• the solicitation appeared to be from Google — it appeared in my iPhone’s messenger list under the name “Google” beneath a series of historical text messages that had in fact come from Google

• the goo.gl link took a redirect chain via royalstargazer.info:80 (172.67.194.183) a Cloudflare protected whose point of presence is San Francisco. (Of course, I didn’t know this at the time, only later when I went digging).
• that site then initiates a redirect to accounts.google.com as part of what looks like a normal authentication request. The continue address is a path on sites.google.com address (/u/0/d/1_1-DTlya3jU_DQsGibFqcM2-hBwFTse7/preview). The ip address of the accounts.google.com site was 172.253.118.84 which resolves to Mountain View, CA and is no doubt a legitimate address. But notice that both addresses share the same prefix, so wouldn’t necessarily stand out when eyeballing long lists of addresses in a network trace.
• if the user is already logged into Google, accounts.google.com automatically redirects to the sites.google.com site
• this page is where the phishing prompt appears — under the Google domain sites.google.com

I should have smelled the rat at the“visit this link” to secure your account stage, but fortunately I grok’ed that the login prompt URL I was presented with (on: sites.google.com) was somewhere where it shouldn’t have been and I started to dig into the authenticity of the entire chain — many people would not be so lucky or have the skills to do so and would fall for this trap.

Apparently Google has sunset the goo.gl URL shortener but it will not be switched off entirely until August 2025.

I presume that the attacker has acquired control of the SMS numbers that Google has used in the past and this was how the message appeared in the same group as previous authentic messages from Google. Either that, or they used whatever mechanism is used to allow senders to set the sender name. Since I don’t understand how that mechanism works I am hesitant to use the “Report Spam” option lest I forever block myself from receiving authentication messages from Google.

This attack was scary because every part of it (other than the solicitation to act by visiting a link) looked like it was authored by Google — the message arrived from what look like a trusted source, it used a URL shortener operated by Google, it redirected via accounts.google.com and thence to sites.google.com with an authentic looking login prompt for Google. The IP addresses used were all in Northern California and most of them were, in fact, Google addresses. The dodgy redirect via royalstargazer.info was invisible to me when I was using the browser.

I have tried to report this issue to Google via this link: https://safebrowsing.google.com/safebrowsing/report_phish/ but who knows if/when they will act on it.

I have also lodged an abuse complaint with the domain registrar of royalstargazer.info

I wrote this message to my local MP. If you have been affected by, or know anyone who has been affected by, this outrage of corporate dumbfuckery, then please do too

I’d like to draw your attention to a pernicious problem that has been occurring since at least May 2023 [1] and maybe as far back as October 2022.

This is the FaceBook suspended account scam — a scam which I was unaware of until I fell victim to it this Saturday at 2023–02–24 22:43:00+11:00.

The scam works like this:

• somehow, the attackers exfiltrate a valid session cookie from a user’s browser to a machine operating in the US (the mechanism by which this occurs is not understood — the fact that it occurs is a demonstrable fact). The IP address used is a subnet within a block allocated to a provider in Vietnam.
• the attackers use this cookie to resume a session with Facebook and then masquerade as the victim
• the attackers use this capability to link the victim’s FaceBook account to a Meta account and an Oculus account which they control
• all this activity happens without the knowledge of the victim
• the attackers contrive to use the Oculus account, of which they are in full control, in such a way that Meta/Facebook suspends not only the offending Oculus account but also all accounts associated with the aggregating Meta account, including the victim’s FaceBook account
• when the victim next attempts to use the FaceBook/Meta account, they are presented with a “We’ve suspended your account message”

• note that the message advises that appeal is available via the “linked auth.oculus.com account” but note that until the point, the victim did not have an Oculus account.
• there is no way for the suspended FaceBook user to exercise FaceBook’s customer service mechanism because FaceBook, in their infinite wisdom, does not provide a mechanism for suspended users to exercise the help system and in a act of mind-blowing stupidity they direct the email that provides the only mechanism of appeal to the email associated with the offending Oculus account INSTEAD of to the email associated with the SUSPENDED account (e.g the victim)

This is unbelievably frustrating and for users whose social life or liveilhood depends on FaceBook it is understandable that many find the Orwellian situation they have been thrown into to be extremely disorientating and threatening. Many reach out onto other forms of social media like X and Linked In to express their grievances &/or to seek help . Instantly their gripes are met by offers of a “solution” from scammers who clearly stand to benefit from the situation that someone who has paid them benefited from. [4]

FaceBook has been aware of this situation since at least May 2023 [1] yet still insists on being fully complicit with the activities of the scammers by:

a) failing to fix the underlying vulnerability that allows users’ session tokens to be exfiltrated
b) failing to implement any mitigating measure such as reducing the lifetime of session tokens or making it much, much harder to link a FaceBook account to Meta accounts until such time as a) could be properly addressed
c) failing to direct appeal emails to the email of the SUSPENDED account
d) failing to provide other mechanisms for suspended users to appeal their suspension
e) failing to disintiguish between the activities of a legitimate user and those of a user who has hijacked the legitimate users account because of the aforemention uncorrected and unmitigated flaw a) in their own system’s software
f) failing to take any action whatsoever to address the grievances of the users that ultimately fund their own shareholders
g) failing to recognise that their own stupendously incompetent inaction is playing directly into the hands of merciless scammers that are victimising their own user population

Having fallen victim to the mindless stupidity of psychopathic corporations in the past (Telstra Premium SMS, circa 2016) I am fully aware that the only way these pathetic organisations can be brought to account is for Governments to act (as the ACCC did in 2018 when it was able to convince Telstra to shutdown the terminally flawed Premium SMS system).

As my local member and a Minister in the current government I urge you to act.

For more information about the dislocation this is causing please read: [2], [3]

[1] https://siliconangle.com/2023/05/28/facebook-users-lose-accounts-due-unknown-linked-spam-instagram-accounts/
[2] https://tinyurl.com/hackedatmetathread
[3] https://twitter.com/MemeRepos/status/1761549048803410396
[4] https://twitter.com/MemeRepos/status/1761549089043570834/photo/2

Hundreds of thousands of viruses have come from zoonotic origins. Only a handful from labs. There’s your math problem. QED

(QED added by this author).

