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Probing the Improbable: Methodological Challenges for
Risks with Low Probabilities and High Stakes
Toby Ord, Rafaela Hillerbrand, Anders Sandberg
*
Some risks have extremely high stakes. For example, a worldwide pandemic or
asteroid impact could potentially kill more than a billion people. Comfortingly,
scientific calculations often put very low probabilities on the occurrence of such
catastrophes. In this paper, we argue that there are important new
methodological problems which arise when assessing global catastrophic risks
and we focus on a problem regarding probability estimation. When an expert
provides a calculation of the probability of an outcome, they are really providing
the probability of the outcome occurring, given that their argument is
watertight. However, their argument may fail for a number of reasons such as a
flaw in the underlying theory, a flaw in the modeling of the problem, or a
mistake in the calculations. If the probability estimate given by an argument is
dwarfed by the chance that the argument itself is flawed, then the estimate is
suspect. We develop this idea formally, explaining how it differs from the
related distinctions of model and parameter uncertainty. Using the risk estimates
from the Large Hadron Collider as a test case, we show how serious the problem
can be when it comes to catastrophic risks and how best to address it.
1. Introduction
Large asteroid impacts are highly unlikely events.
1
Nonetheless, governments spend
large sums on assessing the associated risks. It is the high stakes that make these
otherwise rare events worth examining. Assessing a risk involves consideration of
both the stakes involved and the likelihood of the hazard occurring. If a risk
threatens the lives of a great many people it is not only rational but morally
imperative to examine the risk in some detail and to see what we can do to reduce it.
This paper focuses on low-probability high-stakes risks. In section 2, we show that
the probability estimates in scientific analysis cannot be equated with the likelihood
of these events occurring. Instead of the probability of the event occurring, scientific
analysis gives the event’s probability conditioned on the given argument being
sound. Though this is the case in all probability estimates, we show how it becomes
crucial when the estimated probabilities are smaller than a certain threshold.
To proceed, we need to know something about the reliability of the argument. To do
so, risk analysis commonly falls back on the distinction between model and
parameter uncertainty. We argue that this dichotomy is not well suited for
*
Future of Humanity Institute, University of Oxford.
1
Experts estimate the annual probability as approximately one in a billion (Near-Earth Object
Science Definition Team 2003).
2
incorporating information about the reliability of the theories involved in the risk
assessment. Furthermore the distinction does not account for mistakes made
unknowingly. In section 3, we therefore propose a three-fold distinction between an
argument’s theory, its model, and its calculations. While explaining this distinction
in more detail, we illustrate it with historic examples of errors in each of the three
areas. We indicate how specific risk assessment can make use of the proposed
theory-model-calculation distinction in order to evaluate the reliability of the given
argument and thus improve the reliability of their probability estimate for rare
events.
Recently concerns have been raised that high-energy experiments in particle physics,
such as the RHIC (Relativistic Heavy Ion Collider) at Brookhaven National
Laboratory or the LHC (Large Hadron Collider) at CERN, Geneva, may threaten
humanity. If these fears are justified, these experiments pose a risk to humanity that
can be avoided by simply not turning on the experiment. In section 4, we use the
methods of this paper to address the current debate on the safety of experiments
within particle physics. We evaluate current reports in the light of our findings and
give suggestions for future research.
The final section brings the debate back to the general issue of assessing low-
probability risk. We stress that the findings in this paper are not to be interpreted as
an argument for anti-intellectualism, but rather as arguments for making the noisy
and fallible nature of scientific and technical research subject to intellectual
reasoning, especially in situations where the probabilities are very low and the stakes
very high.
2. Probability Estimates
Suppose you read a report which examines a potentially catastrophic risk and
concludes that the probability of catastrophe is one in a billion. What probability
should you assign to the catastrophe occurring? We argue that direct use of the
report’s estimate of one in a billion is naïve. This is because the report’s authors are
not infallible and their argument might have a hidden flaw. What the report has told
us is not the probability of the catastrophe occurring, but the probability of the
catastrophe occurring given that the included argument is sound. Even if the argument
looks watertight, the chance that it contains a critical flaw may well be much larger
than one in a billion. After all, in a sample of a billion apparently watertight
arguments you are likely to see many that have hidden flaws. Our best estimate of
the probability of catastrophe may thus end up noticeably higher than the report’s
estimate.
