Confidence levels inside and outside an argument

While searching creationist websites for the half-remembered argument I was looking for, I found what may be my new favorite quote: "Mathematicians generally agree that, statistically, any odds beyond 1 in 10 to the 50th have a zero probability of ever happening."

That reminds me of one of my favourites, from a pro-abstinence blog:

When you play with fire, there is a 50/50 chance something will go wrong, and nine times out of ten it does.

Note that someone just gave a confidence level of 10^4478296 to one and was wrong. This is the sort of thing that should never ever happen. This is possibly the most wrong anyone has ever been.

I was in some discussion at SIAI once and made an estimate that ended up being off by something like three hundred trillion orders of magnitude. (Something about giant look-up tables, but still.) Anyone outdo me?

I'm a bit irked by the continued persistence of "LHC might destroy the world" noise. Given no evidence, the prior probability that microscopic black holes can form at all, across all possible systems of physics, is extremely small. The same theory (String Theory[1]) that has led us to suggest that microscopic black holes might form at all is also quite adamant that all black holes evaporate, and equally adamant that microscopic ones evaporate faster than larger ones by a precise factor of the mass ratio cubed. If we think the theory is talking complete nonsense, then the posterior probability of an LHC disaster goes down, because we favor the ignorant prior of a universe where microscopic black holes don't exist at all.

Thus, the "LHC might destroy the world" noise boils down to the possibility that (A) there is some mathematically consistent post-GR, microscopic-black-hole-predicting theory that has massively slower evaporation, (B) this unnamed and possibly non-existent theory is less Kolmogorov-complex and hence more posterior-probable than the one that scientists are currently using[2], and (C) scientists have completely overlooked this unnamed and possibly non-existent theory for decades, strongly suggesting that it has a large Levenshtein distance from the currently favored theory. The simultaneous satisfaction of these three criteria seems... pretty f-ing unlikely, since each tends to reject the others. A/B: it's hard to imagine a theory that predicts post-GR physics with LHC-scale microscopic black holes that's more Kolmogorov-simple than String Theory, which can actually be specified pretty damn compactly. B/C: people already have explored the Kolmogorov-simple space of post-Newtonian theories pretty heavily, and even the simple post-GR theories are pretty well explored, making it unlikely that even a theory with large edit distance from either ST or SM+GR has been overlooked. C/A: it seems like a hell of a coincidence that a large-edit-distance theory, i.e. one extremely dissimilar to ST, would just happen to also predict the formation of LHC-scale microscopic black holes, then go on to predict that they're stable on the order of hours or more by throwing out the mass-cubed rule[3], then go on to explain why we don't see them by the billions despite their claimed stability. (If the ones from cosmic rays are so fast that the resulting black holes zip through Earth, why haven't they eaten Jupiter, the Sun, or other nearby stars yet? Bombardment by cosmic rays is not unique to Earth, and there are plenty of celestial bodies that would be heavy enough to capture the products.)

[1] It's worth noting that our best theory, the Standard Model with General Relativity, does not predict microscopic black holes at LHC energies. Only String Theory does: ST's 11-dimensional compactified space is supposed to suddenly decompactify at high energy scales, making gravity much more powerful at small scales than GR predicts, thus allowing black hole formation at abnormally low energies, i.e. those accessible to LHC. And naked GR (minus the SM) doesn't predict microscopic black holes. At all. Instead, naked GR only predicts supernova-sized black holes and larger.

[2] The biggest pain of SM+GR is that, even though we're pretty damn sure that that train wreck can't be right, we haven't been able to find any disconfirming data that would lead the way to a better theory. This means that, if the correct theory were more Kolmogorov-complex than SM+GR, then we would still be forced as rationalists to trust SM+GR over the correct theory, because there wouldn't be enough Bayesian evidence to discriminate the complex-but-correct theory from the countless complex-but-wrong theories. Thus, if we are to be convinced by some alternative to SM+GR, either that alternative must be Kolmogorov-simpler (like String Theory, if that pans out), or that alternative must suggest a clear experiment that leads to a direct disconfirmation of SM+GR. (The more-complex alternative must also somehow attract our attention, and also hint that it's worth our time to calculate what the clear experiment would be. Simple theories get eyeballs, but there are lots of more-complex theories that we never bother to ponder because that solution-space doesn't look like it's worth our time.)