The problem is, even if the premise is true, it is a specious argument that does not take account of all of the available evidence.

I am going to stipulate, for the purposes of this argument, that there are 100,000 documented cases of zoonotic transfer and also that there have been no more than 10 laboratory leak incidents using the numbers provided to me in the argument above. Perhaps you think I should permit millions of cases of zoonotic transfer, but trust me doing so only makes the argument worse for the natural origin case.

From these numbers, we can derive the Bayesian prior for the hypothesis (H) that: “SARS-CoV-2 is the result of a natural origin and not due to a leak in Wuhan” as 99990/100000 or 99.99%

Now let’s consider, sequentially, two different pieces of evidence:

• the fact that the first outbreak occurred in Wuhan rather than anywhere else in China (E1)
• the fact that the first outbreak occurred 1800 km from the natural reservoir of the closest known relatives of SARS-CoV-2 — RaTG13, RmYN02, et al. That is, 1800 km away from Yunnan province in southwest China. (E2)

Bayesian analysis is a technique for revising a belief in a hypothesis on the basis of the probability that evidence would be present if the hypothesis is true and the probability the evidence would be present if the hypothesis is false. The revised or posterior probability P(H|E) of the hypothesis is given by the formula where Hbar represents the case where the hypothesis is false and P(Hbar) = 1-P(H).

So, let’s see how we revise our belief in H to take account of the chance that the outbreak occurred in Wuhan and not anywhere else in China.

We can estimate this likelihood P(E1|H) as 0.01 on the basis that Wuhan has about 1% of China’s population and there seems no reason to believe that the average Wuhanese would be any more vulnerable to being exposed to a zoonotic transfer event that causes an outbreak in Wuhan than any other Chinese citizen. Given that Wuhan is a highly urbanized, modern city, we might also argue that the average Wuhanese has less chance, however, for the purposes of argument, let’s assume that a Wuhanese is no more or less likely than any other person in China to be exposed in this way.

On the other hand, if the natural origin hypothesis is false, that is the proximal origin actually was a laboratory in Wuhan, then it seems reasonable to assign a probability to E1|Hbar at least as high as 90% since the person first infected is likely to have either worked at the lab or lived in the close vicinity and is more likely to have spread it to others in Wuhan than to people from other places.

This yields a posterior probability of P(H|E1) = 99.1% which, apparently, only slightly weakens the 99.99% certain prior belief in the natural origin hypothesis.

However, we have still not considered all the evidence. Namely, we have not yet considered the evidence of the likely distance of the natural reservoir of the virus from Wuhan.

It is true that we don’t know where the natural reservoir of SARS-CoV-2 or its close progenitors is, but we do know that all the viruses known to be most closely related to SARS-CoV-2 come from Yunnan province and that there is known to be a strong correlation between the geographic location of a host’s habitat and the phylogenetic lineages of the viruses that those hosts contain, so to assume that the natural reservoir is in Yunnan is reasonable, based on the available evidence. We also know that the bat viruses of Hubei province (close to Wuhan) are phylogenetically distinct from those in Yunnan, so a natural reservoir local to Wuhan is unlikely.

How likely are the first recorded cases of a zoonotic transfer event to first occur in a city over 1000km from the most likely natural reservoir of the virus species? In the “hundreds of thousands” of times zoonotic transfer has occurred, has this happened even once? For the purposes of argument, I am going to stipulate that it has happened 500 times, although, this may be absurdly generous to the natural origin case — I would be happy if someone could provide me with a single instance where this has occurred, and we need a lot more than 500 such instances to reduce the serious dent this consideration puts in the natural origin case.

So, P(E2|H) = 500/100000 = 1/200.

Again, if the hypothesis is false, then it is quite likely that the natural origin of the closest known relative is from Yunnan since Wuhan Institute Of Virology is well-known to have specialized in exactly these viruses, so P(E2|Hbar) could be assigned a value of 99/100 and this would not be unfair. However, in the interests of being generous, let’s set this to 80/100 and turn the Bayesian crank again, starting with P(H) = 99.1%.

Lo, now P(H|E1+E2) = 41%, the hypothesis of a natural origin is now less likely to be true than not.

Let’s consider a counter-factual — that Kunming, in Yunnan, has a virology laboratory that intensively studies beta coronaviruses and that the outbreak occurred in Kunming, rather than Wuhan. Then we only need to change P(E1|H) to 80% to reflect the chance that the provincial capital of Yunnan might be more likely than other parts of China to experience the outbreak than other places and P(E2|H) to 99.5% to reflect the fact that there is good reason to believe than 99.5% of zoonotic transfer events result in outbreaks that are within 1000km of the natural reservoir of the pathogen. So:

• P(H) = 99.99%
• P(E1|H) = 80%
• P(E1|Hbar) = 90%
• P(E2|H) = 99.5%
• P(E2|Hbar) = 80%

The end result is P(H|E1) = 99.98% and P(H|E1+E2) = 99.99%.

This is evidence that the difference in probability between what actually happened and a counter-factual is entirely due to the fact that an outbreak in Wuhan resulting from a natural zoonotic transfer event is not that likely and also that the likelihood that a zoonotic transfer event would cause an outbreak over 1000 km from the natural reservoir of the pathogen is also unlikely.

Yes, if the outbreak had occurred in Kunming, then even if Kunming had a highly active research program into beta coronaviruses from the local area, the chance that one could ascribe the outbreak to a virology lab in Kunming would only be 0.01%. However, that this is not where the outbreak occurred. The outbreak occurred in Wuhan, and there is no reason to assume that a natural outbreak of a pathogen whose natural reservoir is in Yunnan is equally likely to occur in Wuhan as it is to occur in Kunming.

This Bayesian argument would produce exactly the outcome the proponents of the natural origin hypothesis expect — if the outbreak had occurred in Kunming. However, the fact is, the outbreak didn’t occur in Kunming — it occurred in Wuhan, and that the fact that it did occur in Wuhan has very real and consequential implications for the analysis of the likelihood of the natural origin hypothesis that must be addressed if it is to be taken seriously.

