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CoupleFlow: Null-Calibrated Auditory Attention Decoding With Time-Locked Brain-Speech Coupling and Flow-Matching Likelihood Ratios

Ali Alavi, Donald S. Williamson

arXiv:2610.32950v1eess.AS

TL;DR

Auditory attention decoding seeks to identify the attended talker from noninvasive brain recordings, but evaluations must distinguish speech-locked neural information from shortcuts. CoupleFlow uses temporal coupling between EEG and speech, with an alternative PMI decision rule, and across nine datasets it outperforms linear and deep baselines while null tests remain at chance. The paper also derives a window-length accuracy law and identifies recording quality as a limit on decoding speed and reliability.

  • Problem

    AAD aims to identify the attended talker from noninvasive brain activity, but reported evaluations can use trial leakage, eye gaze, and acoustic cues rather than speech-evoked neural activity.

    Source: §1 · pp. 1–2

  • Method

    CoupleFlow scores temporal coupling between EEG and speech embeddings and adds a conditional flow-matching head for a PMI-based decision rule.

    Source: §1 · p. 2, §2 · p. 3

  • Results

    Across nine public EEG and MEG datasets, CoupleFlow outperforms linear and deep baselines under content-disjoint evaluation, while null tests remain at chance.

    Source: §1 · p. 2, §5 · p. 7

  • Takeaways & Limitations

    The evidence-accumulation law links decoding accuracy and the window needed for a target accuracy to recording quality and neural speech-tracking strength.

    Source: §5 · p. 7

    The PMI optimality result assumes the recording depends only on the attended stream; AV-GC and NJU are excluded, and MEG is evaluated only by match–mismatch.

    Source: §5 · p. 8

Abstract

from arXiv · show

Auditory attention decoding (AAD) seeks to identify the attended talker from brief noninvasive brain recordings by comparing neural responses with competing speech streams. Recent electroencephalography (EEG) decoders report nearly perfect accuracy from one second of data, yet we show that this performance fails to generalize to unseen speech because models exploit trial leakage, eye gaze, and acoustic cues instead of speech evoked neural activity. We introduce CoupleFlow, which decodes attention through temporal coupling between neural and acoustic embeddings. Its predictions remain at chance when the brain signal contains no informative coupling, preventing shortcuts based on speech content or talker identity. A conditional flow matching head implements a likelihood ratio decision rule based on pointwise mutual information between brain and speech. This rule is Bayes optimal when the recording depends only on the attended talker and removes preferences for speech that is likely a priori. We evaluate CoupleFlow with content disjoint splits recovered from the audio and null tests that distinguish genuine temporal coupling from trial recognition. Across nine EEG and magnetoencephalography (MEG) datasets processed through a common pipeline, spanning four languages, high density EEG recordings, listeners with hearing impairment, and a room with multiple talkers, CoupleFlow outperforms linear and deep baselines while all null tests remain at chance. Finally, we derive an evidence accumulation law that predicts accuracy across window lengths from a single parameter per dataset and predicts that parameter from the strength of neural speech tracking. This law identifies recording signal to noise ratio as a limit on decoding speed and reliability and estimates the window length required to reach a target accuracy.

1 Introduction

Auditory attention decoding aims to identify the attended talker from noninvasive brain activity, but weak, listener-dependent neural responses and shortcuts complicate reliable evaluation. CoupleFlow targets time-locked brain–speech coupling and is evaluated with content-disjoint data and null tests.

1 Introduction

  • The neural response to speech is weak, varies across listeners, and is mixed with unrelated activity, while speech candidates can differ in loudness, voice, or session identity.The paper reports reconstruction correlations of r = 0.07–0.17 on its two-talker EEG datasets.Source: §1 · p. 1
  • Prior direct classifiers reported 95–97% accuracy at 1 s, but their accuracy drops near chance when training and test data come from different trials.The paper also identifies trial leakage, eye gaze, and audio cues as potential shortcuts.Source: §1 · p. 2
  • CoupleFlow scores candidates through temporal covariance between learned acoustic and EEG embeddings, with equal candidate scores when the EEG lacks time-varying information.This design is intended to keep decoding at chance when the brain signal contains no informative speech coupling.Source: §1 · p. 2
  • Across nine public EEG and MEG datasets under one preprocessing pipeline, CoupleFlow outperforms linear and deep baselines when test speech is withheld from training.Control tests assess whether decoding accuracy comes from the brain.Source: §1 · p. 2
  • A single dataset-specific parameter predicts accuracy across window lengths and is itself predicted from neural speech-tracking strength.The law relates decoding speed and reliability to recording quality and estimates the window needed for a target accuracy.Source: §1 · p. 2

2 CoupleFlow

CoupleFlow embeds EEG and candidate speech as time series and makes decisions from their temporal coupling. A conditional flow head supplies an alternative PMI-based rule, while training losses target decoding, density estimation, and shortcut reduction.

