Uniform Single-Peak Log-Likelihood Variance Bound
Abstract
The variance of a uniformly started single-peak log-likelihood path is bounded by twice its accumulated one-step information.
Theorem 1.1 (Single-peak path variance is uniformly controlled by information).
Proof. Machine-checked in Lean as D5/S3/Estimation/TimeArrow/SinglePeakLogLikelihoodVarianceBound.uniform_single_peak_log_likelihood_variance_bound (✓ std3). ∎
Source. Repository-derived.
Commentary.
Let X be a finite sign space with equally many positive and negative signs, and let z be a distinguished positive state. For a peak strength between zero and one, the compensating background strength is r divided by M minus one.
The one-edge second moment psi is at most twice the mean function phi throughout the open unit interval. The odd correction J is nonpositive, the one-edge variance v is at most 2I, and I is nonnegative.
Consequently, for every nonnegative path length, both the forward path and the reversed path have log-likelihood variance at most 2sI. Reversing the path preserves the distribution of the log-likelihood sum, and the zero-length sum has zero variance.