Of course, a Bayesian argument is only as good as the priors and likelihoods brought to the table. Maybe the true posterior probability of a natural origin is higher than 41%, but those who insist that it is must argue why any or all of the following assignments are unreasonable or introduce additional evidence that strengthens their case.

• P(H) = 99.99%
• P(E1|H) = 1%
• P(E1|Hbar) = 90%
• P(E2|H) = 0.5%
• P(E2|Hbar) = 80%

I am, of course, happy to listen to arguments about why these assignments should be adjusted but what this article shows is that the natural origin argument is not invulnerable on a Bayesian basis — the initial 9999/10000 certainty implicit in the strong prior can easily be eroded if two lines of evidence emerge that the natural origin hypothesis can only explain with a probability of 1 in 100 and that the laboratory leak hypothesis can explain with greater confidence.

Also, note that this argument relied entirely on simple mechanistic reasoning about the location in which the outbreak occurred — that outbreaks, whether natural or via a laboratory leak, tend to occur very close to the reservoir of the pathogen involved and that Wuhan just isn’t that likely to be the first place where a natural outbreak of any kind, let alone one that originated in Yunnan province would naturally occur but is quite likely to be a place where an outbreak caused by a leak from a laboratory that studies exactly those viruses may have occurred.

It hasn’t considered at all the following lines of evidence:

• how well adapted the first samples of the virus were to their human hosts (in comparison to SARS-CoV)
• the history of gain-of-function research and the participation of the WIV or its personnel in gain-of-function research, specifically that involving the insertion of a furin cleavage site into the RBD of a spike protein
• the presence of scientifically interesting furin cleavage sites in SARS-CoV-2 RBD
• the fact that some signatories to the original Lancet and Nature Medicine correspondence that argued the case for a natural origin have since effectively rescinded their support for that correspondence.
• the fact that one of the authors (Baric) of the letter to Science which argued that more investigation was needed, has been heavily involved in gain-of-function research himself and has even authored papers with Shi Zhengli, one of the lead scientists at WIV. This is not to say Baric thinks a laboratory leak is the most likely hypothesis, but he at least thinks it is a plausible enough hypothesis to join his name to the call for more investigation of it.

It is also true that this argument has not incorporated any other evidence favorable to the natural origin case and not favorable to the lab leak case. Proponents of the natural origin case are, of course, entitled to do so and present their own Bayesian arguments and in the interests of a full and frank exchange of ideas, I encourage them to do so.

However, what this article has shown is that a simplistic Bayesian argument like:

Hundreds of thousands of viruses have come from zoonotic origins. Only a handful from labs. There’s your math problem. QED

which starts and stops with the assumption of a strong prior is extremely vulnerable to consideration of even the simplest mechanistic evidence about the relative location of Wuhan to the most likely natural reservoir of the virus and the low probability that a citizen of Wuhan, in particular, is to be patient zero of a disease outbreak caused by a zoonotic transfer event that occurs somewhere else in China but is quite likely to be patient zero of a disease outbreak caused by a laboratory leak that occurs in the same city that they live in.

In the next story, we looked more closely at the properties of parabolas which are tangential to a circle, radius r, at circle radii elevated at ±𝛼 from the x-axis.

We showed that these parabolas are described by equations of the form:

with parameters, a and k, determined as follows:

where 𝛼 is the angle between the radius from the center of the circle to the point of tangential intersection with the parabola and r is the radius of the circle.

We stated, without proof, that the same partial parabola can also be represented with an equation in polar coordinates:

where rp is the distance from the center of the circle, to the parabola, at an elevation of 𝜃 above (or below) the x-axis, 𝛼 is the angle of the radius which intersects the circle at the point of tangential intersection with the parabola and r (or r_c in the image above) is the radius of the circle. For the rest of the article r will refer to the radius of the circle unless specified otherwise.

This article presents the derivation of the equation above and also investigates the properties of the ratio:

Derivation Of Equation in Polar Coordinates

The point (x,y) can be expressed in terms of rp and 𝜃

Because (x,y) also lies on the parabola, parameterized as above we have:

Rearranging we have:

The equation has a trivial solution where 𝜃 = 0, namely:

Solving the quadratic equation for rp, we find:

Next, we make the following substitutions:

which yields:

Multiplying top and bottom by:

which yields:

Factoring out cos²𝛼 and applying trigonometric identities, yields:

For the range of interest:

then the following are all true:

The following limits:

It is clear that the additive arm of the solution equation diverges as 𝜃→0 since the numerator approaches a finite value, 2 cos 𝛼, but the denominator approaches 0 indicating that the ratio increases arbitrarily as 𝜃→0

We know, however, that at 𝜃=0:

so we expect that the other solution must converge to this value. In other words, we expect that the following is true, although we cannot yet demonstrate this:

The truth of this statement is at least plausible since both the numerator and denominator approach the same limit (0).

So, we will put aside the divergent arm of the solution and concentrate on simplifying the remaining arm.

Multiplying numerator and denominator by:

yields:

Simplifying the numerator yields:

Factoring out (cos²𝜃 -1) yields:

Applying trigonometric identities and canceling common terms yields:

which is the equation we intended to derive.

Also, it is now readily apparent that the limit 𝜃→0 is k as expected:

Conclusion And Future Work

By expressing the coordinates of the points of parabola in polar coordinates and substituting those expressions into an equation of the parabola in the cartesian plane, we derived a quadratic equation for rp in terms of rc, 𝛼, and 𝜃. Solving that equation and discarding the divergent solutions yielded an expression for rp in terms of rc, 𝛼, and 𝜃 for the domain 0 ≤ 𝜃 ≤ 𝛼 < π/2

The curve can be extended beyond the domain -𝛼 ≤ 𝜃 ≤ 𝛼 with a somewhat clumsy expression involving use for a floor operator:

Next steps might be to find a way a more natural way to express the symmetry that does not rely on use othe f the floor operator.

One property of these partial parabolas is the tangents of adjacent partial parabolas were identical to each other and also to the circle of radius √2/4, centered on (1/4,/1,4) that passes through the corners of the 1/2 unit square.