2 CoupleFlow

  • The EEG path applies listener correction, virtual sensors, and a lag-window encoder, using EEG that follows the corresponding sound.The encoder uses a lag δ ∈{0, 125} ms; EEG before the sound serves as a lag control.Source: §2 · p. 2
  • A shared acoustic encoder maps each candidate to an embedding, and the coupling score compares its time-varying dimensions with the neural embedding.Each fraction is a Pearson correlation over time; learned weights and temperature scale the score.Source: §2 · p. 3
  • The time-centered coupling score gives every candidate score 0 when the neural embedding does not vary over time, preventing audio alone from favoring a candidate.The decision selects the candidate with the highest score and is at chance when EEG carries no information about the speech.Source: §2 · p. 3
  • A conditional flow-matching head estimates how much the recording lowers the error in generating each candidate envelope, approximating pointwise mutual information.The PMI rule is Bayes-optimal when the recording depends only on the attended talker, and cancels the prior preference for typical speech.Source: §2 · p. 3
  • Training combines decision and density losses with anti-shortcut and auxiliary terms, including an audio-only adversary and an EEG-only side head used during training.The side head and envelope read-out do not enter reported decisions; checkpoints are selected using validation contribution, defined as accuracy minus across-trial null accuracy.Source: §2 · pp. 3–4

3 Theory

The theory establishes conditions for chance-level performance under a null and for PMI-based Bayes-optimal decisions. It also derives an evidence-accumulation law linking accuracy to window length and a dataset-specific parameter.

3 Theory

  • Under candidate-shared scoring, an independent brain recording, and permutation-invariant candidate speech, null accuracy is at most 1/K.The across-trial and circular nulls instantiate the independence condition.Source: §3 · p. 4
  • When the recording depends on candidates only through the attended stream and attention is uniform and independent of candidate envelopes, maximizing PMI gives the MAP decision.Scoring log p(ak | x) alone adds a term based on audio likelihood, which can favor typical speech.Source: §3 · p. 4
  • The evidence-accumulation law models accuracy across window lengths from the gap between attended and competing correlations, assuming effectively independent samples and independent Fisher-transformed correlations.It also gives the window length needed for a target accuracy; κ is a window-independent figure of merit whose value reflects model, training-data, and recording quality.Source: §3 · p. 4

4 Experiments

Across content-disjoint evaluations, CoupleFlow outperforms strong baselines on most datasets, while its accuracy follows a simple window-length accumulation law. The experiments also test the flow decision rule, multiple-talker decoding, and whether recent side classifiers generalize beyond within-trial splits.

Data and protocol

  • The evaluation groups trials sharing speech into content-disjoint folds and tests across nine datasets processed through a common pipeline.The datasets include high-density recordings, hearing-impaired listeners, single-talker tasks, and a room with four attendable talkers; every accuracy is paired with EEG-uninformative nulls.Source: §4 · p. 4

Main results

  • At 5 s, CoupleFlow is best on seven of eight datasets, with margins of +2.3 to +8.9 points over the strongest baseline; on Broderick it ties the linear decoder.On unseen listeners, it retains 2.0–8.3 points over the pooled linear model wherever either transfers.Source: §4 · p. 5, Table 1 · p. 6

Accumulation law

  • One κ per curve predicts accuracy across 14 dataset–task curves from 1 to 60 s with 0.2–1.8-point mean absolute error; EEG κ is 0.13–0.63 versus 1.02 for MEG.The corresponding T95 is 7–170 s for EEG and about 2.6 s for MEG; prediction of held-out windows has 0.4–2.0-point error.Source: §4 · pp. 5–6, Figure 2 · p. 6

Task difficulty and κ

  • On six two-talker datasets, the task-based prediction κcomp/κmm = 1 − γ matches observed values with mean error 0.02 and r = 0.95.The competing-talker task raises κ by a factor of 1.05–1.38.Source: §4 · p. 5

