Residual E1 | Feasibility E2 | Noise E3 | Mixture |
|---|---|---|---|---|
| 1 | 1.6 | 1.1 | 0.9 | 1.31 |
| 2 | 8.0 | 1.8 | 1.5 | 4.84 |
| 3 | 31.0 | 4.0 | 2.5 | 17.20 |
| 4 | 38.0 | 7.0 | 3.0 | 21.70 |
Channel E1 exceeded 20 earlier, but the mixture rule intentionally demands enough weighted evidence. A separate emergency channel would need its own multiplicity allocation and declared boundary.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Calibrate weights without consuming the monitoring stream
Use historical or simulated failure modes to compare expected detection delay and missed-change cost. Select weights on that calibration set, then freeze them for the prospective e-process.
Adapting weights predictably during monitoring can be valid in specially constructed strategies, but choosing whichever channel currently looks largest is not the same as a fixed mixture and requires its own proof.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Guard against dilution and model mismatch
A broad mixture can be slow because evidence is spread across weak channels. Remove channels that cannot be validated, and use a small protected weight for rare but serious failure modes rather than adding every available diagnostic.
Component validity matters more than attractive mixture behaviour. If one channel’s conditional e-value construction fails under feedback or overlap, repair or quarantine it instead of assuming the mixture will dilute the error.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Changed-context transfer: sparse support unlocks
Transfer the method to sparse optimisation, mixing inactive-gradient, validation-loss and feasibility evidence for a coefficient that may need to re-enter the active set.
The transfer check specifies the common null, filtration, predictable updates and reset policy. It evaluates false unlocks and detection delay under serial dependence, regime shifts and near-degenerate supports.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connect the anytime-valid branch
The alarm-dependence guide introduces e-process unlock control. The dual-window guide separates detection from confirmation.
Use the false-lock escape guide to define the operational state machine around the evidence threshold.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
Mixture weights are not posterior probabilities that a channel is correct. They are design allocations inside an evidence process.
Dependence-robust mixture validity does not excuse an invalid component. Every component must satisfy its own null calibration relative to the joint history.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Practical evidence checklist
Archive the common null, filtration, component constructions, shared inputs, fixed weights, calibration data, crossing boundary, emergency overrides, resets and every stopping time.
Report false-unlock rates, detection-delay distributions and sensitivity to weight perturbations. Stress overlapping windows, feedback-driven sampling and simultaneous channel spikes.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Why not multiply the channel e-values?
A product can amplify shared evidence and generally needs stronger conditional validity. A fixed convex mixture offers a safer hedge when dependence is present.
Should equal weights be the default?
They are transparent but not automatically optimal. Consequence, sensitivity and calibration evidence can justify unequal weights.
Can the weights change after a restart?
Yes only under a predeclared renewal design that preserves the intended error guarantee. Post-hoc reweighting should not be treated as the original test.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Vovk studies merging sequential e-values via martingales. Howard and colleagues develop time-uniform confidence sequences. Ramdas and colleagues survey safe anytime-valid inference and e-processes. The mixture ledgers above are original safeguard examples.
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