To further investigate the partial parabolas of this kind, particularly parabolas arrayed around the edges of a circle, I thought it would be useful to generalize the derivation of their characteristic parameters.

To start with, we will derive the equation of a parabola that subtends an arc of ±𝛼 radians of a circle of radius r, centered on the origin. The parabola’s axis of reflection is the x-axis, specifically:

The chief constraint is that at the point of intersection of the circle of radius r, the parabola must be tangent to the circle and thus perpendicular to the circle radius at angle 𝛼 to the x-axis

The coordinates of the point of intersection are:

The slope of the tangent to the circle at angle 𝛼 is:

The slope of the parabola at y (in terms of y) is:

Given the constraint, we have, at y=r.sin(𝛼)

Re-arranging, we get:

Substituting for x, a, y in the equation of the parabola, we get:

Rearranging and simplifying several times:

The working is not shown, but it is possible to express k as a function of the curvature parameter a, as follows:

Conclusion

This article has investigated the parameters of a parabola that intersects tangentially with a circle of radius r, centered on the origin. The point of intersection is the point at which a radius which is elevated at angle 𝛼 from the x-axis intersects with the circumference of the circle

The parameters fit the equation:

The parameters are:

The diagram below shows the parabolas that intersect the unit circle for radii at various 𝛼 = π/n, for n in the range 3 to 12.

Additional Representations

I will save the derivation of this equation for another article, but here is a representation of the partial parabola in terms of polar coordinates:

here r_p and r_c represent the radial coordinates of the parabola and the reference circle respectively, and 𝛼 is the parameter used to derive the equation of the parabola.

I eventually decided on this Python procedure which exploits a quirk of complex arithmetic whereby a counter-clockwise turn of 𝛑/2 radians can be simply implemented by multiplication by the complex number 0+i. The other insight the code uses is that the length of each side of a spiral increases by 1 for every half-turn.

The outer loop of the function iterates across the turns of the spiral until a maximum number of points has been visited. The inner loop enumerates all the points on a single quarter-turn of the spiral.

def spiral(max):
 “””
     Yields an iteration through a counter-clockwise, 
     square-spiral starting at 0,0 and extending in 
     increments of 1 heading east initially and turning 
     counter-clockwise at each corner of the spiral.
 “”” 
 position = complex(0,0) # origin
 motion = complex(1,0)   # initially, head east
 turn = complex(0,1)     # always turn counter-clockwise
 extent = 0              # initially, zero extent
 
 count = max
 while count > 0:
    if motion.imag == 0:   # if about to head east or west...
      extent = extent + 1  # ...extend by one
    k = min(count,extent)
    count = count — k
    while k > 0:
      yield (position.real, position.imag)
      position = position + motion
      k = k — 1
    motion = motion * turn

Here’s the result of running plotting the output of spiral(625).

For some reason, I was curious as to how the average position of the points in the spiral would change as the number of points in the spiral increased. My initial intuition was that the average would converge to the origin, however, my intuition would prove to be wrong.

The following graph shows how the average position of all points previously plotted evolves:

There are some interesting points to note about this graph:

• it becomes smoother as the number of points in the spiral increases
• it seems to regularly pass through the fixed points (0,0), (0,1/2), (1/2, 0), (1/2, 1/2)
• the curve does seem to converge to a smooth curve, but what is that curve — it looks vaguely elliptical, but is it? (spoiler: it isn’t)

The fixed points in the 1/2 unit square centered on (1/4, 1/4) are interesting, and perhaps not too surprisingly correspond to points where the number of points in the curve has some simple relationship to a perfect square. Let N be the number of points in the spiral, then:

• if N=n² and n is odd, then the average is always (0,0)
• if N= n² and n is even, then the average is always (1/2,1/2)
• if N=(n+1)*n and n is odd, then the average is always (1/2, 0)
• if N=(n+1)*n and n is even, then the average is always (0, 1/2)

In the end, it is no surprise that the average doesn’t converge — perfect squares of odd numbers are pulling the average in one way, perfect squares of even numbers are pulling the average in a different way and there is a transition between the two extremes on each turn of the spiral. The curve of convergence is becoming smoother as the number of points in each quarter-turn of the spiral increases, and the general symmetry of the spiral is preventing arbitrary divergence from the observed curve.

Interestingly, the average of the averages does converge to (1/4, 1/4) as seen with this plot. Once we discover that parametric equation for the curve of convergence, it will become obvious why this convergence happens.

So What Is The Curve Of Convergence?

The curve of averages appears to be converging towards an ellipse-like curve. When I originally plotted the curve, it appeared very ellipse-like, but that was an artifact of scaling decisions made by the graphing software I was using.

How to calculate the curve of convergence?

We need an expression for each point of the curve as N, the number of points in the curve approaches infinity.

Let’s start with a spiral of N points, where N=n² and n is an odd number,

We know that the average for this spiral is 0. This is a consequence of all the points in the spiral being evenly distributed around the origin, so the contribution from each point is canceled out by a point exactly opposite to that point through the origin of the spiral.

Extending the spiral involves adding n points to the spiral starting at ((n+1)/2,-(n-1)/2) and extending to ((n+1)/2,(n-1)/2). These points can be described by the set: ((n+1)/2, -(n-1)/2+k-1) : for k in [1,n] which can be simplified as: ((n+1)/2, -(n+1)/2+k) : for k in [1,n].

The sum of all points of the first N+k points in the spiral is thus:

This simplifies to:

which further simplifies to:

The average for each in point the spiral is obtained by dividing these points by n²+k:

A quick sanity check, if we substitute k=0, the average is (0,0) which is the average at the starting position. If we substitute k=n, the average is (1/2,0), so we have preserved the expected results at the limits.

Now, we define a ratio r=k/n and rewrite the equations in terms of r and n.

Simplifying this we get:

Taking limits as n->∞:

Now, substituting x=r/2, we get:

Or, in the form of a quadratic equation:

Overlaying this over the plot of averages for finite spirals we get the orange curve segment, indicating that the equation is a good fit for the observed curve.

With symmetry arguments, you can derive the 3 equations for the other parts of the curve of convergence.

