Continuous-posterior singular-fraction PAC-Bayes bounds #
This module mixes the actual lower-tail predictable-mean empirical-Bernstein
processes jointly over an arbitrary measurable hypothesis prior and the
singular fraction family u ↦ exp (-1 / u). One Ville event then controls
every reporting time, every eligible posterior chosen after observing the
path, and every admissible target fraction.
The final corollary is an observable one-sided LIL-order bound for the posterior-averaged conditional prefix mean encountered along the monitored stream. It is not a population, future, deployment, or stationary-risk statement. The scalar fraction mixture is not a posterior over arbitrary predictable betting strategies, and no such strategy-uniform claim is made.
The two joint-measurability arguments are a deliberate caller interface.
hjoint_ambient n must establish strong measurability of the actual process
on the ambient product sigma algebra of paths, fractions, and hypotheses;
hjoint_filtered n must establish the corresponding fact using F n on the
path coordinate. Per-hypothesis adaptedness does not imply either joint fact
for an arbitrary measurable hypothesis space. Concrete callers are
responsible for deriving them from their jointly measurable observation and
conditional-mean families. The measurable singular-fraction map is already
composed into the exact function displayed in both hypotheses.
Joint hypothesis--fraction prior mixture of the actual lower-tail predictable-mean empirical-Bernstein processes.
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The one Ville crossing event for the joint hypothesis--fraction mixture.
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Explicit posterior deviation boundary at one target fraction. The hypothesis KL cost and the log price of the positive-mass singular window are both paid inside the continuous mixture.
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Confidence level obtained by absorbing the posterior KL cost into the confidence parameter of the scalar singular-fraction boundary.
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The KL-adjusted confidence level remains positive.
A probability-level confidence budget remains at most one after the KL adjustment.
Absorbing KL into the confidence level exactly reproduces the PAC-Bayes logarithmic price.
Bounded adapted observations and explicit joint measurability make the joint hypothesis--fraction prior mixture an actual e-process.
The caller supplies both product-space obligations: hjoint_ambient for the
ambient path sigma algebra and hjoint_filtered for F n. These assumptions
refer to the actual process after composition with singularFraction; they
are not inferred from separate per-hypothesis measurability claims.
The joint hypothesis--fraction mixture has one exceptional event of outer
mass at most delta. Its joint-measurability arguments are exactly the two
caller-supplied obligations consumed by
continuousSingularFractionBesselMasterProcess_eProcess.
A failure of one posterior boundary forces the single joint hypothesis--fraction mixture across its Ville threshold.
Outside the one joint crossing event, every eligible posterior satisfies the explicit singular-fraction boundary at every time and every admissible target fraction. The posterior is quantified after the path is observed.
Outside the joint crossing event, substituting the observable tuned fraction gives a one-sided LIL-order bound for every posterior selected from the observed path and every reporting time.
One outer-probability event controls the monitored posterior-averaged
conditional prefix mean for every observed path in the event, every eligible
path-selected posterior, and every reporting time n >= 2. The radius is
one-sided and LIL-order; the claim does not identify the conditional prefix
mean with population, future, deployment, or stationary risk.
The final two hypotheses remain explicit because arbitrary measurable hypothesis spaces need not turn separate section measurability into joint measurability. They are propagated unchanged to the master e-process and its exceptional-event bound.