2
Let us use the following notation:
2
Scientific arguments are also sometimes erroneous due to deliberate fraud, however we
shall not address this particular concern in this paper.
3
X = the catastrophe occurs,
A = the argument is sound.
While we are actually interested in P(X), the report provides us only with an estimate
of P(X|A), since it can’t take into account the possibility that it is in error.
3
From the
axioms of probability theory, we know that P(X ) is related to P(X|A)!by the
following formula:
(1) P(X)!=!P(X|A)!P(A)!+!P(X|¬A)!P(¬A) .
To use this formula to derive P(X) we would require estimates for the probability
that the argument is sound, P(A), and the probability of the catastrophe occurring
given that the argument is unsound, P(X|¬A). We are highly unlikely to be able to
acquire accurate values for these probabilities in practice but we shall see that even
crude estimates are enough to change the way we look at certain risk calculations.
A special case, which occurs quite frequently, is for reports to claim that X is
completely impossible. However, this just tells us that X is impossible given that all
our current beliefs are correct, i.e. that P(X|A)!=!0. By equation (1) we can see that
this is entirely consistent with P(X)!>!0, as the argument may be flawed.
Figure 1 is a simple graphical representation of our main point. The square on the
left represents the space of probabilities as described in the scientific report, where
the black area represents the catastrophe occurring and the white area represents it
not occurring. The normalized vertical axis denotes the probabilities for the event
occurring and not occurring. This representation ignores the possibility of the
argument being unsound. To accommodate this possibility, we can revise it in the
form of the square on the right. The black and white areas have shrunk in proportion
to the probability that the argument is sound and a new grey area represents the
possibility that the argument is unsound. Now the horizontal axis is also normalized
and represents the probability that the argument is sound.
3
An argument can take into account the possibility that a certain sub-argument is in error. For
example, it could offer two alternative sub-arguments to prove the same point. We encourage
such practice and look at an example in section 4. However, no argument can fully take into
account the possibility that it is itself is flawed — this would require an additional higher-
level argument.
4
Figure 1: The left panel depicts a report’s view on the probability of an event
occurring. The black area represents the chance of the event occurring, the white area
represents it not occurring. The right hand panel is the more accurate picture, taking
into account the possibility that the argument is flawed and that we thus face an grey
area containing an unknown amount of risk.
To continue our example, let us suppose that the argument made in the report looks
very solid, and that our best estimate of the probability that it is flawed is one in a
thousand, (P(¬A)!=!10
-3
). The other unknown term in equation!(1), P(X|¬A), is
generally even more difficult to evaluate, but lets suppose that in the current
example, we think it highly unlikely that the event will occur even if the argument is
not sound, and that we also treat this probability as one in a thousand. Equation (1)
tells us that the probability of catastrophe would then be just over one in a million —
an estimate which is a thousand times higher than that in the report itself. This
reflects the fact that if the catastrophe were to actually occur, it is much more likely
that this was because there was a flaw in the report’s argument than that a one in a
billion event took place.
Flawed arguments are not rare. One way to estimate the frequency of major flaws in
academic papers is to look at the proportion which are formally retracted after
publication. While some retractions are due to misconduct, most are due to
unintentional errors.
4
Using the MEDLINE database
5
(Cokol, Iossifov et al. 2007)
found a raw retraction rate of 6.3!!!10
-5
, but used a statistical model to estimate that
the retraction rate would actually be between 0.001 and 0.01 if all journals received
the same level of scrutiny as those in the top tier. This would suggest that P(¬A) >
0.001, making our earlier estimate rather optimistic. We must also remember that an
argument can easily be flawed without warranting retraction. Retraction is only
called for when the underlying flaws are not trivial and are immediately noticeable
by the academic community. The retraction rate for a field would thus provide a
lower bound for the rate of serious flaws. Of course, we must also keep in mind the
possibility that different branches of science may have different retraction rates and
different error rates. In particular, the hard sciences may be less prone to error than
the more applied sciences.