Plotting these curves yields:

Overlaid over the average of a spiral of finite length. we get:

The derivatives of the curve segments at (0,0) and (1/2,1/2) are -1 and at (1/2,0) and (0, 1/2) are +1 indicating that the complete curve is continuously differentiable and suggesting that it may be possible to derive a single equation for the complete curve, perhaps by rendering it into polar coordinates.

The parabolic equations can be expressed in this form:

which leads to the following 4 equivalent equations:

The focal points of these parabolas are: (1/4, 0), (1/4, 1/2), (0,1/4), (1/2, 1/4)

It is interesting to consider how close these jointed parabolas are to a pure circle centred on (1/4,1/4) that passes through each of the fixed points of the curve. That is a circle with this equation:

It turns out, it is quite close:

The curve of convergence is bounded on the outside by:

Or, shown together:

Approximating The Equation of the Curve

With a little bit of fiddling, I was able to find what appears to be a single equation of the curve that apparently fits 4 parabolic equations almost exactly.

The basic technique was to calculate the ratio of the interval between the parabola and the point (1/4,1/4) and √2/4 for 𝜃 in the range π/4 and 3π/4 and then plot this ratio against 𝜃. This resulted in this plot:

The next step was to fit the equation to the above curve:

The 4𝜃 is related to the 4-fold symmetry of the parabolic approximation and the fact that the ratio between the circle and one parabola cycles from 1 back to 1 again over the 90 degrees between 45 degrees and 135 degrees and does it 3 more times for each of the other parabolas.

The terms (3/(2√2)–1)/2 and (3/(2√2)+1)/2 are related to key dimensions derived from the parabola and the circle it approximates. The first is half the maximum difference of the ratio of the distances between the parabola and circle and the center of the circle along a radius of the circle. The second is the midpoint between these two ratios.

Folding the equation into the equation of a circle of radius 2√2/4 centered on (1+i)/4 and simplifying that results in the final equation.

Note that I have not yet derived this equation directly from the parabolic equations themselves. To assess possible errors between my guesstimate and the parabolic curves, I calculated the distance between the parabolic curves and the point (1/4,1/4) and also between the equation above and (1/4+1/4i) and then calculated the difference between the two values. These are plotted below. It is possible that these errors are due to a higher-order term that the equation above omits, but it may also be due to numerical errors however, I haven’t yet been able to rule either explanation out.

So, still to be done is to derive the final equation of the curve from the parabolic equations and so rule either in or out the presence of higher-order terms.

Another interesting thing to explore is constructing parabolic approximations with a larger number of parabolas. The key will likely be to ensure that the curve is continuously differentiable at the junction between each parabola.

This article applies Bayesian analysis to the hypothesis that the proximal origin of SARS-CoV-2 was an uncontrolled release from a laboratory using, as evidence, one aspect of the SARS-CoV-2 origin story — where the first recorded outbreak occurred.

The goal is not to derive a single definitive estimate of the probability that the proximal origin of SARS-CoV-2 was a laboratory or was natural — any given Bayesian analysis is only as unbiased as its priors and the priors are only as unbiased as those selecting them.

Instead, we sample 2 of the parameters across 2 orders of magnitude and the 3rd across a plausible range, and show how the posterior probabilities vary as the parameters vary across these orders of magnitude. The article shows that even if you do not believe the laboratory origin hypothesis, the location of the first recorded outbreak is still highly relevant to a rational revision of your initial belief in that hypothesis.

Bayes Theorem

The eponymously named Bayes Theorem was discovered by the Reverend Thomas Bayes in the 1700’s and saved for posteriority by an archivist of his papers who discovered the work posthumously. In common language, it provides a rational technique for revising a prior belief in light of new evidence. The equation for Bayes Theorem is given below:

where:

• H is the statement of the hypothesis of interest
• P(H) is the prior probability that the hypothesis is true, independent of the evidence.
• E is the evidence being used to revise the belief in hypothesis
• P(E) is the marginal likelihood of the evidence, independent of the hypothesis
• P(E|H) is the likelihood the evidence, given that the hypothesis is true
• P(H|E) is the posterior probability of the hypothesis, given the evidence.

P(E) is sometimes difficult to estimate, but the following identity must hold:

Here P(E|^H) is the probability of the evidence, assuming the hypothesis is false and P(^H) is the probability the hypothesis is false which is the same as 1-P(H). Estimating the two conditional probabilities P(E|H) and P(E|^H) is generally easier than estimating the unconditional probability, P(E).

The Scenarios

The Hypothesis

The hypothesis, H, for the main scenario is:

That the outbreak of SARS-CoV-2 in the human population was caused by an uncontrolled release of the virus from a virology laboratory in China.

By using the term uncontrolled release, we are specifically excluding from consideration the possibility that the pathogen was deliberately released from the laboratory.

The hypothesis is stated without reference to the particular city in which the outbreak occurred nor to a particular laboratory. The hypothesis only assumes the same state of knowledge that would be shared by anyone learning, for the first time, about the outbreak of a novel coronavirus in China in early 2020.

The Evidence

The evidence, E, for the main scenario is:

The first recorded outbreak of SARS-CoV-2 in the human population occurred in a city that is also home to a virology laboratory that actively performs research on closely related viruses.

In this case, the city is Wuhan and the virology laboratory is run by the Wuhan Institute of Virology.

Supporting Facts

These are the facts of the matter that are relevant to the analysis. These are by no means all the relevant facts to the origin of SARS-CoV-2, but are sufficient to conduct this Bayesian analysis of the factual scenario and a selected counter-factual scenario.

• COVID-19 is a disease known to by caused by SARS-CoV-2, a species of betacoronavirus.
• Wuhan is home to Wuhan Institute Of Virology which is a well-known centre of expertise about bat coronaviruses in general and of one of the closest known relatives of SARS-CoV-2, RaTG13, in particular.
• the two closest known wild relatives of SARS-CoV-2, RaTG13 and RmYN02, were discovered in Yunnan province, near Mojiang
• the population of China is 1.3b people
• population of Wuhan prefecture, 11m ~ 0.8% of Chinese population
• area of Wuhan prefecture, 8494km²
• distance from Mojiang to Wuhan, capital of Hubei province ~ 1800km
• area of Kunming prefecture, 21000km²
• population of Kunming prefecture, 6.6m ~ 0.5% of Chinese population
• distance from Mojiang to Kunming, capital of Yunnan province ~ 250km

The Parameters

The three parameters to estimate are: P(H), P(E|H) and P(E|^H). P(E) can be calculated from P(H), P(E|H) and P(E|^H) given the identity above.

