Using the Blackwell-Cover Guessing Strategy to Separate Unobserved Bernoulli Trials - II
Abstract
In a prior paper, we investigated what could be said about the number in the pair that was not observed when we employed the BlackwellCover Guessing Strategy. In this paper, we generalize prior results to arbitrary distributions. We use a variant of the BlackwellCover strategy ro take pairs of coins chosen from a mixture of two different distributions G and H such that one of the coins is flipped, and as a result of that flip the unobserved (unflipped) coin is placed in one of two sets A and B. We show that, regardless of which distributions are used for G and H, there are situations in which this can be done so that the mean of the heads probabilities of the unobserved coins in A is less than the mean of the heads probabilities of the unobserved coins in B. What makes this result surprising, in addition to its generality, is a comparison with a corresponding result for marbles. If one knows that one will always be weighing one of two marbles with different specific weights, such as 10g and 20g, knowing the weight of the marble being weighed automatically determines the weight of the other. However, even if one knows that one will be flipping one of two coins with specific heads probabilities such as 1/3 and 2/3, the outcome of the flip of one of the coins will not immediately enable one to conclude anything about the heads probability of the unflipped coin, which is where the BlackwellCover Guessing Strategy comes into play.
Related
- Using the Blackwell-Cover Guessing Strategy to Separate Unobserved Bernoulli Trials
- Adaptive Conformal Inference through the Lens of Blackwell Approachability
- Redundancy and synergy in multivariate Gaussians via the Blackwell order
- Robust Design and Evaluation of Predictive Algorithms under Unobserved Confounding
- Strong failures of club guessing at the successor of a regular cardinal
BibTeX
@misc{stein2026using,
title = {{Using the Blackwell-Cover Guessing Strategy to Separate Unobserved Bernoulli Trials - II}},
author = {James D. Stein},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36016},
eprint = {2610.36016},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36016},
note = {CC0},
}
RIS
TY - UNPB
TI - Using the Blackwell-Cover Guessing Strategy to Separate Unobserved Bernoulli Trials - II
AU - James D. Stein
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.36016
UR - https://arxiv.org/abs/2610.36016
AB - In a prior paper, we investigated what could be said about the number in the pair that was not observed when we employed the BlackwellCover Guessing Strategy. In this paper, we generalize prior results to arbitrary distributions. We use a variant of the BlackwellCover strategy ro take pairs of coins chosen from a mixture of two different distributions G and H such that one of the coins is flipped, and as a result of that flip the unobserved (unflipped) coin is placed in one of two sets A and B. We show that, regardless of which distributions are used for G and H, there are situations in which this can be done so that the mean of the heads probabilities of the unobserved coins in A is less than the mean of the heads probabilities of the unobserved coins in B. What makes this result surprising, in addition to its generality, is a comparison with a corresponding result for marbles. If one knows that one will always be weighing one of two marbles with different specific weights, such as 10g and 20g, knowing the weight of the marble being weighed automatically determines the weight of the other. However, even if one knows that one will be flipping one of two coins with specific heads probabilities such as 1/3 and 2/3, the outcome of the flip of one of the coins will not immediately enable one to conclude anything about the heads probability of the unflipped coin, which is where the BlackwellCover Guessing Strategy comes into play.
ER -