[3] Even if they were stable on the order of seconds to minutes, they wouldn't destroy the Earth: the resulting black holes would be smaller than an atom, in fact smaller than a proton, and since atoms are mostly empty space the black hole would sail through atoms with low probability of collision. I recall that someone familiar with the physics did the math and calculated that an LHC-sized black hole could swing like a pendulum through the Earth at least a hundred times before gobbling up even a single proton, and the same calculation showed it would take over 100 years before the black hole grew large enough to start collapsing the Earth due to tidal forces, assuming zero evaporation. Keep in mind that the relevant computation, t = (5120 × π × G^2 × M^3) ÷ (ℏ × c^4), shows that a 1-second evaporation time is equal to 2.28e8 grams[3a] i.e. 250 tons, and the resulting radius is r = 2 × G × M ÷ c^2 is 3.39e-22 meters[3b], or about 0.4 millionths of a proton radius[3c]. That one-second-duration black hole, despite being tiny, is vastly larger than the ones that might be created by LHC -- 10^28 larger by mass, in fact[3d]. (FWIW, the Schwarzschild radius calculation relies only on GR, with no quantum stuff, while the time-to-evaporate calculation depends on some basic QM as well. String Theory and the Standard Model both leave that particular bit of QM untouched.)

[3a] Google Calculator: "(((1 s) h c^4) / (2pi 5120pi G^2)) ^ (1/3) in grams"

[3b] Google Calculator: "2 G 2.28e8 grams / c^2 in meters"

[3c] Google Calculator: "3.3856695e-22 m / 0.8768 femtometers", where 0.8768 femtometers is the experimentally accepted charge radius of a proton

[3d] Google Calculator: "(2.28e8 g * c^2) / 14 TeV", where 14 TeV is the LHC's maximum energy (7 TeV per beam in a head-on proton-proton collision)

One might be tempted to respond "But there's an equal chance that the false model is too high, versus that it is too low."

I'm not sure why one might be tempted to make this response. Is the idea that, when making any calculation at all, one is equally likely to get a number that is too big as one that is too small? But then, that's before you have looked at the number.

Yet another counter-response is that even if the response were true, the false model could be much too high, but it can only be slightly too low, since 1-10^-9 is quite close to 1.

First, great post. Second, general injunctions against giving very low probabilities to things seems to be taken by many casual readers as endorsements of the (bad) behavior "privilege the hypothesis" - e.g. moving the probability from very small to moderately small that God exists. That's not right, but I don't have excellent arguments for why it's not right. I'd love it if you wrote an article on choosing good priors.

Cosma Shalizi has done some technical work that seems (to my incompetent eye) to be relevant:

http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.ejs/1256822130&page=record

That is, he takes Bayesian updating, which requires modeling the world, and answers the question 'when would it be okay to use Bayesian updating, even though we know the model is definitely wrong - e.g. too simple?'. (Of course, making your model "not obviously wrong" by adding complexity isn't a solution.)

Added to Absolute certainty LW wiki page.

This raises the question: Should scientific journals adjust the p-value that they require from an experiment, to be no larger than the probability (found empirically) that a peer-reviewed article contains a factual, logical, methodological, experimental, or typographical error?

I don't think the lottery is an exception. There's a chance that you misheard and they said "million", not "billion".

There are really two claims here. The first one -- that if some guy on the Internet has a model predicting X with 99.99% certainty, then you should assign less probability to X, absent other evidence -- seems interesting, but relatively easy to accept. I'm pretty sure I've been reasoning this way in the past.

The second claim is exactly the same, but applied to oneself. "If I have come up with an argument that predicts X with 99.99% certainty, I should be less than 99.99% certain of X." This is not something that people do by default. I doubt that I do it unless prompted. Great post!