Predicting κ from tracking

  • A shared constant f fits κ from the per-sample tracking margin ∆ζ across all 14 dataset–task values, with r = 0.996, including EEG and MEG.The figure defines ∆ζ as the Fisher-z margin between attended and negative-candidate correlations with the reconstruction.Source: §4 · p. 5, Figure 4 · p. 7

Flow rule and per-source residuals

  • The flow head’s prior-free PMI rule beats conditional likelihood by 5–25 points on every dataset, while the audio-prior-only rule is near chance at 47.0–50.9%.The attended source has lower brain-conditioned residuals on all datasets, whereas unconditional residuals are nearly equal; the attention effect is mainly in the delta and theta bands.Source: §4 · p. 5, Table 3 · p. 7

Four talkers in a room (MAESTRO)

  • In MAESTRO’s four-talker task, latency-corrected CoupleFlow reaches 71.2% per trial within listeners and 56.9% on unseen listeners, against 25% chance.All 5-s nulls lie in [24.8, 25.3]%.Source: §4 · pp. 5, 7, Table 4 · p. 7

Recent side classifiers

  • Recent side classifiers reproduce 96.8–98.1% at 1 s under within-trial splits but fall to 40–62% on several content-disjoint datasets and remain flat from 1 to 5 s.Fuglsang’20 is an exception at 67–71%; the KU Leuven side task is ill-posed under content-disjoint folds.Source: §4 · p. 7, Table 5 · p. 7

5 Discussion and Limitations

Leakage-free evaluation leaves time-locked AAD signal across nine datasets, while the discussion relates decoding speed to recording SNR and states the attended-stream assumption. The paper also reports how AI tools were used and how results were verified.

5 Discussion and Limitations

  • Leakage-free evaluation leaves time-locked signal across nine datasets; a 5-s target requires κ ≈0.74, reached only by MEG (κ = 1.02).Sequential decisions on calibrated 1-s scores gain 5.3–7.9 points at the same mean decision time.Source: §5 · p. 7
  • On MAESTRO, removing any one safeguard leaves every null at chance, while the time-centred score closes the loudest-talker shortcut by construction.The attended talker is always loudest in this setting.Source: §5 · p. 7
  • Prop. 2 assumes recordings depend only on the attended stream; weak tracking of ignored speech makes that assumption approximate.Matched and same-talker sets partly address this; MEG is evaluated only by match–mismatch, and AV-GC and NJU are excluded.Source: §5 · p. 8

AI use statement

  • The authors used Claude to write and run data-processing, modeling, baseline, experiment, table, figure, text, and proof code, but not to collect data.They report script-generated results, tested code and data checks, and author verification of proofs and references.Source: AI use statement · p. 8

A Extended Related Work

The related work grounds AAD in speech-envelope tracking and contrasts reconstruction, match–mismatch, direct classification, and generative approaches. It highlights evaluation confounds and CoupleFlow’s use of likelihood ratios to cancel audio priors.

Neural basis

  • Auditory cortex tracks the attended speech envelope more strongly than ignored speech, motivating envelope-based noninvasive AAD with EEG or MEG.The ignored talker is still weakly represented, especially under masking.Source: A Extended Related Work · p. 11

Stimulus reconstruction

  • Backward models and CCA reconstruct speech envelopes but need tens of seconds for high accuracy; deep reconstruction raises correlations modestly.Under content-disjoint folds, VLAAI stays below the per-listener linear decoder on every two-talker dataset; CoupleFlow learns joint representations with per-listener correction.Source: A Extended Related Work · p. 11

Match–mismatch

  • Match–mismatch tests whether speech evoked a given EEG segment, avoids spatial confounds, and can be applied to single-talker data.Here it is used with held-out stimuli as a same-talker candidate set and for single-talker datasets.Source: A Extended Related Work · p. 11

Direct classifiers

  • Direct side classifiers report 95–97% accuracy at 1 s on KU Leuven, but fall to 40–62% on DTU, ESAA, and the 255-channel set with held-out stimuli.EEG-only classifiers can exploit gaze, lateralized muscle activity, or trial-specific drifts; KU Leuven’s side and story are confounded.Source: A Extended Related Work · p. 11

Brain decoding with learned representations and generative models

  • Generative likelihood-ratio classification cancels the audio prior that would otherwise favor acoustically typical candidates, while InfoNCE is restricted to windows of one listener.The likelihood ratio is built from conditional speech generation and used here to compare brain–speech association.Source: A Extended Related Work · p. 11

B Proofs

The proofs establish content-disjoint folds, chance-level behavior without informative coupling, and a likelihood-ratio decision rule. They also derive how evidence accumulation and tracking strength relate to accuracy and recording SNR.

