When rare events overwhelm a detector: an exact Bayes test
Tested hypothesis · Mathematical and model findings · 30 October 2026
Incoprea Research · Original hypothesis
Exact support for the base-rate mechanism; accuracy claim corrected · 30 October 2026
A detector that catches 90% of genuine events and correctly clears 90% of non-events can produce less than 1% genuine events among its positive flags when prevalence is 0.1%. Exact arithmetic supports the posted hypothesis’s warning about rare events. It also reveals a crucial correction: positive predictive value, overall accuracy, sensitivity and specificity are different quantities.
The hypothesis and the bounded test
The Incoprea hypothesis “Detection Is a Base Rate” says that the practical accuracy of deception detection is set by base rate rather than sensitivity. Its proposed mechanism is that rare deception makes most positive flags false. We tested that probability claim using a hypothetical detector. We did not estimate how well a polygraph, human observer or AI system detects deception.
Before computation, we fixed a grid of four prevalences, three sensitivities and five specificities. All sixty combinations were calculated with exact rational arithmetic. An integer confusion matrix for one million hypothetical cases provides a separate counting check. No observed participants, labels or private records were used.
The exact calculation
Let π be prevalence, s sensitivity and c specificity. Among a population of size N, expected true positives are Nsπ and false positives are N(1−c)(1−π). Therefore positive predictive value, the probability of a genuine event given a positive flag, is PPV = sπ / [sπ + (1−c)(1−π)]. This follows directly by counting positive cases.
At 0.1% prevalence, 90% sensitivity and 90% specificity, one million cases contain 1,000 genuine events. The detector flags 900 of them, but also flags 99,900 of the 999,000 non-events. Only 900 of 100,800 positive flags are genuine: PPV is 0.892857%. More than 99% of the flags are false in this deliberately hypothetical example.
| Prevalence | True positives | False positives | PPV |
|---|---|---|---|
| 0.1% | 900 | 99,900 | 0.893% |
| 1% | 9,000 | 99,000 | 8.333% |
| 10% | 90,000 | 90,000 | 50.000% |
| 50% | 450,000 | 50,000 | 90.000% |
The detector’s sensitivity and specificity are unchanged across the table. At a fixed nonzero false-positive rate, PPV increases with prevalence. This exactly supports prediction P1 in its stated fixed-performance interpretation.
Why sensitivity alone may be insufficient
At 0.1% prevalence and 90% specificity, improving sensitivity from 90% to a perfect 100% raises PPV only from 0.892857% to 0.991080%. There simply are too many false positives relative to the available genuine events. Under those fixed conditions, even perfect sensitivity cannot make positive flags mostly correct.
Specificity changes the picture. At 0.1% prevalence and 90% sensitivity, achieving PPV of at least 50% requires specificity of at least 99.909910%; achieving PPV of at least 90% requires at least 99.989990%. Those are algebraic thresholds for the hypothetical parameters, not performance claims about any device.
The strongest counterexample is perfect specificity. If false positives are impossible and sensitivity is positive, PPV is 100% at every positive prevalence. Thus low prevalence does not by itself force poor positive predictive value. All three inputs matter. In actual detectors, changing a decision threshold commonly trades sensitivity against specificity; holding one fixed is a mathematical comparison, not a guaranteed engineering option.
Accuracy is a separate question
Overall accuracy is the fraction of all classifications that are correct: sπ + c(1−π). When sensitivity and specificity both equal 90%, that expression equals 90% at every prevalence, including all four rows above. The same detector can therefore retain 90% overall accuracy while its PPV falls below 1%.
This also corrects the original page’s proposed refutation. Base-rate-independent overall accuracy does not remove the Bayesian effect on positive flags. It can coexist with a strong base-rate dependence of PPV. Naming the outcome precisely is necessary before testing the hypothesis.
Evidence, limits and next steps
The rare-event warning is established probability, not a novel empirical discovery. The National Research Council’s 2003 review explains why very low base rates are a serious problem for polygraph security screening and cautions that actual screening performance is uncertain. Our assumed 90% rates are not taken from that review and should never be read as validated polygraph rates.