Stylistic nitpick, though: things like "999,999,999 in a billion" are tricky to parse, especially when compared to "999,999,999,999 in a trillion" (which I initially read as approximately 1 in 1000 before counting the 9s) or "1/1 billion". Counting the 9s is part of the problem, the other that the numerator is a number and the denominator is a word. What's wrong with writing 99.99% and 99.9999%? These are different from the original values in the post, but still carry the argument, and are easier to read.

But my external level of confidence should be lower, even if the model is my only evidence, by an amount proportional to my trust in the model.

Only to the extent you didn't trust in the statement other than because this model says it's probably true. It could be that you already believe in the statement strongly, and so your external level of confidence should be higher than the model suggests, or the same, etc. Closer to the prior, in other words, and on strange questions intuitive priors can be quite extreme.

Another voting example; "Common sense and statistics", Andrew Gelman:

A paper* was published in a political science journal giving the probability of a tied vote in a presidential election as something like 10^-90**. Talk about innumeracy! The calculation, of course (I say “of course” because if you are a statistician you will likely know what is coming) was based on the binomial distribution with known P. For example, Obama got something like 52% of the vote, so if you take n=130 million and P=0.52 and figure out the probability of an exact tie, you can work out the formula etc etc.

From empirical grounds that 10^-90 thing is ludicrous. You can easily get an order-of-magnitude estimate by looking at the empirical probability, based on recent elections, that the vote margin will be within 2 million votes (say) and then dividing by 2 million to get the probability of it being a tie or one vote from a tie.

The funny thing—and I think this is a case for various bad numbers that get out there—is that this 10^-90 has no intuition behind it, it’s just the product of a mindlessly applied formula (because everyone “knows” that you use the binomial distribution to calculate the probability of k heads in n coin flips). But it’s bad intuition that allows people to accept that number without screaming. A serious political science journal wouldn’t accept a claim that there were 10^90 people in some obscure country, or that some person was 10^90 feet tall.

...To continue with the Gigerenzer idea [of turning probabilities into frequencies], one way to get a grip on the probability of a tied election is to ask a question like, what is the probability that an election is determined by less than 100,000 votes in a decisive state. That’s happened at least once. (In 2000, Gore won Florida by only 20-30,000 votes.***) The probability of an exact tie is of the order of magnitude of 10^(-5) times the probability of an election being decided by less than 100,000 votes...Recent national elections is too small a sample to get a precise estimate, but it’s enough to make it clear that an estimate such as 10^-90 is hopelessly innumerate.

1. * "Is it Rational to Vote? Five Types of Answer and a Suggestion", Dowding 2005; fulltext: https://pdf.yt/d/5veaHe6F5j-k6oNQ / https://www.dropbox.com/s/fxgfa04hmpfntgh/2005-dowding.pdf / http://libgen.org/scimag/get.php?doi=10.1111%2Fj.1467-856x.2005.00188.x
2. ** 1/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 ; or to put it in context, 'inside' the argument, the claim is that you could hold a presidential election for every atom in the universe, and still not ever have a candidate win by one vote

3. *** From the comments:

   It happened five or six times in 2000 alone, depending on how you think about electoral–college ties. http://en.wikipedia.org/wiki/United_States_presidential_election,_2000#Votes_by_state.

This one seems pretty relevant here:

Probing the Improbable: Methodological Challenges for Risks with Low Probabilities and High Stakes - Toby Ord, Rafaela Hillerbrand, Anders Sandberg

• http://arxiv.org/abs/0810.5515

This is not a fully general argument against giving very high levels of confidence:

It seems to me we can use the very high confidence levels and our understanding of the area in question to justify ignoring, heavily discounting, or accepting the arguments. We can do this on the basis that it takes a certain amount of evidence to actually produce accurate beliefs.

In the case of the creationist argument, a confidence level of 10^4,478,296 to 1 requires (really) roughly 12,000,000 bits of evidence. (10^4,000,000 =~ 2^12,000,000). The creationist presents these twelve million bits in the form of observations about cells. Now, using our knowledge of biology and cells (specifically, their self-assembling nature, restrictions on which proteins can combine, persistence and reproduction) we can confidently say that observations of cells do not provide 12,000,000 bits of evidence.