Content disjointness

  • Whole-component test folds cannot share linked speech with training or validation streams, although shared content in chunks under 10 s or below the linking threshold is not covered.The folds are unions of connected components built from envelope links.Source: B Proofs · p. 13

Chance-level decoding

  • When EEG is conditionally independent of candidate speech given attention, permutation symmetry bounds accuracy at 1/K; a time-constant embedding ties all candidates and cannot exceed chance.The assumptions hold exactly for same-talker negatives and approximately for competing talkers in counterbalanced designs; the null tests instantiate the independence condition.Source: B Proofs · p. 13

Likelihood-ratio decision

  • Under the attended-stream-only assumption and a uniform attention prior, the MAP decision maximizes pointwise mutual information between each candidate speech stream and the brain signal.Using log p(a_k | x) instead adds the candidate’s audio prior to its score.Source: B Proofs · p. 13

Evidence accumulation and SNR

  • The evidence-accumulation proof models the attended-minus-ignored tracking statistic as Gaussian with variance set by effective sample count; its noise model predicts tracking strength rises with SNR and saturates.The noise-model approximation assumes weak correlations, and the two envelopes are exact independent negatives or approximately weakly correlated.Source: B Proofs · p. 13

C Datasets and Preprocessing

The appendix documents a unified preprocessing pipeline and nine public datasets, then details audio-derived content grouping, alignment checks, and exclusions. It also records a post-experiment timing correction that was not evaluated in the reported experiments.

Datasets

  • The dataset collection spans nine releases and includes EEG and MEG, four languages, high-density EEG, hearing-impaired listeners, and multi-talker room recordings.Examples include 255-channel EEG, a hearing-impaired cohort, Mandarin speech, 208-channel MEG, and six loudspeakers in a room.Source: Table 7 · p. 13, C Datasets and Preprocessing · p. 14

Timing limitation

  • Version 2.1 corrects a 70-ms EEG–stimulus misalignment discovered after the experiments, and the experiments were not rerun on that version.The reported lag window spans about 1 s and was selected on validation data.Source: C Datasets and Preprocessing · p. 15

Content groups

  • Trials are grouped by audio-envelope similarity, and test folds comprise whole groups, so no test stream is linked to a training or validation stream.The two-stage search confirms candidate links at 64 Hz; connected components define groups, and folds are unions of whole groups.Source: C Datasets and Preprocessing · p. 15, Table 9 · p. 16

Preprocessing

  • EEG preprocessing includes bad-channel removal, average reference, resampling, filtering, canonical-channel mapping, and per-trial, per-channel z-scoring.MEG retains 208 sensors; the envelope is computed from 28 gammatone bands spanning 50 Hz–5 kHz.Source: C Datasets and Preprocessing · pp. 15–16

Alignment checks

  • MAESTRO talkers are aligned using one global 0.781-s latency, with EEG tracking of the six-talker mixture peaking at zero lag after correction.Without correction, a nominally post-sound window covers EEG from 0.78 s before sound; pre-sound EEG reaches 78.0% per-trial accuracy versus 70.2% causal.Source: C Datasets and Preprocessing · p. 16

Excluded datasets

  • NJU is excluded because released audio timing cannot be restored, while AV-GC is excluded because its re-paired podcasts leave no content-disjoint split.NJU also lacks attended-envelope tracking under the tested alignment checks.Source: C Datasets and Preprocessing · pp. 14, 16

D Network Details

CoupleFlow uses listener-adapted neural and shared acoustic encoders to score time-locked brain–speech coupling, with a flow-matching head providing a likelihood-ratio rule. Its architecture constrains the decision to temporal neural information and includes separate training or reporting branches that do not enter reported coupling accuracies.

