From the
UK electoral commission and the
police data:
There were 22 reports (0.7 ppm) of voter impersonation at poll stations in 2017 UK parliamentary election.
One conviction, two under investigation, one locally resolved, four false positives, one is unclear, 13 lack evidence.
No information on false negative rate i.e. out of all fraudulent votes, how many go unreported or undetected. In other words, there's no information on how insensitive the voter fraud detection system is.
What we really want to know, is how many successfully cast votes were fraudulent? (AKA the false omission rate)
Let F = Fraudulent vote
Let D = Fraud detected
Bayes law:
P(F|not D) = P(not D | F)*P(F)/P(not D)
Law of total probability:
P(F) = P(F | not D)*P(not D) + P(F | D)*P(D)
Thus:
P(F|not D) = P(not D | F)*(P(F | not D)*P(not D) + P(F | D)*P(D))/P(not D)
= P(not D | F)*P(F | not D) + P(not D | F)*P(F | D)*P(D)/P(not D)
<=>
(1 - P(not D | F))*P(F | not D) = P(not D | F)*P(F | D)*P(D)/P(not D)
<=>
P(F | not D) = P(F | D) * [P(not D | F)/(1 - P(not D | F))] * [P(D)/(1 - P(D))]
In other words,
False omission rate = Precision * False Negative Rate/Sensitivity * Detection Rate/Undetection Rate
Useful definitions(Detection rate = (TP + FP)/(P + N), Undetection rate = (FN + TN)/(P + N))
So far:
Precision >= 1/22 (=1/5 if inconclusive or unclear reports are excluded)
False Negative Rate and Sensitivity are unknown. (97% and 3% respectively if using
NYC DOI data). Generally difficult to acquire due to the legality of the means of information collection required.
Detection rate is about 0.7 ppm
Undetection rate is about 1 - 0.7 ppm or approximately 1.
Throwing caution to the wind and using the "conclusive" precision and NYC data, the rate of successfully cast votes (AKA false omission rate) is approximately 4 ppm.
A more conservative measure using the lower bound for precision gives a false omission rate of 1 ppm.
Assuming furthermore a false negative rate of 50% and 10% gives a false omission rate of 0.03 ppm and 0.003 ppm.
If all cases that lack evidence were also fraudulent, then using the 97%, 50%, and 10% false negative rates, the false omission rates are 20 ppm, 0.6 ppm, 0.06 ppm.
Which value should we believe? Well, assuming that all poll books are accurate, there is no voter ID law, the probability that the poll worker knows the voter is negligible, and the poll worker makes no mistakes (no false positives), then the probability of getting detected is equal to the probability of the fraudulent voter using an identity that has already voted (and gotten marked in the poll books).
Assume for simplicity that all sincere voters have already voted. Then the probability of getting detected is equal to #Voters/#Eligible voters AKA the turnout. Sort of, since we're not accounting for fraudulent voters using the same non-voter false identities. It makes it simpler, however, and the difference is negligible for almost all elections.
This is all assuming the fraudulent voter is using random names from that district (which they probably won't).
In any case, the probability of a fraudulent voter successfully casting a vote is thus 1 - Turnout, and this we can use as an estimate of the False Negative Rate and consequentially the Sensitivity = Turnout.
In 2017, the turnout was 68.8% and thus the false negative rate is estimated to be 31.2%.
The precision varies from 1/22 to 18/22. We can thus estimate it to be 19/44 +- 17/44.
The false omission rate in 2017 is estimated to be 0.1 ppm +- 90% (utter garbage).
Or to put it another way, the expected value of the number of successfully cast fraudulent votes in the 2017 UK parliamentary election is zero to eight votes, rounded to closest integer.
Until proper research is done on the false negative rate, this is the best garbage estimate that I can make.
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