I'm not knowledgeable about biology, so I can't say how many bits of evidence for a creator they provide in this manner but I gather it's not many. We then adjust the argument's strength down to that many bits of evidence. In effect, we are discounting the creationist argument for a lack of understanding, and discounting it by exactly how much it lacks understanding.

Applying this to the LHC argument: the argument specifies odds of 10^25 to 1, and the evidence is in the form of cosmic ray interactions not destroying the world. Based on our understanding of the physics involved (including our understanding of the results of Giddings and Mangano), we can say that cosmic ray interactions don't provide quite as much evidence as the argument claims - but they provide most of the evidence they claimed to (even if we have to resort to our knowledge about white dwarf stars).

I think we should prefer to downgrade the argument from our knowledge about the relevant area rather than hallucinations or simple error, because the prior for 'our understanding is not complete' is higher than 'hallucinating' and 'simple error' put together - and, to put it bluntly, in the social process of beliefs and arguments, most people are capable of completely dismissing arguments from hallucination and simple error, but are far less capable of dismissing arguments from incomplete knowledge.

As for inside / outside the argument, I found it helpful while reading the post to think of the outside view as a probability mass split between A and ~A, and then inside the argument tells us how much probability mass the argument steals for its side. This made it intuitive, in that if I encountered an argument that boasted of stealing all the probability mass for one side, and I could still conceive of the other side having some probability mass left over, I should distrust that argument.

This was predictable: this was a simple argument in a complex area trying to prove a negative, and it would have been presumptous to believe with greater than 99% probability that it was flawless. If you can only give 99% probability to the argument being sound, then it can only reduce your probability in the conclusion by a factor of a hundred, not a factor of 10^20.

As I recall, there was a paper in 2008 or 2009 about the LHC problem which concluded effectively that the tiny errors that an analysis was incorrectly carried out cumulatively put a high floor on what small risk we could conclude the LHC posed.

Unfortunately, I can't seem to refind it to see whether it's a better version of this argument, so perhaps someone else remembers specifics.

Very interesting principle, and one which I will bear in mind since I very recently had a spectacular failure to apply it.

What happens if we apply this type of thinking to Bayesian probability in general? It seems like we have to assign a small amount of probability to the claim that all our estimates are wrong, and that our methods for coming to those estimates are irredeemably flawed. This seems problematic to me, since I have no idea how to treat this probability, we can't use Bayesian updating on it for obvious reasons.

Anyone have an idea about how to deal with this? Preferably a better idea than "just don't think about it" which is my current strategy.

Great post!

The moment the topic came up, I also thought back to something I once heard a creationist say. Most amusingly, not only did that probability have some fatuously huge order of magnitude, its mantissa was quoted to about 5 decimal places.

One gets 'target confusion' in such cases - shall I point out that no engineer would ever quote a probability like that to their boss, on pain of job loss? Shall I ask if my interlocutor even knows what a "power" IS?

This is at best weakly related to the statistics of error in a communications channel. Here, simulations are often used to run trillions of trials to simulate (monte carlo calculate) the conditions to get bit error rates (BER) of 10^-7, 10^-8, and so on. As an engineer more familiar with the physical layer (transistor amplifiers, thermal noise in channels, scattering of RF etc), I know that the CONDITIONS for these monte carlo calculations to mean something in the real circuits are complex and not as common as the new PhD doing the calculation thinks they are. Further, the lower the BER calculated, the more likely something else has come along to bite you on the arse and raise the actual error rate in an actual circuit. STILL, in engineering presentation after presentation, people put these numbers up and other people nod gravely when they see them.

Amazingly, I"m finding the feeling of the post but in error rates which are gigantic compared to the probabilities discussed in the article. We get wiggly when a 1 has 6 zeros in front of it, you are using exponential notation to avoid writing much longer strings of zeros.

Maybe the "great filter" that prevents us seeing a universe filled with at least a few other intelligent species is that finally, one of the big physics experiments large smart civilizations build finally does destroy the local solar system. Maybe we should ban successors to the Large Hadron Collider until we are ensconced in at least one other solar system.