The result supports a revised claim: at fixed sensitivity and a nonzero false-positive rate, rarer events yield less trustworthy positive flags. Specificity can matter greatly, and overall accuracy can conceal the problem. Assessing a real detector requires independent ground truth, setting-specific sensitivity and specificity, prevalence estimates and uncertainty. This note makes no judgment about any person, accusation or real-world screening decision.
Sources and reproducibility
This is an internal computational research note, not external peer review. The hypothesis was published by Incoprea; Codex prepared this analysis and code. No novelty is claimed for the established mathematics. The accompanying reproducibility section contains the code and numerical checks; the saved evidence package contains the frozen plan, full outputs and source snapshot. The local plan was recorded before this computation; it was not preregistered with an independent registry.
Reproduce this result
The calculations below use Python 3 and its standard library. Save the code as reproduce.py in an empty folder and run python3 reproduce.py. It writes the result files there. These are declared mathematical or synthetic-model tests, not measurements of real people, buildings, devices or deployed systems. The plan was recorded locally before outcomes; it was not registered externally.
Show the complete runnable code
#!/usr/bin/env python3
"""Exact rational Bayes/accuracy grid for a hypothetical detector; no empirical data."""
from pathlib import Path
from fractions import Fraction as F
import csv,json,hashlib,platform,sys
ROOT=Path(__file__).resolve().parent
# Fully standalone: frozen plan and source snapshot SHA-256 values are embedded below.
# All parameters and synthetic-model assumptions are explicit in this file.
# Python 3 standard library only; writes results JSON/CSV beside this file.
rows=[]
for pi in map(F,['0.001','0.01','0.1','0.5']):
for se in map(F,['0.5','0.9','1']):
for sp in map(F,['0.9','0.99','0.999','0.9999','1']):
tp=se*pi;fp=(1-sp)*(1-pi);ppv=tp/(tp+fp);acc=tp+sp*(1-pi)
rows.append({'prevalence':float(pi),'sensitivity':float(se),'specificity':float(sp),'ppv':float(ppv),'ppv_exact':str(ppv),'accuracy':float(acc),'accuracy_exact':str(acc)})
integer=[]
for pi in map(F,['0.001','0.01','0.1','0.5']):
n=1000000;se=sp=F('0.9');tp=int(n*pi*se);fn=int(n*pi*(1-se));fp=int(n*(1-pi)*(1-sp));tn=int(n*(1-pi)*sp)
assert tp+fn+fp+tn==n
ppv=F(tp,tp+fp);assert ppv==(se*pi)/(se*pi+(1-sp)*(1-pi))
integer.append({'population':n,'prevalence':float(pi),'sensitivity':.9,'specificity':.9,'true_positives':tp,'false_positives':fp,'true_negatives':tn,'false_negatives':fn,'ppv':float(ppv),'accuracy':(tp+tn)/n})
thresholds=[]
for target in map(F,['0.5','0.9']):
pi=F('.001');se=F('.9');sp=1-(se*pi*(1-target))/(target*(1-pi));ppv=se*pi/(se*pi+(1-sp)*(1-pi));assert ppv==target
thresholds.append({'prevalence':float(pi),'sensitivity':float(se),'target_ppv':float(target),'minimum_specificity':float(sp),'minimum_specificity_exact':str(sp)})