4
Between 1982 and 2002, 62% of retractions were due to unintentional errors rather than
misconduct (Nath, Marcus et al. 2006).
5
A very extensive database of biomedical research articles from over 5,000 journals.
5
It is important to note the particular connection between the present analysis and
high-stakes low-probability risks. While our analysis could be applied to any risk, it
is much more useful for those in this category. For it is only when P(X|A) is very low
that the grey area has a relatively large role to play. If P(X|A) is moderately high,
then the small contribution of the error term is of little significance in the overall
probability estimate, perhaps making the difference between 10% and 10.001% rather
than the difference between 0.001% and 0.002%. The stakes must also be very high to
warrant this additional analysis of the risk, for the adjustment to the estimated
probability will typically be very small in absolute terms. While an additional one in
a million chance of a billion deaths certainly warrants further consideration, an
additional one in a million chance of a house fire does not.
One might object to our approach on the grounds that we have shown only that the
uncertainty is greater than previously acknowledged, but not that the probability of
the event is greater than estimated: the additional uncertainty could just as well
decrease the probability of the event occurring. When applying our approach to
arbitrary examples, this objection would succeed; however in this article, we are
specifically looking at cases where there is an extremely low value of P(X|A), so
practically any value of P(X|¬A) will be higher and will thus drive the combined
probability estimate upwards. The situation is symmetric with regard to extremely
high estimates of P(X|A), where increased uncertainty about the argument will
reduce the probability estimate, the symmetry is broken only by our focus on
arguments which claim that an event is very unlikely.
Another possible objection is that since there is always a nonzero probability of the
argument being flawed, the situation is hopeless: any new argument will be unable
to remove the grey area completely. It is true that the grey area can never be
completely removed, however if a new argument (A
2
) is independent of the previous
argument (A
1
) then the grey area will shrink, for P(¬A
1
,!¬A
2
)!<!P(¬A
1
). This can
allow for significant progress. A small remaining grey area can be acceptable if
P(X|¬A)P(¬A) is estimated to be sufficiently small in comparison to the stakes.
3. Theories, Models and Calculations
The most common way to assess the reliability of an argument is to distinguish
between model and parameter uncertainty and assign reliabilities to these choices.
While this distinction has certainly been of use in many practical cases, it is
unnecessarily crude for the present purpose, failing to account for potential errors in
the paper’s calculations or a failure of the background theory.
In order to account for all possible mistakes in the argument, we look separately at
its theory, its model, and its calculations. The calculations evaluate a concrete model
representing the processes under consideration, e.g. the formation of black holes in a
particle collision, the response of certain climate parameters (such as mean
temperature or precipitation rate) to changes in greenhouse gas concentrations, or
the response of economies to changes in the oil price. These models are mostly
derived from more general theories. In what follows, we do not restrict the term
‘theory’ to well-established and mathematically elaborate theories like
6
electrodynamics, quantum chromodynamics or relativity theory. Rather, theories are
understood to include theoretical background knowledge such as specific research
paradigms or the generally accepted research practice within a field. An example is
the efficient market hypothesis which underlies many models within economics,
such as the Black-Scholes model.
Even incorrect theories and models can be useful, if their deviation from reality is
small enough for the purpose at hand. Hence we consider adequate models or
theories rather than correct ones. For example, we wish to allow that Newtonian
mechanics is an adequate theory in many situations, while recognizing that in some
cases it is clearly inadequate (such as for calculating the electron orbitals). We thus
call a representation of some system adequate if it is able to predict the relevant
system features at the required precision. For example, if climate modellers wish to
determine the implications our greenhouse gas emissions will have on the well-being
of future generations; their model/theory will not be adequate unless it tells them
the changes in the local temperature and precipitation. In contrast, a model might
only need to tell them changes in global temperature and precipitation to be adequate
for answering less sensitive questions. On a theoretical level, much more could be
said about this distinction between adequacy and correctness, but for the purposes of
evaluating the reliability of risk assessment, the explanation above should suffice.