P(H)

P(H) is the prior probability that the hypothesis is true, independent of the evidence.

Estimating the prior probability of the hypothesis is difficult to do accurately and care must be taken to discount, for now, any knowledge one might actually have about the relative location of the Wuhan Institute of Virology to the location of the outbreak, since that “knowledge” will be provided by the statement of evidence.

Is a laboratory outbreak impossible? No, clearly not. Laboratory outbreaks have occurred before and they will occur in the future. For the purposes of illustration, I have selected 3 different values to be considered: 1/10000, 1/1000 and 1/100 that span 3 orders of magnitude — in other words the prior probability of the hypothesis is no more than 1% but may be as small as 0.01%.

Ardent believers in the laboratory origin hypothesis might argue that the prior probability of the hypothesis is more than 1%. Likewise, ardent sceptics might argue that it can’t possibly be greater than 0.01%. However, the range chosen does cover 2 orders of magnitude and those that wish to argue that this range does not include the actual prior probability of the hypothesis being true will need to make their own arguments about why this is so.

Is there a rational basis for excluding smaller or larger probabilities? I don’t see the need to include possibilities less than 1/10000 in the absence of an argument that demonstrates the need to do so. Likewise, I can’t provide an argument for considering possibilities of more than 1 in 100, since laboratory outbreaks are far from everyday occurrences.

So, for the purposes of this analysis, we are assuming that:

P(E|^H)

P(E|^H) is the conditional probability of the evidence, given that the hypothesis is actually false.

If a laboratory wasn’t responsible for the uncontrolled release of SARS-CoV-2, then what is the chance that the first recorded outbreak of the virus would occur in Wuhan, home of the Wuhan Institute of Virology?

There might be many ways to answer that. With perfect knowledge and unlimited computing resources, one could enumerate all possible transmission chains from the most likely natural reservoir of SARS-CoV-2 (or its close predecessors) and then count up how many of those end up causing the first recorded outbreak in Wuhan rather than elsewhere in China (or, indeed, the rest of the world). This would be a pretty good estimate of the chance that the first recorded outbreak occurs in Wuhan.

In the absence of perfect knowledge, we can find other ways to produce an estimate.

One way that springs to mind is to consider the relative area of Wuhan prefecture within the 1800km radius circle centred on Mojiang caves in Yunnan.

So, if estimated the probability that the first recorded SARS-CoV-2 outbreak were to occur in Wuhan rather than anywhere else in the 1800km radius around Mojiang caves, then that probability is the vanishingly small 0.00083 or close to 1/1000.

However, that estimate is not fair to the natural origin hypothesis because the population is not uniformly distributed across the larger area, indeed, some of that area is indubitably over the ocean.

An option that is fairer to the natural origin hypothesis is estimate the probability based on the chance that everyone in China had equal chance of being the initial vector of SARS-CoV-2 and then count how many of those people live in Wuhan.

This estimate has the advantage of better taking account of population density but is biased against the laboratory origin hypothesis because it assumes that everyone in Wuhan was just as likely as everyone in Kunming to be the first to contract SARS-CoV-2 even though Kunming is at least 6 times closer to Mojiang than Wuhan is. However, we will select P(E|^H) = 0.01 as the centre point of our estimates for this parameter and choose estimates of 0.001 and 0.1 to account for the extremes on either side.

P(E|H)

P(E|H) is the conditional probability of the evidence, given the hypothesis is true.

In other words, if there was an uncontrolled release of a virus from a laboratory, what are the chances that the resulting outbreak occurs in the same city as the laboratory itself?

If the disease caused by the virus had an incubation period of years and the infectious agent was primarily transmitted by contact with bodily fluids such as during relatively infrequent episodes of sexual contact, one could imagine scenarios where a travelling scientist catches it in a laboratory in one city and then travels to another city and only then do circumstances arise where an outbreak occurs in a city remote to the lab. For hypothetical diseases of this kind, you might assign a low to moderate value of P(E|H) to reflect the possibility that evidence might escape detection in the vicinity of the laboratory where zoonotic transfer first occurred.

In most other cases, such as this one, where the infectious agent is known to be highly contagious and the incubation period is relatively short, the chance that first outbreak occurs in the immediate vicinity of the laboratory where the zoonotic transfer first occurred is much greater.

One way to estimate this number more rigorously might be, for example, tabulating all the known cases of zoonotic transfer involving laboratory animals that have caused outbreaks and identifying which percentage of those cases involved outbreaks in the same city as the city where zoonotic transfer occurred. We can use this percentage as an estimate of P(E|H)

For the reasons given above, it seems likely that this number is greater than 0.5 for SARS-CoV-2 and must be less than or equal to 1. To obtain a middle estimate which is fairer to the natural origin case than not, we will use the geometric mean of 0.51 and 1, so 0.71.

Populating The Cube

With 3 estimates for the 3 unknown parameters of posterior probability equation, we can calculate a cube of estimates for P(H|E) as follows.

Each of the green squares can be considered as a layer of a cube. The values of P(E|H) correspond to a coordinate of the top-bottom axis, the values of P(H) correspond to a coordinate of the left-right axis and the values of P(E|^H) correspond to the coordinates of a back-front axis.

Parameter values are arranged so that values that increase the value of P(H|E) range from left-to-right or top-to-bottom or back-to-front, so sceptical estimates tend to be in the back, upper, left corner of the cube, susceptible estimates tend to be in the front, lower, right corner cube.

The ardent sceptic’s revised estimate is no more than 0.05% or 5/10000. It applies to someone who was initially very sceptical about a lab origin, who believes there is no more than 51% chance that an uncontrolled release of a highly contagious disease would lead to a local outbreak and who thinks there was at least a 10% chance that a natural outbreak of a virus native to Yunnan would have occurred in Wuhan before any place else.