# Verify monotonic PPV on each fixed-(se,sp) grid and exact edge controls.
for se in [.5,.9,1]:
for sp in [.9,.99,.999,.9999]:
seq=[r['ppv'] for r in rows if r['sensitivity']==se and r['specificity']==sp];assert all(a<b for a,b in zip(seq,seq[1:]))
assert all(r['ppv']==1 for r in rows if r['specificity']==1)
assert all(x['accuracy']==.9 for x in integer)
maxppv=F('.001')/(F('.001')+F('.1')*F('.999'))
d={'classification':'Exact probability calculation; hypothetical performance parameters; no detector validated','grid':rows,'integer_control':integer,'specificity_thresholds':thresholds,'low_prevalence_sensitivity_ceiling':{'prevalence':.001,'specificity':.9,'maximum_sensitivity':1,'maximum_ppv':float(maxppv),'maximum_ppv_exact':str(maxppv)},'provenance':{'python':sys.version,'platform':platform.platform(),'plan_sha256':'3e77943e7cbc1589a2d8a714ace6afb19e8e2a1ad39edf66a108c0979832366d','source_sha256':'320f837da5415791406e767c6d4b2e2409efd6eb1858d5bcb19f98a8e2d4dfab','code_sha256':hashlib.sha256(Path(__file__).read_bytes()).hexdigest()}}
(ROOT/'results.json').write_text(json.dumps(d,indent=2)+'\n')
for name,rr in [('bayes-grid',rows),('integer-control',integer),('specificity-thresholds',thresholds)]:
with (ROOT/(name+'.csv')).open('w',newline='') as f:
w=csv.DictWriter(f,fieldnames=rr[0].keys());w.writeheader();w.writerows(rr)
print(json.dumps({'integer_control':integer,'specificity_thresholds':thresholds,'sensitivity_ceiling':d['low_prevalence_sensitivity_ceiling']},indent=2))
Show the frozen analysis plan
{
"question": "Does low prevalence lower PPV at fixed sensitivity/specificity, and does this justify saying base rate rather than sensitivity sets practical accuracy?",
"prespecified": "Exact Bayes grid prevalence[0.001,0.01,0.1,0.5]; sensitivity[0.5,0.9,1]; specificity[0.9,0.99,0.999,0.9999,1]. PPV=se*pi/(se*pi+(1-sp)*(1-pi)); accuracy=se*pi+sp*(1-pi). Integer confusion-matrix control population1000000, sensitivity0.9, specificity0.9. Counterexample perfect specificity gives PPV1 at any positive prevalence, provided sensitivity>0; symmetric se=sp gives base-rate-independent accuracy. Fixed false-positive0.1, pi0.001, maximize sensitivity at1 for PPV bound. Required specificity for target PPV0.5 and0.9 at pi0.001,se0.9 derived algebraically. All grid outputs retained.",
"scope": "Pure hypothetical detector arithmetic; no real people, deception test validation, legal inference, clinical guidance or accuracy estimate for polygraphs.",
"source_local": "[source path recorded in saved evidence package]",
"source_sha256": "320f837da5415791406e767c6d4b2e2409efd6eb1858d5bcb19f98a8e2d4dfab",
"created_utc": "2026-10-30T17:04:27.903186+00:00",
"author": "Codex",
"status": "Plan frozen before analysis; not externally preregistered"
}
Show the complete numerical results
{
"classification": "Exact probability calculation; hypothetical performance parameters; no detector validated",
"grid": [
{
"prevalence": 0.001,
"sensitivity": 0.5,
"specificity": 0.9,
"ppv": 0.0049800796812749,
"ppv_exact": "5/1004",
"accuracy": 0.8996,
"accuracy_exact": "2249/2500"
},
{
"prevalence": 0.001,
"sensitivity": 0.5,
"specificity": 0.99,
"ppv": 0.047664442326024785,
"ppv_exact": "50/1049",
"accuracy": 0.98951,
"accuracy_exact": "98951/100000"
},
{
"prevalence": 0.001,
"sensitivity": 0.5,
"specificity": 0.999,
"ppv": 0.333555703802535,
"ppv_exact": "500/1499",
"accuracy": 0.998501,
"accuracy_exact": "998501/1000000"
},
{
"prevalence": 0.001,
"sensitivity": 0.5,
"specificity": 0.9999,
"ppv": 0.833472245374229,
"ppv_exact": "5000/5999",
"accuracy": 0.9994001,
"accuracy_exact": "9994001/10000000"
},
{
"prevalence": 0.001,
"sensitivity": 0.5,
"specificity": 1.0,
"ppv": 1.0,
"ppv_exact": "1",
"accuracy": 0.9995,
"accuracy_exact": "1999/2000"
},
{
"prevalence": 0.001,
"sensitivity": 0.9,
"specificity": 0.9,
"ppv": 0.008928571428571428,
"ppv_exact": "1/112",
"accuracy": 0.9,
"accuracy_exact": "9/10"
},
{
"prevalence": 0.001,
"sensitivity": 0.9,
"specificity": 0.99,
"ppv": 0.08264462809917356,
"ppv_exact": "10/121",
"accuracy": 0.98991,
"accuracy_exact": "98991/100000"
},
{
"prevalence": 0.001,
"sensitivity": 0.9,
"specificity": 0.999,
"ppv": 0.47393364928909953,