Input and listener correction

  • A rank-2 listener correction adapts the shared EEG model, followed by eight virtual sensors; the decision path has 25,689 shared parameters plus 256 per listener.The correction starts at zero, so each listener departs from the shared model only as far as the data support.Source: D Network Details · pp. 17–18, Figure 5 · p. 18

Candidate features and quantile matching

  • Candidate features combine broadband, onset, sub-band, glimpsed, and masked channels; quantile matching equalizes candidate envelope distributions to remove loudness and level cues.The 54-channel candidate input uses 8- and 28-band envelopes, while the two envelope-only datasets use four channels.Source: D Network Details · p. 19, Figure 7 · p. 20

Encoders and lag window

  • The causal audio encoder uses the 0.48 s of audio up to time t, while the EEG encoder reads a 0.98 s window after t, fixing the brain-response lag direction.The neural lag is 0 or 125 ms, chosen on validation; each decision window includes 100 context samples on each side to avoid padding-dependent outputs.Source: D Network Details · pp. 18–19, Figure 6 · p. 19

Coupling score

  • The coupling score combines weighted, time-centered correlations between EEG and candidate embeddings, yielding zero for a time-constant neural embedding.Centring the scores over candidates makes the decision depend on how the signals co-vary over time, not on embedding scale or offset.Source: D Network Details · p. 19, Figure 8 · p. 20

Flow head

  • The flow head estimates each candidate’s pointwise mutual information with the recording from shared conditional and unconditional flow-matching residuals, cancelling the audio-only likelihood term.If the network ignores the neural embedding, every candidate receives a score of zero.Source: D Network Details · pp. 19–21, Figure 9 · p. 21

Auxiliary heads and compute

  • Separate side, audio-adversary, and envelope-readout branches support training or diagnostics but do not enter reported coupling decisions; all modules run in time linear in window length.The side head’s contribution is reported separately, and the envelope read-out is not used by any decision rule.Source: D Network Details · pp. 21–22, Figure 10 · p. 22

E Training Objective

The training objective combines candidate classification with contrastive brain–speech matching and constraints intended to discourage shortcuts or collapsed representations. Auxiliary heads shape or measure representations without entering reported decisions, and the additional-results table covers performance across window lengths and nulls.

E Training Objective

  • The task cross-entropy ranks candidates using the coupling score plus a side-head score during training, but every reported decision uses the coupling score alone.Label smoothing reflects that the attended-side label is an instruction and may not match attention during every window.Source: E Training Objective · p. 22
  • InfoNCE matches each recording to its attended speech among 32 windows from one listener, providing more supervision than the two-candidate task.Because the batch comes from one listener, listener identity cannot solve the matching problem; the time course of the response must do so.Source: E Training Objective · pp. 22–23
  • The null hinge requires the attended speech to score higher with its own recording than with another or a constant recording; the audio adversary removes cues that predict the attended candidate.VICReg guards against collapsed neural embeddings; the adversary is at chance at the first and selected epochs on every dataset.Source: E Training Objective · pp. 23–24
  • Flow matching trains the envelope-generation head and encourages the neural embedding to predict the attended envelope, while separate side and envelope read-outs provide auxiliary measurements.Lateralised EEG can also reflect gaze or listener identity, so side-head credit is measured separately and excluded from reported decisions.Source: E Training Objective · pp. 23–24
  • Loss weights were fixed across datasets rather than tuned per dataset, and removing any single loss term changes 5-s accuracy by at most 0.8 points on KU Leuven, DTU, and MEG-MASC.On MAESTRO, removing InfoNCE changes per-trial accuracy by -3.6 points.Source: E Training Objective · p. 24

F Additional Results

  • Table 13 reports CoupleFlow and baseline accuracy at each decision window on content-disjoint folds; its caption reports CoupleFlow nulls of 49.7–50.5% through 10 s and 42.5–61.1% at 20–60 s.The caption also states that DECAF stays near chance on the two-talker sets.Source: F Additional Results · p. 25, Table 13 · p. 26

F.1 Ablations

Ablations identify the per-listener correction as the main accuracy-bearing component in the reported 5-s tests, while lag controls lose accuracy. The leakage analyses distinguish limited gains from shared stories across trials from much larger gains when training and test windows share trial identity.