We have hypothesis H and evidence E, and we dutifully compute

P(H) * P(E | H) / P(E)

It sounds like your advice is: don't update yet! Especially if this number is very small. We might have made a mistake. But then how should we update? "Round up" seems problematic.

One might be tempted to respond "But there's an equal chance that the false model is too high, versus that it is too low." Maybe there was a bug in the computer program, but it prevented it from giving the incumbent's real chances of 999,999,999,999 out of a trillion.

I have a different response to this than the one you gave.

Consider your meta ("outside") uncertainty over log-odds, in which independent evidence can be added, instead of probabilities. A distribution that averages out to the "internal" log-odds would, when translated back into probabilities, have an expected probability closer to 1/2 than the "inside" probability.

If you apply this to your prior probability as well as the evidence, this should generally move your probabilities towards 1/2.

Splitting it by internal/external is a nice system.

I think people do this instinctively in real life. Exhibit A: people buy lottery tickets. My theory for this is that they know that the odds of winning are too low to justify buying a ticket assuming it is actually fully random. However, most people are willing to put the probability that karma, divine justice, God's plan or their lucky ritual might swing the lottery in their direction at some nonzero value. If they believe in one of these things with even 1% certainty then the ticket is a good deal for them. 

On the LHC black holes vs cosmic ray black holes, both kinds of black holes emerge with nonzero charge and will very rapidly brake to a halt. And there's cosmic rays hitting neutron stars, as well, and cosmic rays colliding in the magnetic field of neutron stars, LHC style. Bottom line is, the HLC has to be extremely exceptional to destroy the earth. It just doesn't look this exceptional.

The thing is that a very tiny black hole has incredibly low accretion rate (quite reliable argument here; it takes a long time to push Earth through a needle's eye, even at very high pressure) and even if we had many of those inside stars, planets, etc. we would never know. The HLC may have 'doomed' the Earth - to be destroyed in many billions years timespan.

The more interesting example would be PRA - probabilistic risk analysis - such as done for space shuttle, nuclear reactors, et cetera. The risk is calculated based on a sum of risks over very small selection of events (picked out of the space of possible events), and the minuscule risk figures that get calculated is representative not of low probability of failure but of low probability that the failure will be among the N guesses.

At same time, we have no good reason to believe PRA works at all, and a plenty of examples (Space Shuttle, nuclear reactors) where PRA was found off by a factor of 1000 (high confidence result 'cause its highly unlikely space shuttle PRA was correct yet two were lost).

The way I'd describe PRA is as estimating failure rate of a ball bearing in a car by adding up failure rates of the individual balls and other components. That's obviously absurd; the balls and their environment interact in such complicated, non-linear ways that you can't predict their failure rates by adding up component failure rates.

If a method clearly won't work for something as simple as a ball bearing, why would anyone assume it'd work for space shuttle or nuclear power plant which are much much more complex than ball bearing? My theory is that those things are so complicated that a person has such difficulty of reasoning about them as to be unable to even see that they are too complex for PRA to work; while ball bearing is simple enough. At same time there's a demand for some number to be given; this demand creates pseudoscience.

The map being distinct from the territory, you must go outside your map to discount your probability calculations made in the map. But how to do this? You must resort to a stronger map. But then the calculations there are subject to the errors in designing that map.

You can run this logic down to the deepest level. How does a rational person adopt a Bayesian methodology? Is there not some probability that the choice of methodology is wrong? But how do you conceive of that probability, when Bayesian considerations are the only ones available to evaluate truth from given evidence?

Why don't these considerations prove that Bayesian epistemology isn't the true account of knowledge?

In order for a single cell to live, all of the parts of the cell must be assembled before life starts. This involves 60,000 proteins that are assembled in roughly 100 different combinations. The probability that these complex groupings of proteins could have happened just by chance is extremely small. It is about 1 chance in 10 to the 4,478,296 power. The probability of a living cell being assembled just by chance is so small, that you may as well consider it to be impossible. This means that the probability that the living cell is created by an intelligent creator, that designed it, is extremely large. The probability that God created the living cell is 10 to the 4,478,296 power to 1.

