The cost of an exact position: tolerant losses, integrated positions and detection thresholds for change-points
Abstract
In exact Bayes rules for change-points, the penalty of a change-point decomposes into a localisation cost of 2 log n, for specifying its position among n candidates, and a dimension cost of order log n per changing coefficient. We show that, for small jumps, the localisation cost is determined by the loss. We introduce losses that tolerate an error of w positions on each change-point, bridging the exact 0–1 loss and a counting loss whose Bayes rule is the posterior mode of the number of change-points. For one small change in a Gaussian mean, the counting rule sums the Bayes factors over all positions; the effective number of positions has a limit law involving a functional of a Brownian motion with drift, and the detection threshold on the squared standardised jump falls from 3 log n to log n. In exchange, the false alarm rate, which still vanishes, is bounded by n^(-1/2) rather than n^(-1), up to logarithmic factors. For several change-points, we prove a band, stated with joint energies, between the thresholds of the two rules. In regression with q changing coefficients, the exact rule needs (2+q) log n, and we conjecture that the counting rule relaxes this to q log n. Simulations compare the rules with other criteria.
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BibTeX
@misc{lyazrhi2026cost,
title = {{The cost of an exact position: tolerant losses, integrated positions and detection thresholds for change-points}},
author = {Faouzi Lyazrhi},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05792398v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The cost of an exact position: tolerant losses, integrated positions and detection thresholds for change-points
AU - Faouzi Lyazrhi
PY - 2026
DA - 2026/10/11
PB - HAL
UR - https://hal.science/hal-05792398v2
AB - In exact Bayes rules for change-points, the penalty of a change-point decomposes into a localisation cost of 2 log n, for specifying its position among n candidates, and a dimension cost of order log n per changing coefficient. We show that, for small jumps, the localisation cost is determined by the loss. We introduce losses that tolerate an error of w positions on each change-point, bridging the exact 0–1 loss and a counting loss whose Bayes rule is the posterior mode of the number of change-points. For one small change in a Gaussian mean, the counting rule sums the Bayes factors over all positions; the effective number of positions has a limit law involving a functional of a Brownian motion with drift, and the detection threshold on the squared standardised jump falls from 3 log n to log n. In exchange, the false alarm rate, which still vanishes, is bounded by n^(-1/2) rather than n^(-1), up to logarithmic factors. For several change-points, we prove a band, stated with joint energies, between the thresholds of the two rules. In regression with q changing coefficients, the exact rule needs (2+q) log n, and we conjecture that the counting rule relaxes this to q log n. Simulations compare the rules with other criteria.
ER -