With the following notation:
T = the involved theories are adequate
M = the derived model is adequate
C = the calculations are correct
we break down A in the way indicated above and replace P(X|A) in equation (1) by
P(X|T,M,C) and P(A) by P(T ,M,C ). From the laws of conditional probability it
follows that:
(2) P(T,M,C) = P(T)!P(M|T)!P(C|M,T)
We may assume C to be independent of M and T, as the correctness of a calculation is
independent of whether the theoretical and model assumptions underpinning it
were adequate. Given this independence, P(C|M,T) = P(C), so the above equation
can be simplified:
(3) P(T,M,C) = P(T) P(M|T) P(C).
Substituting this back into equation (1), we obtain a more tractable formula for the
probability that the event in question occurs.
We have already made a rough attempt at estimating P(A) from the paper retraction
rates. Estimating P(T), P(M|T) and P(C) is more accurate and somewhat easier,
though still of significant difficulty. Though estimating the various terms in equation
(3) must ultimately be done on a case by case basis, the following elucidation of what
we mean by calculation, model and theory will shed some light on how to pursue
7
such an analysis. By incorporating our threefold distinction, it is straightforward to
apply findings on the reliability of theories from philosophy of science — based, for
example, on probabilistic verification methods (e.g. (Reichenbach 1938)) or
falsifications as in (Hempel 1950) or (Popper 1959). Often, however, the best we can
do is to put some bounds upon them based on the historical record. We thus review
typical sources of error in the three areas.
3.1. Calculation: Analytic and Numeric
Estimating the correctness of the calculation independently from the adequacy of the
model and the theory seems important whenever the mathematics involved is non-
trivial. Most cases where we are able to provide more than purely heuristic and
hand-waving risk assessments are of this sort. Consider climate models evaluating
runaway climate change and risk estimates for the LHC or for asteroid impacts.
When calculations accumulate, even trivial mathematical procedures become error-
prone. A particular difficulty arises due to the division of labour in the sciences:
commonly in modern scientific practice, various steps in a calculation are done by
different individuals who may be in different working groups in different countries.
The Mars Climate Observer spacecraft was lost in 1999 because a piece of control
software from Lockheed Martin used Imperial units instead of the metric units the
interfacing NASA software expected (NASA 1999).
Calculation errors are distressingly common. There are no reliable statistics on the
calculation errors made in risk assessment or, even more broadly, within scientific
papers. However, there is research on errors made in some very simple calculations
that performed in hospitals. Dosing errors give an approximate estimate of how
often mathematical slips occur. Errors in drug charts occur at a rate of 1.2% to 31%
across different studies (Prot, Fontan et al. 2005; Stubbs, Haw et al. 2006; Walsh,
Landrigan et al. 2008), with a median of roughly 5% of administrations. Of these
errors 15-40% were dose errors, giving an overall dose error rate of about 1–2%.
What does this mean for error rates in risk estimation? Since the stakes are high
when it comes to dosing errors, this data represents a serious attempt to get the right
answer in a life or death circumstance. It is likely that the people doing risk
estimation are more reliable at arithmetic than health professionals and have more
time for error correction, but it appears unlikely that they would be many orders of
magnitude more reliable. Hence a chance of 10
-3
for a mistake per simple calculation
does not seem unreasonable. A random sample of papers from Nature and the British
Medical Journal found that roughly 11% of statistical results were flawed, largely due
to rounding and transcription errors (García-Berthou and Alcaraz 2004).
Calculation errors include more than just the ‘simple’ slips which we know from
school, such as confusing units, forgetting a negative square root, or incorrectly
transcribing from the line above. Instead, many mistakes arise here due to numerical
implementation of the analytic mathematical equations. Computer based simulations
and numerical analysis are rarely straightforward. The history of computers contains
a large number of spectacular failures due to small mistakes in hardware or software.