Note that someone just gave a confidence level of 10^4478296 to one and was wrong. This is the sort of thing that should never ever happen. This is possibly the most wrong anyone has ever been.

Particularly in the light of the fact that he seems to have got the numbers the wrong way round from what he intended in the final sentence.

I speculate there's at least two problems with the creationism odds calculation. First, it looks like the person doing the calculation was working with maybe 60,000 protein molecules rather than zillions of protein molecules.

The second problem I'm having trouble putting precisely in words, concerning the use of the uniform distribution as a prior. Sometimes the use of the uniform distribution as a prior seems to me to be entirely justified. An example of this is where there is a well-constructed model as to subsequent outcomes.

Other times, when the model for subsequent outcomes is sketchy, the uniform distribution is used as a prior simply as a default. Or, as in this case, it's clearly not an appropriate prior. In this case, the person is probably assuming that all combinations of proteins are equally likely (I suspect this assumption is false.)

The argument was that since cosmic rays have been performing particle collisions similar to the LHC's zillions of times per year, the chance that the LHC will destroy the world is either literally zero,

This argument doesn't work for anthropic reasons. It could be that in the vast majority of Everett branches Earth was wiped out by cosmic ray collisions.

"This person believes he could make one statement about an issue as difficult as the origin of cellular life per Planck interval, every Planck interval from the Big Bang to the present day, and not be wrong even once" only brings us to 1/10^61 or so."

Wouldn't that be 1/ 2^(10^61) or am I missing something?

I'm a bit irked by the continued persistence of "LHC might destroy the world" noise. Given no evidence, the prior probability that microscopic black holes can form at all, across all possible systems of physics, is extremely small. The same theory (String Theory[1]) that has led us to suggest that microscopic black holes might form is also quite adamant that all black holes evaporate, and just as adamant that microscopic ones evaporate faster than larger ones, by a precise factor of the mass ratio cubed. If we think the theory is talking complete nonsense, then the posterior probability of an LHC black hole forming in the first place goes down, because we slide back to the prior of a universe without microscopic black holes.

Thus, the "LHC might destroy the world" noise boils down to the possibility that (A) there is some mathematically consistent post-GR, microscopic-black-hole-predicting theory that has massively slower evaporation, (B) this unnamed and possibly non-existent theory is less Kolmogorov-complex and hence more posterior-probable than the one that scientists are currently using[2], and (C) scientists have completely overlooked this unnamed and possibly non-existent theory for decades, strongly suggesting that it has a large Levenshtein distance from the currently favored theory. The simultaneous satisfaction of these three criteria seems... pretty fing unlikely, since each tends to reject the others. A/B: it's hard to imagine a theory that predicts post-GR physics with LHC-scale microscopic black holes that's more Kolmogorov-simple than String Theory, which can actually be specified pretty damn compactly. B/C: people already have explored the Kolmogorov-simple space of post-Newtonian theories pretty heavily, and even the simple post-GR theories are pretty well explored, making it unlikely that even a theory with large edit distance from either ST or SM+GR has been overlooked. C/A: it seems like a hell of a coincidence that a large-edit-distance theory, i.e. one extremely dissimilar to ST, would just happen to also predict the formation of LHC-scale microscopic black holes, then* go on to predict that they're stable* on the order of hours or more by throwing out the mass-cubed rule[3], then* go on to explain why we don't see them by the billions despite their claimed stability. (If the ones from cosmic rays are so fast that the resulting black holes zip through Earth, why haven't they eaten Jupiter, the Sun, or other nearby stars yet? Bombardment by cosmic rays is not unique to Earth, and there are plenty of celestial bodies that would be heavy enough to capture the products.)

[1] It's worth noting that our best theory, the Standard Model with General Relativity, does not predict microscopic black holes at LHC energies. Only String Theory does: ST's 11-dimensional compactified space is supposed to suddenly decompactify at high energy scales, making gravity much more powerful at small scales than GR predicts, thus allowing black hole formation at abnormally low energies, i.e. those accessible to LHC. And GR without the SM doesn't predict microscopic black holes. At all. Naked GR only predicts supernova-sized black holes and larger.

