Improving public understanding of probability and risk with special emphasis on its application to the law. Why Bayes theorem and Bayesian networks are needed

It's been a while since we last highlighted the difference between Covid-19 'case' numbers (and by extension this means also hospitalisation numbers and death numbers) and actual Covid-19 illness.

The NHS pathways coronavirus triages website (see https://digital.nhs.uk/dashboards/nhs-pathways) provides an accurate representation of actual illness due to Covid-19 as it combines all 999, 111, online and ambulance calls relating to Covid-19 triages. Previous articles (see links) make clear what the caveats are.

The triage data confirms the real pandemic of spring 2020. I've still yet to see any better evidence that the (vast) majority of 'cases' (i.e. positive PCR test results) since the summer of 2020 have been false positives.

Our paper about the “1 in 3 people with Covid-19 have no
symptoms” claim has had (at time of writing this) 4093 reads since we posted it on researchgate on
Friday, and 336,755 impressions to the tweet about it. The video summary
has been watched by 7,422 people in 2 days.

But, this was the response we got less than 24 hours after
we submitted it to the BMJ:

“Thank you for sending us your paper. We read it with
interest but I am sorry to say that we do not think it is right for the BMJ. In
comparison with the many other papers we have to consider, this one is a lower
priority for us. We do not send out for external peer review manuscripts whose
subject matter, design or topic do not meet our current priorities and are
unlikely to make it through our process.”

Even more bizarrely, neither the medRxiv or arXiv
sites (where we routinely post pre-prints of our research) would accept the
paper. MedRxiv said:

“Thank you for submitting your manuscript to medRxiv. We
regret to inform you that your manuscript is inappropriate for posting. medRxiv
is intended for research papers, and our screening process determined that this
manuscript fell short of that description.”

while arXiv (which initially said: “Your article is
currently scheduled to be announced at Fri, 9 Apr 2021 00:00:00 GMT”) quietly
changed the status of the article to “on hold” as the submission “was identified
by arXiv administrators or moderators as needing further attention.” UPDATE: we now have the following response:

"Our moderators have determined that your submission is not
of sufficient interest for inclusion within arXiv. The moderators have rejected
your submission after examination, having determined that your article does not
contain sufficient original or substantive scholarly research.

As a result, we have removed your submission.Please note that our moderators are not referees and provide
no reviews with such decisions. For in-depth reviews of your work, please seek
feedback from another forum.

Please do not resubmit this paper without contacting arXiv
moderation and obtaining a positive response. Resubmission of removed papers
may result in the loss of your submission privileges"

Compare this with what happened in April 2020 when we
first investigated the Covid-19 data. Whereas our latest work shows that Covid-19 'case' numbers have been exaggerated and that mass testing of asymptomatic people is counter-productive, at that time we were actually concerned
that

a) the numbers infected were being UNDERESTIMATED and b) the data was
being skewed by the fact that ONLY people with extreme covid symptoms
were being tested (and hence we argued for the need for more random
testing).

These views were not considered threatening to the "official narrative"
and of course random testing WAS widely implemented after August. We had no
problem getting those articles published in academic journals (see e.g. here and here are some of our other articles on Covid-19)

But things are completely different when you challenge
the "official narrative". Given that even researchgate has been
censoring such articles it is possible that our latest paper may be
removed. If so a copy can be found here.

Fenton, N. E., Neil, M., & McLachlan, G. S. (2021). "What
proportion of people with COVID-19 do not get symptoms?"
https://doi.org/10.13140/RG.2.2.33939.60968

One of the most
persistent and widely publicised claims made by the UK government and its
scientific advisers about SARS-Cov-2 is that "1 in 3 people who have the
virus have no symptoms”.

However, using
data from a study of asymptomatics at Cambridge [1] we show that
both the “1 in 3” claim and the Office
for National Statistics (ONS) infection rate estimates are exaggerated. The
full analysis is provided in [2], but here we provide a simplified summary and
explanation.

