Cross-entropy rare-event importance sampling

⚠️ Curated Limitations & Negative Results

• Proposal support is required on every event used in a change of measure, including lower pilot thresholds, not only the final target event. The ideal conditional density requires ell>0. Valid support and an unbiased final mean do not imply finite variance, low relative error or guaranteed variance reduction: a finite second moment of the event likelihood ratio is an additional condition. Restricted-family KL fitting is not generally the same as minimizing that second moment.
• The elementary final-estimator argument requires new draws with their stated conditional proposal law. Reusing pilot observations with a denominator fitted from those same observations need not satisfy it, even when all fitted proposals have full support. Adaptive final-sample stopping also needs separate analysis. This does not claim that all reuse or adaptive importance-sampling schemes are invalid; it limits the particular unadjusted averaging argument.
• The weighted likelihood must have a suitable maximizer in the chosen admissible domain; zero usable weights, singular fits, support collapse and numerical likelihood ratios need explicit policies. The source calls a maximized weighted log likelihood convex on PDF p. 11, but a zero-gradient condition is not a generic maximum certificate. In regular natural exponential-family coordinates the usual log likelihood is concave; other parametrizations change curvature. Its following blanket raw-coordinate mean statement on PDF p. 13 must be read through the relevant sufficient statistics and parametrization, rather than transferred from the independent-exponential example to every family.
• The empirical quantile convention, ties, finite pilot sizes and fitted-family restrictions can affect the trajectory. Inclusive score thresholds need not select exactly rho N occurrences. The sample levels need not improve monotonically or reach gamma in a finite run, and no universal fit, variance or convergence guarantee is inferred. The exact-expectation deterministic counterpart is separate from this finite-sample procedure; noisy scores and stochastic-process extensions require additional evidence and assumptions.
• This is the static rare-event construction, not a generic identity for every importance sampler, annealed importance-sampling path estimator, EDA or information-geometric optimizer. Some fitting substeps can overlap without making the complete procedures equivalent. The source’s problem-specific timings and error estimates do not establish universal speedups, sample efficiency or confidence guarantees for arbitrary applications.
• The selected source is the retained 49-page published 2005 tutorial; the separate 47-page 2003 manuscript and earliest papers are not silently treated as equivalent or newly read. Source acquisition, PDF build dates, publication year and later selective readings are distinct. The research inspected the defining static method and selected context, not all application pages or all cited proofs. Author code, calculator-based validation and benchmark reproduction were not part of this evidence. Historical owner 244 and the dated identity assessment are research context, not a statement about uninspected future repository states.