F.1 Ablations

  • Removing the per-listener correction lowers 5-s accuracy by 3.0 points on KU Leuven, 4.2 on DTU, and 2.0 on MEG-MASC; the lag control loses 18.8, 13.7, and 15.3 points, respectively.Other individual components change 5-s accuracy by at most 0.8 points; MEG remains above chance under the lag control.Source: §F.1 · p. 27, Table 14 · p. 27

F.2 Two kinds of leakage

  • Sharing stories across trial-disjoint training and test folds changes accuracy by no more than +2.0 points across the evaluated models.The trial split holds out whole trials, but stories from test trials also occur in training trials.Source: §F.2 · p. 27, Table 15 · p. 28
  • Within-trial splits inflate earlier-decoder accuracy by 16.7–31.5 points and yield 96.8–98.1% at 1 s for KU Leuven side classifiers.Such test windows share trial-specific structure and labels with training windows; the compared earlier decoders used a 50%-overlap window split.Source: §F.2 · p. 27, Table 16 · p. 28, Table 17 · p. 29
  • NJU reached 86.0% under a within-trial split despite having no EEG–audio time-locking; content-disjoint folds rule out both story-sharing and trial-identity leakage.Source: §F.2 · p. 28

F.3 Ablation on MAESTRO

On MAESTRO, where loudness offers an audio shortcut, ablations attribute accuracy to components shaping brain–speech coupling rather than to several safeguards. Further analyses test the evidence-accumulation law, examine the frequency bands carrying the attention effect, and report results for hearing-impaired listeners.

F.3 Ablation on MAESTRO

  • On MAESTRO, removing the per-listener correction changes per-trial accuracy by -8.7 points, compared with -3.6 for InfoNCE, -1.9 for sub-band features, and -1.4 for glimpse channels.Unmatched and matched candidates give accuracy within 0.3 points, every null stays at chance, and removing several safeguards changes per-trial accuracy by at most 0.8 points.Source: §F.3 · p. 29, Table 18 · p. 29

F.4 Variants and the accumulation law

  • The accumulation law’s task prediction has mean error 0.02 for the linear decoder and 0.07 for CoupleFlow; its fit to 14 linear values has mean error 3.7%.At matched task and listening condition, MEG tracking is 2.10× SparrKULee EEG tracking; the observed κ ratios are 2.03 for the linear decoder and 1.62 for CoupleFlow.Source: §F.4 · p. 29, Table 20 · p. 30
  • The variant table includes lag-window, input-feature, listener-adaptation, and channel variants; larger rank and per-listener fine-tuning were tested on KU Leuven, DTU, and ESAA but did not explain the ESAA gap.Source: Table 19 · p. 30, §F.4 · p. 30

F.5 Further analyses

  • The delta band carries 71–91% of the PMI attention effect across datasets, theta carries 7–22%, and alpha plus beta carry at most 9%.The reported band profile matches known cortical speech tracking for both MEG and EEG.Source: §F.5 · p. 30
  • Five-second accuracy on Fuglsang’20 is 80.6% for 22 hearing-impaired listeners and 78.5% for 22 normal-hearing listeners; the reported across-seed Mann–Whitney p-values are 0.06–0.46.The evaluation uses the release’s envelopes of unamplified stimuli for every listener.Source: §F.5 · p. 30

F.6 Leave-one-listener-out evaluation

On leave-one-listener-out folds, CoupleFlow generally transfers better than a pooled linear model, though an unseen listener reduces accuracy and Broderick does not transfer. The appendix develops calibrated evidence accumulation and sequential decisions, showing how the evidence rate governs decoding accuracy and decision time.

F.6 Leave-one-listener-out evaluation

  • CoupleFlow stays ahead of the pooled linear model where both transfer, but unseen listeners cost 3.4–9.2 points at 5 s except on Broderick, where neither model transfers.Folds hold out both a listener and content; Broderick's failure is shared by the pooled linear model.Source: §F.6 · pp. 30–31, Table 21 · p. 31

G.1 Theory

  • Under additive-evidence assumptions, summed log-likelihood ratios give the minimum-error decision, while majority voting is optimal only when all window margins have equal magnitude.The model's single-window accuracy follows Φ(κ√T), with κ linked to the information rate.Source: §G.1 · pp. 31–32
  • Summing log-odds uses more evidence than majority voting, which needs about π/2 times as many windows as summing when per-window information is small.The comparison is asymptotic as µ → 0.Source: §G.1 · p. 32
  • The sequential rule reaches the same error rate in less mean time than a fixed-length decision, with E[TA]/Tfix ratios of 0.59, 0.54, 0.49, and 0.42 at α = 0.2, 0.1, 0.05, and 0.01.The comparison assumes the same κ and the theory's sequential-decision assumptions.Source: §G.1 · p. 32
  • A stopping decision at |S_N| ≥ A has posterior correctness Λ(|S_N|), so its accuracy is at least Λ(A) regardless of its length.The stopping decision reads the sign of accumulated calibrated log-odds.Source: §G.1 · pp. 31–32
  • Summing overlapping-window log-odds preserves fixed-length decisions but overstates evidence by the overlap factor unless scaled by h/w.With this correction, sequential checking more often only reduces threshold overshoot.Source: §G.1 · p. 32