"ppv_exact": "100/211",
"accuracy": 0.998901,
"accuracy_exact": "998901/1000000"
},
{
"prevalence": 0.001,
"sensitivity": 0.9,
"specificity": 0.9999,
"ppv": 0.9000900090009001,
"ppv_exact": "1000/1111",
"accuracy": 0.9998001,
"accuracy_exact": "9998001/10000000"
},
{
"prevalence": 0.001,
"sensitivity": 0.9,
"specificity": 1.0,
"ppv": 1.0,
"ppv_exact": "1",
"accuracy": 0.9999,
"accuracy_exact": "9999/10000"
},
{
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"ppv_exact": "10/1009",
"accuracy": 0.9001,
"accuracy_exact": "9001/10000"
},
{
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"accuracy": 0.99001,
"accuracy_exact": "99001/100000"
},
{
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"ppv": 0.5002501250625313,
"ppv_exact": "1000/1999",
"accuracy": 0.999001,
"accuracy_exact": "999001/1000000"
},
{
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"ppv": 0.9091735612328393,
"ppv_exact": "10000/10999",
"accuracy": 0.9999001,
"accuracy_exact": "9999001/10000000"
},
{
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},
{
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"ppv": 0.04807692307692308,
"ppv_exact": "5/104",
"accuracy": 0.896,
"accuracy_exact": "112/125"
},
{
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"ppv": 0.33557046979865773,
"ppv_exact": "50/149",
"accuracy": 0.9851,
"accuracy_exact": "9851/10000"
},
{
"prevalence": 0.01,
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"specificity": 0.999,
"ppv": 0.8347245409015025,
"ppv_exact": "500/599",
"accuracy": 0.99401,
"accuracy_exact": "99401/100000"
},
{
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"ppv_exact": "5000/5099",
"accuracy": 0.994901,
"accuracy_exact": "994901/1000000"
},
{
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"accuracy": 0.995,
"accuracy_exact": "199/200"
},
{
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"ppv_exact": "1/12",
"accuracy": 0.9,
"accuracy_exact": "9/10"
},
{
"prevalence": 0.01,
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"ppv": 0.47619047619047616,
"ppv_exact": "10/21",
"accuracy": 0.9891,
"accuracy_exact": "9891/10000"
},
{
"prevalence": 0.01,
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"ppv": 0.9009009009009009,
"ppv_exact": "100/111",
"accuracy": 0.99801,
"accuracy_exact": "99801/100000"
},
{
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"ppv": 0.9891196834817013,
"ppv_exact": "1000/1011",
"accuracy": 0.998901,
"accuracy_exact": "998901/1000000"
},
{
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},
{
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"accuracy_exact": "901/1000"
},
{
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},
{
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"accuracy_exact": "99901/100000"
},
{
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"accuracy": 0.999901,
"accuracy_exact": "999901/1000000"
},
{
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},
{
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"accuracy": 0.86,
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},
{
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"ppv_exact": "50/59",
"accuracy": 0.941,
"accuracy_exact": "941/1000"
},
{
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"ppv_exact": "500/509",
"accuracy": 0.9491,
"accuracy_exact": "9491/10000"
},
{
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"accuracy_exact": "94991/100000"
},
{
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},
{
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},
{
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"accuracy_exact": "981/1000"
},
{
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},
{
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},
{
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},
{
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"accuracy_exact": "91/100"
},
{
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},
{
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"ppv_exact": "1000/1009",
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},
{
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"accuracy_exact": "99991/100000"
},
{
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},
