Exchangeable Testing Against an Unknown Benchmark
Abstract
We generate infinite binary exchangeable sequences by sequential comparison of data points against a latent benchmark. Assuming a prior distribution of the benchmark rank R0 within an unobserved group of players, we set up the Bayesian machinery which determines the posterior distribution of the running rank Rn in purely combinatorial terms, by counting insertion slots in a uniformly random ranking. This yields an explicit predictive probability of winning against the benchmark, which is a rational function of the counts. We identify some cases where the predictive probability has particularly simple form, notably a tournament related to the Topp--Leone distribution. The running benchmark rank is a Pólya urn with random initial composition, and its normalised value converges to a latent strength variable with polynomial density. In a projective time scale, the fluctuations of the normalised rank about this limit are Brownian, with the Wright--Fisher coefficient frozen at the limit, while the associated recurrent up-down Markov chain on a high level n of the Pascal graph is approximated by a Wright--Fisher diffusion with a nonlinear drift expressed through the prior density.
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BibTeX
@misc{gnedin2026exchangeable,
title = {{Exchangeable Testing Against an Unknown Benchmark}},
author = {Alexander Gnedin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.04838},
eprint = {2608.04838},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.04838},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exchangeable Testing Against an Unknown Benchmark
AU - Alexander Gnedin
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2608.04838
UR - https://arxiv.org/abs/2608.04838
AB - We generate infinite binary exchangeable sequences by sequential comparison of data points against a latent benchmark. Assuming a prior distribution of the benchmark rank R_0 within an unobserved group of players, we set up the Bayesian machinery which determines the posterior distribution of the running rank R_n in purely combinatorial terms, by counting insertion slots in a uniformly random ranking. This yields an explicit predictive probability of winning against the benchmark, which is a rational function of the counts. We identify some cases where the predictive probability has particularly simple form, notably a tournament related to the Topp--Leone distribution. The running benchmark rank is a Pólya urn with random initial composition, and its normalised value converges to a latent strength variable with polynomial density. In a projective time scale, the fluctuations of the normalised rank about this limit are Brownian, with the Wright--Fisher coefficient frozen at the limit, while the associated recurrent up-down Markov chain on a high level n of the Pascal graph is approximated by a Wright--Fisher diffusion with a nonlinear drift expressed through the prior density.
ER -