On the other extreme, is the ardent believer who started with at least a 1% belief in a laboratory outbreak, is 100% certain that an uncontrolled laboratory release would result in a local outbreak and believes that the probability that a natural outbreak of a virus native to Yunnan would occur in Wuhan before any place else is less than 0.1%. The ardent believer’s revised belief is that the probability that the Wuhan outbreak was caused by an uncontrolled laboratory release is at least 91%.

In the centre, is the so-called “central” observer who accepts that the central values for each of the parameter ranges are reasonable estimates of the true values of the probability being estimated. The central observer started with an initially sceptical belief in the hypothesis of 0.1%, believes that average citizen in Wuhan was a likely as any other citizen of China to be the initial vector of the virus into the human population and believes that there is no more or less than a 71% chance that an uncontrolled release from a laboratory of a highly contagious pathogen such as SARS-CoV-2 would result in a local outbreak as opposed to an outbreak in some other location. The central observer’s revised belief in the hypothesis is 6.6%.

It should be noted that the “central” observer isn’t necessarily the same thing as the unbiased or neutral observer. In particular, the central values for all 3 parameters are likely, in the opinion of this author, to be at least slightly biased towards the natural origin position.

A Counter Factual Scenario

To illustrate how important the parameter P(E|^H) is to the posterior probability of the hypothesis, we can apply the same analysis to a counter factual scenario in which the probability of the evidence being true given the hypothesis is false is far higher — a scenario where the first outbreak occurred in Kunming.

Kunming is the capital city of Yunnan province. It doesn’t necessarily have a world-leading bat coronavirus research lab, but it does host Kunming Institute of Zoology (KIZ) and, at least in principle, might conduct research into viruses.

However, the key fact that increases P(E|^H) in this case is the sheer proximity of Kunming to the natural reservoir of the two closest known relatives (RaTG13, RmYN02) of SARS-CoV-2 (~300 km) compared to Wuhan which is 1900 km away. It is quite easy to believe there is a multitude of pathways from the caves of Mojiang to the first recorded outbreak in the provincial capital. For this reason, we can assign a range of probabilities between 50% and 100% with a central assignment of 71%.

If we do this, we get a probability cube with a much reduced probability of P(H|E) as shown below:

Strength Of Belief Revision

The point of the counter factual is to illustrate that the sheer unlikelihood of an outbreak occurring in Wuhan if the hypothesis is false is enough to revise the initial strength of one’s belief in the hypothesis by between 1 and 2 orders of magnitude as is illustrated by this table which shows the ratio between P(H|E) and P(H) for the Wuhan scenario

Discussion

This post has not attempted to derive the one true and absolute probability that the first outbreak of SARS-CoV-2 in Wuhan originated from an uncontrolled release from a laboratory.

It has presented a Bayesian analysis, with sceptical priors, and a plausible range of likelihood estimates that should encompass the positions of most sceptics, believers and neutral observers. By tabulating these values and the ratio between P(H|E) and P(H), it is possible to see how important the fact the first outbreak occurred in Wuhan is to a rational revision to any prior belief about the likelihood of a uncontrolled release from a laboratory.

As a sanity check that the analysis is reasonable, this post has also presented a counterfactual which shows how weak a posterior belief in a lab origin would remain if the outbreak had instead occurred in Kunming. The only change of relevance in the counter factual analysis, was the far greater likelihood that the evidence would occur, even if the hypothesis was false, in Kunming due to its relative proximity to the natural reservoirs of the closest known relatives to SARS-CoV-2.

Postscript

Since beginning to write this post in late November, the work [1] of Gilles Demaneuf and Rodolphe De Maistre has come to my attention. At the time of writing this postscript, I had not read past the abstract of that work, but I understand those authors also selected, using different arguments and counter argumentts, a conservative value of 1% for the parameter P(E|^H). I decided not to read their paper until I had finally published this post so that I could truthfully claim that my analysis had not been influenced by theirs, but now look forward to reading what they have written.

[1] Outlines of a probabilistic evaluation of possible SARS-CoV-2 origins

I have recently come to the conclusion that now I really don’t need to lose any more weight. A lot of other people came to that conclusion quite a while ago (hi Mum!), but I was prepared to push it a little further just in case a 6-pack popped out at some point. It never did.

The lowest daily reading I got to was 63.8kg, the lowest weekly average was 64.7kg.

So I decided to have a second DEXA scan on Thursday (October 29, 2020) to understand what minimum weight looks like and also see what changed since March 2019.

Some basic numbers:

Bathroom scales weight on day of scan (7am):
2019: 70.2kg
2020: 65.9kg
delta: -4.3kg

DEXA Scan Weight:
2019: 69.9kg (-0.3kg)
2020: 66.3kg (+0.4kg)
delta: -3.6kg
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The differences between the scales weight and the DEXA scan weight are perhaps not that surprising. March 29, 2019 was quite hot, so I likely lost some water on the walk to the clinic. On October 29, 2020 I could well have been hydrated to excess.

Time between scans (days): 580
Total weight loss (g): 3641
Average total weight loss (g/day): 6.2
Equivalent average calorie deficit (kcal/day): 56

Of course, a DEXA scan do more than tell you to total weight loss — it can tell you where the weight loss occurred.

Lean mass loss (g): 2138 (4%) (3.69g/day)
Fat mass loss (g): 1514 (12%) (2.6g/day)
Bone/Mineral mass loss (g): -10 (0%)

So, while in percentage terms, I lost more fat that lean mass, in absolute terms I lost more lean than fat mass.

As a result, there was a small change (-1.3%) in body fat percentage, but it was not huge.

Body fat percentage:
2019: 17.6
2020: 16.3

Making the slightly questionable assumption that energy gained from weight loss varies by tissue type as follows: 4 cal/g (lean mass), 9 cal/g (fat mass)

Equivalent average calorie deficit (kcal/day): 38

So, clearly, I haven’t exactly been starving myself.

Two other ratios are used to summarise body composition. These are similar to the BMI, but break out the lean mass and fat mass separately.