The Cambridge study
uses PCR tests on asymptomatic people, and a person is classified as having the
virus if an initial positive test is confirmed in a follow-up test. Their data
shows that, during a period when the ONS estimated the infection rate was 0.71%,
an average of only 1 in 4,867 people (0.00205%) with no symptoms had the virus
at any time.Although this does not tell
us how many people with the virus had no symptoms, we can conclude the following (the formal
proofs are provided in [2]):

Conclusion 1: If the ONS reported infection rate, 0.71%, is correct, then at
most 2.9% (1 in 34) of people with the virus have no symptoms, and not 1 in 3
as claimed by the government.

Informal explanation: The population of Cambridge is 129,000. So, since only 1 in 4867
asymptomatics have the virus, the maximum possible number of asymptomatics with
the virus is 27. If the ONS claimed infection rate is correct, then 0.71% of
people in Cambridge would have the virus. This is about 916 people. Hence, at
most 27 out of 916 with the virus had no symptoms. That is a maximum of 2.9%
(27/916), 1 in 34.

But, on the
other hand, it tells us:

Conclusion 2: If the government claim that “1 in 3 people with the virus has no
symptoms” is correct, then the ONS reported infection rate must be at most
0.06%. This would mean the reported rate of 0.71% is at least 11 times greater
than the true infection rate.

Informal explanation: We already noted that in Cambridge the maximum number of
asymptomatics with the virus is 27. But if 1 in 3 people with the virus have no
symptoms, then the maximum total number of people with the virus is three times
that number, 81. That means a maximum of 81 out of 129,000 have the virus.
Thus, the maximum infection rate consistent with the government’s claim is
0.06% (81/129,000) and not 0.71%.

Hence, the UK
government claim “1 in 3” claim and the ONS infection rate claim cannot both
be simultaneously true.

What is the
explanation for the government and ONS claims being mutually incompatible, and
what is the likely virus infection rate and proportion of people with
the virus who have no symptoms? The explanation lies in the impact of false positives.
Unlike in the Cambridge study, the government assumes that a person who tests
positive in a single PCR test has SARS-Cov-2 and they are very unlikely to be
subject to confirmatory testing to determine whether this is a true or false
positive. So, an estimated 0.71% infection rate is essentially an estimate that,
if we randomly tested 100,000 people, we would expect 710 to test positive.

However, we
also know from the Cambridge study that 3 in every 1000 people without symptoms
will falsely test positive. Taking this, and all the other observed data into
account in [2] we show that

·The infection rate, for the period in question,
was most likely to be 0.379% (95% confidence interval is 0.372% to 0.387%). We
also performed a sensitivity analysis that showed that the results are robust
against a range of prior assumptions; specifically, there is no reasonable
scenario under which the maximum value for the infection rate was greater than
0.44%.

·The government’s
“1 in 3” claim overestimates the percentage of people with SARS-Cov-2 who have
no symptoms by at least 560%.

·The ONS’s estimate of the infection rate
(0.71%) is exaggerated by at least 80%.

In [2] we also considered the extent to which the
government and ONS claims are exaggerated for different possible infection
rates. As the infection rate drops, the extent to which the estimated infection
rate is exaggerated increases, because the proportion of false positives
inevitably increases. At the current estimated infection rate of 0.35%, it is
likely that the actual infection rate is only around 0.1% meaning that the
current estimated infection rate exaggerates the true rate by 250%. The percentage
of people with SARS-Cov-2, but no symptoms is most likely to be 9.62%, about 1
in 10, not 1 in 3 as claimed.

One other interesting observation we found was that, when the ONS
estimated infection rate is 1% the proportion of people testing positive who
have no symptoms is approximately 1 in 3. Hence, the claim that “1 in 3 people
with the virus have no symptoms” claim is approximately true in this case only if
we replace “with the virus” with “test positive”.

While the focus was on just one UK city there is reason to believe
that the results apply generally throughout the UK, and it is therefore also
reasonable to conclude that SARS-Cov-2 case numbers have been systematically
exaggerated. It is also reasonable to conclude that mass testing of
asymptomatic people (who have not been in recent contact with a person
confirmed as having the virus) may be causing unnecessary anguish for minimal
benefit at a very high societal and economic cost.

[2]N. E. Fenton, M. Neil, and G. S.
McLachlan, “What proportion of people with COVID-19 do not get symptoms?,”
2021. [Online]. Available: http://www.doi.org/10.13140/RG.2.2.33939.60968.