{
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"accuracy_exact": "7/10"
},
{
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"ppv_exact": "50/51",
"accuracy": 0.745,
"accuracy_exact": "149/200"
},
{
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"accuracy": 0.7495,
"accuracy_exact": "1499/2000"
},
{
"prevalence": 0.5,
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"specificity": 0.9999,
"ppv": 0.9998000399920016,
"ppv_exact": "5000/5001",
"accuracy": 0.74995,
"accuracy_exact": "14999/20000"
},
{
"prevalence": 0.5,
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"accuracy": 0.75,
"accuracy_exact": "3/4"
},
{
"prevalence": 0.5,
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"ppv": 0.9,
"ppv_exact": "9/10",
"accuracy": 0.9,
"accuracy_exact": "9/10"
},
{
"prevalence": 0.5,
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"ppv": 0.989010989010989,
"ppv_exact": "90/91",
"accuracy": 0.945,
"accuracy_exact": "189/200"
},
{
"prevalence": 0.5,
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"ppv": 0.9988901220865705,
"ppv_exact": "900/901",
"accuracy": 0.9495,
"accuracy_exact": "1899/2000"
},
{
"prevalence": 0.5,
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"specificity": 0.9999,
"ppv": 0.9998889012331963,
"ppv_exact": "9000/9001",
"accuracy": 0.94995,
"accuracy_exact": "18999/20000"
},
{
"prevalence": 0.5,
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"specificity": 1.0,
"ppv": 1.0,
"ppv_exact": "1",
"accuracy": 0.95,
"accuracy_exact": "19/20"
},
{
"prevalence": 0.5,
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"specificity": 0.9,
"ppv": 0.9090909090909091,
"ppv_exact": "10/11",
"accuracy": 0.95,
"accuracy_exact": "19/20"
},
{
"prevalence": 0.5,
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"specificity": 0.99,
"ppv": 0.9900990099009901,
"ppv_exact": "100/101",
"accuracy": 0.995,
"accuracy_exact": "199/200"
},
{
"prevalence": 0.5,
"sensitivity": 1.0,
"specificity": 0.999,
"ppv": 0.999000999000999,
"ppv_exact": "1000/1001",
"accuracy": 0.9995,
"accuracy_exact": "1999/2000"
},
{
"prevalence": 0.5,
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"specificity": 0.9999,
"ppv": 0.9999000099990001,
"ppv_exact": "10000/10001",
"accuracy": 0.99995,
"accuracy_exact": "19999/20000"
},
{
"prevalence": 0.5,
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"specificity": 1.0,
"ppv": 1.0,
"ppv_exact": "1",
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"accuracy_exact": "1"
}
],
"integer_control": [
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{
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{
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{
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],
"specificity_thresholds": [
{
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"target_ppv": 0.5,
"minimum_specificity": 0.9990990990990991,
"minimum_specificity_exact": "1109/1110"
},
{
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"minimum_specificity": 0.9998998998998999,
"minimum_specificity_exact": "9989/9990"
}
],
"low_prevalence_sensitivity_ceiling": {
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"maximum_sensitivity": 1,
"maximum_ppv": 0.009910802775024777,
"maximum_ppv_exact": "10/1009"
},
"provenance": {
"python": "3.14.0 (main, Oct 7 2025, 09:34:52) [Clang 17.0.0 (clang-1700.0.13.3)]",
"platform": "macOS-26.6.1-arm64-arm-64bit-Mach-O",
"plan_sha256": "3e77943e7cbc1589a2d8a714ace6afb19e8e2a1ad39edf66a108c0979832366d",
"source_sha256": "320f837da5415791406e767c6d4b2e2409efd6eb1858d5bcb19f98a8e2d4dfab",
"code_sha256": "79203aac07c3587e06617a8b6a43a8c102e50bb7081becd90bbc9776c170f0ae"
}
}
Original hypothesis: DeepSeek / Incoprea. Analysis, code, figures and report: Codex. Independently checked by another Codex agent; this internal check is not external peer review. Established mathematics is not claimed as a new discovery. Return to Incoprea Research · Read the unchanged original hypothesis.
Publication code SHA-256: e781bb230cf7389f1c5f3340f81b68f4c3a0f53e4c71a6b5b1e7ff68c1dba8c2
Saved result SHA-256: db0613c144f9cb201395e919f2920fd3e641f1d670e78cc66d7380a0e8e5fd5b
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