Fat Mass Ratio (kg/m²):
2019: 4.16
2020: 3.65

Lean Mass Ratio (kg/m²):
2019: 18.6
2020: 17.9

It is always interesting to compare oneself to others. The reference ranges used in reporting DEXA scan results are derived from studies done in the US population, so take these numbers with a grain of salt (and some ketchup on the side).

Fat Mass Ratio (aged match percentile):
2019: 8
2020: 4

So, in 2019, my fat mass ratio was in the bottom 8% of all white American males my age, but in 2020 my fat mass ratio is now in the bottom 4% of all white American males my age (smaller is better).

Sadly, a similar trend is evident in the lean mass ratio (but here, larger is better):

Lean Mass Ratio (aged match percentile):
2019: 24
2020: 15

So, while I was in the bottom 24% for lean mass in 2019 which was not ideal then, I am now in a worse position: bottom 15%.

Needless to say, if I want to be “normal” in terms of lean mass, I need to closer to the 50th percentile, which happens to be 20kg/m² according to this graph:

which, if you rearrange the equation, means I need to add 6133g of lean mass (DEXA weight of 72.3kg) . Now, if I did manage to do that without adding any fat, I would hit the magical 15% body fat that was sort of my original goal.

If I had started this project in March 2019 instead, I would have had to add only 3995g of lean mass, for a final weight of 73.9kg and body fat of 16.7%

So, now begins a new project — adding more lean mass, without adding a heap of fat.

This will probably involve more bodyweight exercise and perhaps going to a gym. Fortunately, my swimming membership gives me access to the Ian Thorpe Aquatic Center gym so at least it won’t cost me anything extra.

A question is: what time frame should I try to achieve this in? I like the symmetry of doing it by March 26, 2023 which would represent 5 years after a switch flipped in my brain and I decided to get fit.

So, how many grams of lean mass is that per day? 6133g/879days = 7g/day or 49g per week. Now, this is about 1/2 of what a Google search says is reasonable, but a) I don’t know what I am doing and b) I need to sustain this for 840 days. I think I will set myself a slightly less aggressive target and see what happens after 6–12 months and adjust the program accordingly.

The trickiest problem will be working out if I am adding lean mass or adding weight since I can’t have a DEXA scan each week. I think what I will do is use my waist measurement as a proxy for body fat percentage and try to maintain that within limits while slowly increasing my body weight.

Onward!

This post considers the hypothesis that SARS-CoV-2 evolved from a virus that is endemic to the Indochinese Peninsula (also known as Mainland South East Asia). This is the region of the world that includes the countries: Thailand, Laos, Vietnam, Cambodia, Myanmar and Malaysia.

What is interesting about these countries is that 4 of them (Laos, Vietnam, Cambodia and Myanmar) have experienced very low rates of infection compared to the rest of the world and the other two (Thailand and Malaysia) have experienced rates of infection that are less than Australia and New Zealand which have, themselves, experienced relatively low rates of infection compared to the rest of the world.

[ source: Our World In Data ]

The other interesting property of the Indochinese Peninsula is that it is the centre of the habitat of both Rhinolophus malayanus and the Rhinolophus affinis bat species, which are the species of bat from which the two closest known relatives of SARS-CoV-2 (RmYn02 and RaTG13) were collected. RaTG13 and RmYn02 were collected in 2013 and 2019 from locations in the Yunnan province of China, which is geographically adjacent to the Indochinese peninsula so there is a good chance that the natural reservoir of SARS-CoV-2 is also a near relative of R.malayanus, R.affinis or a closely related species.

[ source: Wikipedia ]

So, how is it that the population that seems most immune to the effects of SARS-CoV-2 happens to live coincidentally to the native habitat of the bat species that are most likely to have hosted SARS-CoV-2?

Could it be that past exposure to a relative of SARS-CoV-2 has conferred some kind of immunity upon these populations? If so, we can we determine how this immunity has been conferred and so derive clues about how to develop an effective treatment and/or vaccine?

Of course, the first thing to do is to try to establish that there is, in fact, a biological basis to the immunity amongst these populations. This might (?) be done with cell cultures and also epidemiological studies amongst at risk medical staff in these countries and overseas (for example: are Vietnamese nurses in the US less likely to be infected than their non-Vietnamese co-workers?)

If it turns out that there is, indeed, a biological basis to the immunity that these populations appear to have, then the next step would be to try work out whether that basis can be reproduced, in some fashion, in other populations.

All of this is speculation, of course, and as a layperson I am in no position to either advance or reject the hypothesis, but it does seem like one worth investigating, in case it does, indeed, lead to a path forward.

So, how about arranging for a poll that is guaranteed to show rising week-on-week support for Trump — a poll of COVID-19 sufferers?

What? Surely they would take exception to the gross indifference and incompetence of the Golfer in Chief?

And, yes, if this was March 30 I would agree with you.

But today is July 6. Demographics have changed. The cohort of current and future COVID-19 sufferers can only become increasingly faithful to Trump as the virus spreads into the Trump heartland. It may kill and maim them, but they have already absolved Trump of the need to care, perform or otherwise act like any other US President is expected to (especially by them).

They never needed Trump to build a wall. They never needed Trump to reform the Supreme Court . They needed Trump to kick the status quo and to give permission for themselves to express and act on their own prejudices and hatreds.

Trump supporters who fall victim to COVID-19 wont’t blame him because hating something, anything feels so damn righteous and good. A debilitating COVID-19 illness can be blamed on literally any other thing, as facts have never mattered since that glorious day when the Electoral College appointed Trump and issued them all with a golden ticket to smirk, to bait and to hate — to poke fun at the political correctness of others while refusing to acknowledge that they themselves are enslaved by a far nastier form of political correctness called political incorrectness.

The number one tenet of political incorrectness that virtually all its adherents believe is simply this: Trump can do no wrong. You know this is true because you never see anyone who despises political correctness make a sustained criticism of Trump. Why would they? Political correctness cast them out of the in-group in the first place, the solidarity they feel by belonging to the out-group of the politically incorrect is way too valuable to be put aside for theoretical concepts like the Rule of Law or good governance.

So, yeah, if Trump wants rising polls, he should start polling COVID-19 sufferers and only COVID-19 sufferers. In that poll, his numbers can only go up.