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We formulate Sampling Decisions as a path-space relative-entropy projection on a growing autoregressive state graph. The unique prior-relative minimizer is a Doob transform governed by a linear backward recursion. A route-resolved formulation then yields a finite-particle algorithm based on conditional self-normalized importance sampling, and we prove convergence of its transition kernels and terminal law as the path budget grows.\n  For binary graphical models, we prove an exact cancellation theorem: all fixed singleton-product priors disappear from the population correction; only the ordering policy survives. Thus, more accurate one-point marginals may not produce a better finite-budget sampler. We therefore introduce a prefix-dependent autoregressive Local-Boltzm","title":"Sampling Decisions: Exact Path-Space Correction, Prior Cancellation and Local-Boltzmann Guidance","url":"https://arxiv.org/abs/2503.14549","vendor":"arxiv_cs_ai"},"summary":"arXiv:2503.14549v3 Announce Type: replace-cross \nAbstract: How can a cheap but biased sequential, finite-horizon sampler over a discrete space be corrected so that its terminal output follows a prescribed Gibbs distribution? We formulate Sampling Decisions as a path-space relative-entropy projection on a growing autoregressive state graph. The unique prior-relative minimizer is a Doob transform governed by a linear backward recursion. A route-resolved formulation then yields a finite-particle algorithm based on conditional self-normalized importance sampling, and we prove convergence of its transition kernels and terminal law as the path budget grows.\n  For binary graphical models, we prove an exact cancellation theorem: all fixed singleton-product priors disappear from the population correction; only the ordering policy survives. Thus, more accurate one-point marginals may not produce a better finite-budget sampler. 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