Reader route · probability
Follow the event, filtration, and kernel.
The probability layer moves from nonnegative supermartingales and e-processes to common high-probability events for posterior families, then to path measures generated by history-dependent or Markov kernels.
Proof spine
- Build an exponential process from conditional moment control.
- Use Ville-style maximal control or a weighted event allocation.
- Apply a PAC-Bayes change of measure on the same outer event.
- Substitute a path-dependent posterior or catalog selector only after the common event is established.
- For stationary targets, add a finite-depth Poisson correction under contraction.
The repository contains both process-level and event-level theorems. A simultaneous event indexed by all integer times is not automatically an e-process statement; the site keeps those routes separate.
Routes through the API
- E-process foundations
Nonnegative supermartingale constructions, Type-I control, and optional continuation at the process layer.
- Sequential umbrella
Filtrations, conditional means, and reusable sequential concentration mechanisms.
- Finite joint model--strategy posterior
A product prior and joint finite posterior place both selections inside one common PAC-Bayes event; the factorized corollary separates their KL costs.
- Countable predictable-strategy e-process
A normalized countable mixture covers a fixed catalog of predictable strategies; the common event permits later selection of an atom with its explicit weight penalty.
- Countable sleeping-expert master
The exact finite active prefix and closed-form unit-wealth tail equal the real infinite mixture; the resulting predictable master is an e-process when the declared expert wealth processes are.
- History-dependent trajectory measure
Kernels may depend on the entire finite prefix; the capstone shown here begins from a deterministic state.
- Growing-prefix trajectory oracle
The observable finite-state specialization selects the exact boundary within the reporting-time tilt prefix, compares monitored conditional loss with empirical prequential loss, and supplies an LIL-order envelope.
- Arbitrary-measurable trajectory oracle
A joint score-measurability contract supports path-selected continuous posterior measures, arbitrary measurable state and hypothesis spaces, and exact selection over a growing countable tilt prefix.
- Poisson bridge to stationary risk
A logarithmic depth schedule balances finite-depth correction and trajectory uncertainty under explicit contraction and a supplied bound on centered row-risk oscillation.
- Empirical transition layer
Coordinate-wise transition confidence transfers candidate contraction to the true kernel on the selected event.
- Countable transition layer
One event covers every atom in a predeclared geometric tilt catalog; vanishing normalized budgets remain conditional on positive row frequencies and candidate discrepancy convergence.
- 20-state worked application
The opt-in finite-state receipt makes the event-membership and numerical-comparison boundaries explicit.
Quantifiers that matter
- One event, many choices: data-dependent substitution is valid only when the choice appears after membership in the shared good event.
- Finite versus countable: hypothesis spaces, tilt catalogs, state spaces, and time indices have different finiteness assumptions.
- Selection versus mixing: the predictable-strategy catalog is mixed before observation; the reported strategy atom and posterior may then be selected on the common event. This does not create an e-process from the selected atom.
- Known versus estimated dynamics: the Poisson endpoint takes a kernel and invariant PMF; the empirical catalog adds a separate transition-confidence budget.
- Existence versus uniqueness: the canonical finite invariant PMF exists without strict contraction, while uniqueness is explicitly conditional on the selected contraction coefficient being below one.
Next step
Inspect the stochastic-dynamics umbrella.
Use doc-gen search for a declaration, then follow its imported mechanism modules to reconstruct the event.