G.2 Calibration and combining windows

  • Calibrated 1-s log-odds have Var/(2 mean) = 0.99–1.13 and lag-1 autocorrelation 0.05–0.05, indicating consecutive windows are close to, but not exactly, independent.One validation-fitted scale brings every dataset to ECE ≤0.028; averaging three seeds' calibrated log-odds adds 1.6–2.1 points at 5 s.Source: §G.2 · p. 33, Table 22 · p. 34

G.3 Sequential decisions

  • At A=1, the sequential rule gains 5.1–5.6 points for CoupleFlow and 3.8–5.5 for the linear backward model; at A=2, gains are 5.3–7.9 and 3.3–8.0, respectively.At equal mean decision time, both rules use the same calibrated scores; observed times are 0.92–1.08 times Wald's prediction.Source: §G.3 · pp. 33–34, Figure 11 · p. 34, Table 23 · p. 35

G.4 Anatomy of the sequential decisions

The threshold largely sets sequential-decision accuracy while decision length adapts to evidence strength. Overlapping windows require correction to keep evidence calibrated, and trial endings can force decisions before the threshold is reached.

G.4 Anatomy of the sequential decisions

  • At A=1, decisions are 74.9–80.2% correct (Λ(1) = 73.1%); at A=2, they are 86.1–90.6% (Λ(2) = 88.1%) across datasets.Accuracy is close to the threshold guarantee, with small deviations linked to overshoot and mild dependence between consecutive windows.Source: §G.4 · p. 35, Figure 12 · p. 35, Table 24 · p. 36
  • At A=1, median decision length is 3–7 s and the 90th percentile is 7–21 s; at A=2, these are 6–17 s and 14–49 s.Mean length grows roughly as A(1 − 2α)/(2κ²), so lower-κ datasets wait longer.Source: §G.4 · p. 35, Figure 12 · p. 35, Table 24 · p. 36
  • Decisions of at most 8 s are at most 1.9 points below Λ(2), but the longest decisions are up to 9.7 points below it.The longest decisions come from stretches with poor speech tracking, where a global calibration scale overstates evidence.Source: §G.4 · p. 35, Figure 13 · p. 37
  • When trials end before the threshold is reached, forced decisions are 58.4–79.6% accurate at A ∈ {1, 2}, and occur on up to 37.6% of Broderick decisions at A=2.Such forced decisions do not arise in continuous listening.Source: §G.4 · p. 35, Figure 12 · p. 35, Table 24 · p. 36

G.5 Overlapping windows

  • For overlapping 1-s windows, scaling summed log-odds by h/w reduces ECE to ≤0.02; the corrected sequential rule gains 0.1–1.6 points at equal mean time.After a decision, restarting at the first window disjoint from it avoids reusing threshold-crossing evidence.Source: §G.5 · p. 36

G.6 Training-window length

The training-window ablation compares CoupleFlow trained on different crop lengths and tests accuracy across window lengths. Mixed crop lengths improve short-window accuracy without materially reducing longer-window accuracy.

G.6 Training-window length

  • Mixed-length training raises 1-s accuracy on all 8 datasets by +0.4 to +3.8 points, while lowering neither 5-s nor 30-s accuracy by more than 0.1 points.It also raises the information-rate parameter κ by a factor of 1.05–1.22 and reduces A=2 sequential decision time to 0.76–0.88 of the paper's 5-s-training time.Source: §G.6 · p. 38, Figure 15 · p. 39, Table 26 · p. 40
  • Training on 1-s crops raises 1-s accuracy by +0.8 to +5.9 points across 8 datasets, while changing 5-s accuracy by -1.3 to +0.9 and 30-s accuracy by -1.8 to +0.8.Training on 10-s crops changes 1-s accuracy by -4.3 to -0.2 and 5-s accuracy by -1.4 to -0.1 points.Source: §G.6 · pp. 37–38, Table 26 · p. 40

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