[2] The biggest pain of SM+GR is that, even though we're pretty damn sure that that train wreck can't be right, we haven't been able to find any disconfirming data that would lead the way to a better theory. This means that, if the correct theory were more Kolmogorov-complex than SM+GR, then we would still be forced as rationalists to trust SM+GR over the correct theory, because there wouldn't be enough Bayesian evidence to discriminate the complex-but-correct theory from the countless complex-but-wrong theories. Thus, if we are to be convinced by some alternative to SM+GR, either that alternative must be Kolmogorov-simpler (like String Theory, if that pans out), or that alternative must suggest a clear experiment that leads to a direct disconfirmation of SM+GR. (The more-complex alternative must also somehow attract our attention, and also hint that it's worth our time to calculate what the clear experiment would be. Simple theories get eyeballs, but there are lots of more-complex theories that we never bother to ponder because that solution-space doesn't look like it's worth our time.)

[3] Even if they were stable on the order of seconds to minutes, they wouldn't destroy the Earth: the resulting black holes would be smaller than an atom, in fact smaller than a proton, and since atoms are mostly empty space the black hole would sail through atoms with low probability of collision. I recall that someone familiar with the physics did the math and calculated that an LHC-sized black hole could swing like a pendulum through the Earth a hundred times before gobbling up even a single proton, and the same calculation showed it would take over 100 years before the black hole grew large enough to start collapsing the Earth due to tidal forces, assuming zero evaporation. Keep in mind that the relevant computation, t = (5120 × π × G^2 × M^3) ÷ (ℏ × c^4), shows that a 1-second evaporation time is equal to 2.28e8 grams[3a] i.e. 250 tons, and the resulting radius is r = 2 × G × M ÷ c^2 is 3.39e-22 meters[3b], or about 0.4 millionths of a proton radius[3c]. That one-second-duration black hole, despite being tiny, is vastly larger than the ones that might be created by LHC -- 10^28 larger in fact[3d]. (FWIW, the Schwarzschild radius calculation relies only on GR, with no quantum stuff, while the time-to-evaporate calculation depends on some basic QM as well. String Theory and the Standard Model both leave that particular bit of QM untouched.)

[3a] Google Calculator: "(((1 s) h c^4) / (2pi 5120pi G^2)) ^ (1/3) in grams" [3b] Google Calculator: "2 G 2.28e8 grams / c^2 in meters" [3c] Google Calculator: "3.3856695e-22 m / 0.8768 femtometers", where 0.8768 femtometers is the experimentally accepted charge radius of a proton [3d] Google Calculator: "(2.28e8 g * c^2) / 14 TeV", where 14 TeV is the LHC's maximum energy (7 TeV per beam in a head-on proton-proton collision)

Finally, consider the question of whether you can assign 100% certainty to a mathematical theorem for which a proof exists

To ground this issue in more concrete terms, imagine you are writing an algorithm to compress images made up of 8-bit pixels. The algorithm plows through several rows until it comes to a pixel, and predicts that the distribution of that pixel is Gaussian with mean of 128 and variance of .1. Then the model probability that the real value of the pixel is 255 is some astronomically small number - but the system must reserve some probability (and thus codespace) for that outcome. If it does not, then it violates the general contract that a lossless compression algorithm should assign a code to any input, though some inputs will end up being inflated. In other words it risks breaking.

On the other hand, it is completely reasonable that it should assign zero probability to the outcome that the pixel value is 300. That all pixels values fall between 0 and 255 is a deductive consequence of the problem definition.

But it's hard for me to be properly outraged about this, because the conclusion that the LHC will not destroy the world is correct.

What is your argument for claiming that the LHC will not destroy the world?

That the world still exists albeit ongoing experiments is easily explained by the fact that we are necessarily living in those branches of the universe where the LHC didn't destroy the world. (On an related side note: Has the great filter been found yet?)

This post raises very similar issues to those discussed in comments here.