Reader route · statistics and ML
Start with the estimand and the selection rule.
FormalSLT’s main public endpoints control posterior risk under explicit observation, dependence, variance, and selection assumptions. This route translates the checked statements into the choices a statistician or learning theorist makes.
First question
What quantity do you want to control?
- IID population risk: use the continuous-posterior empirical-Bernstein event.
- Conditional mean under sequential observations: use the forward predictable-residual result.
- Prequential risk under changing dynamics: use the adaptive trajectory event.
- Stationary Markov risk: use the Poisson depth or empirical catalog layer.
Selection is inside the theorem only where stated. Posterior, time, tilt, candidate, and depth choices are not interchangeable. Follow the endpoint declaration before treating a choice as post-data valid.
Theorem routes
- Continuous-posterior empirical-Bernstein event
One IID event, every sample size from two onward, and every posterior measure satisfying the theorem’s absolute-continuity and integrability conditions.
- Finite model--strategy posterior
Both posterior factors may depend on the observed path and reporting time. The shared-strategy ordinary-risk corollary displays separate model and strategy KL charges.
- Countable predictable-strategy master
One e-process covers a fixed countable catalog of legal history-dependent strategies, every finite model posterior, and every reporting time with positive exposure. Post-data selection pays the declared strategy-atom weight.
- Executable betting master
An exact finite active-prefix computation plus a closed-form sleeping tail equals the real countable expert mixture and competes with every active declared expert.
- Observable growing-prefix trajectory oracle
Finite hypotheses and states, path-selected posteriors, exact minimization over the reporting-time geometric prefix, ordinary monitored conditional-risk semantics, an explicit LIL-order envelope, and selected width tending to zero.
- Trajectory empirical-Bernstein PAC-Bayes
Finite-state prefix-dependent dynamics with observed prequential scores and a selected boundary that converges to zero.
- Arbitrary-measurable trajectory oracle
Arbitrary measurable state and hypothesis spaces, path-selected continuous posterior measures, exact growing-prefix tilt minimization, ordinary monitored conditional risk, and a LIL-order envelope.
- Stationary Poisson depth selection
Known finite-state Markov kernels with an invariant PMF, explicit oscillation contraction, and a supplied nonnegative bound on centered row-risk oscillation.
- Empirical stationary catalog
Finite candidate kernels plus empirical transition confidence, with full-row visitation required.
- Countable transition confidence
A predeclared geometric tilt catalog gives simultaneous coordinate bands; normalized row and kernel-budget convergence retains explicit visit-frequency and candidate-discrepancy premises.
- 20-state worked application
A checked finite-state numerical receipt and baseline comparison, imported through the opt-in Applications umbrella and outside the 19-name compatibility promise.
Read the boundary
The formal statements do not turn every adaptive analysis into an anytime-valid one. In particular:
- The continuous-posterior IID result is simultaneous in sample size but is not presented as a forward e-process.
- The forward and trajectory strategy atoms are predeclared; data may select among them but cannot create a new strategy after its scored outcomes are seen.
- The growing-prefix oracle is exact over its declared finite prefix, not over all real tilts, and its selected boundary is not itself an e-process.
- The executable betting master covers a fixed dyadic sleeping-expert catalog; it is not a parameter-free continuum coin-betting theorem or yet composed with the measurable PAC-Bayes risk endpoint.
- The arbitrary-measurable trajectory oracle starts deterministically; its exact argmin is noncomputable, and vanishing width requires the stated pathwise posterior-KL rate.
- The stationary layer is finite-state. Unknown-kernel claims use a finite candidate catalog and observed transition rows.
- Vanishing width is an asymptotic property of the selected boundary, not a finite-sample sharpness claim for every instance.
Next step
Search by statistical concept, then inspect the declaration.
The concept index finds the relevant family; doc-gen gives the checked type and dependencies.