The June 4 1996 explosion of an Ariane 5 rocket was due to a leftover piece of code
triggering a cascade of failures (ESA 1996). Audits of spreadsheets in real-world use
8
find error rates on the order of 88% (Panko 1998). The 1993 Intel Pentium floating
point error affected 3-5 million processors, reducing their numeric reliability and
hence our confidence in anything calculated with them (Nicely 2008). Programming
errors can remain dormant for a long time even in apparently correct code, only to
emerge under extreme conditions. An elementary and widely used binary search
algorithm included in the standard libraries for Java was found after nine years to
contain a bug that emerges only when searching very large lists (Bloch 2006). A
mistake in data-processing led to the retraction of five high-profile protein structure
papers as the handedness of the molecules had become inverted (Miller 2006).
In cases where computational methods are used in modelling, many mistakes cannot
be avoided. Discrete approximations of the often continuous model equations are
used, and in some cases we know that the discrete version is not a good proxy for the
continuous model (Morawetz and Walke 2003). Moreover, numerical evaluations are
often done on a discrete computational grid, with the values inside the meshes being
approximated from the values computed at the grid points. Though we know that
certain extrapolation schemes are more reliable in some cases than others, we are
often unable to exclude the possibility of error, or to even quantify it.
3.2 Ways of modelling and theorizing
Our distinction between model and theory follows the typical use of the terms within
mathematical sciences like physics or economics. Whereas theories are associated
with broad applicability and higher confidence in the correctness of their description,
models are closer to the phenomena. For example, when estimating the probability
of a particular asteroid colliding with the earth, one would use either Newtonian
mechanics or general relativity as a theory for describing the role of gravity. One
could then use this theory in conjunction with observations of the bodies’ positions,
velocities and masses to construct a model, and finally, one could perform a series of
calculations based on this model to estimate the probability of impact. As this shows,
the errors that can be introduced in settling for a specific model include and surpass
those which are sometimes referred to as parameter uncertainty. As well as questions
of the individual parameters (positions, velocities, masses), there are important
questions of detail (can we neglect the inner structure of the involved bodies?), and
breadth (can we focus on the Earth and asteroid only, or do we have to model other
planets, or the Sun?).
6
As can be seen from this example, one way to distinguish theories from models is
that theories are too general to be applied directly to the problem. For any given
theory, there are many ways to apply it to the problem and these ways give rise to
different models. Philosophers of science will note that our theory/model distinction
6
This question of breadth is closely linked to what (Hansson 1996) refers to as demarcation
uncertainty. But demarcation of the problem involves not only the obvious demarcation in
physical space and time, but also questions of the systems to consider, the scales to consider
etc.
9
is in accordance with the non-uniform notion used by (Giere 1999), (Morrison 1998),
(Cartwright 1999), and others, but differs from that of (Suppes 1957).
We should also note that it is quite possible for an argument to involve several
theories or several models. This complicates the analysis and typically provides
additional ways for the argument to be flawed.
7
For example, in estimating the risk
of black hole formation at the LHC, we not only require quantum chromodynamics
(the theory the LHC is built to test), but also relativity and Hawking’s theory of black
hole radiation. In addition to their other roles, modelling assumptions also have to
explain how to glue such different theories together (Hillerbrand and Ghil 2008).
In risk assessment, the systems involved are most often not as well understood as
asteroid impacts. Often, various models exist simultaneously — all known to be
incomplete or incorrect in some way, but difficult to improve upon. Particularly in
these cases, having an expected or desired outcome in mind while setting up a
model, makes one vulnerable to expectation bias: the tendency to reach the desired
answer rather than the correct one. This bias has affected many of science's great
names (Jeng 2006), and in the case of risk assessment, the desire for a ‘positive’
outcome (safety in the case of the advocate or danger in the case of the protestor)
seems a likely cause of bias in modelling.
Figure 2: Our distinctions regarding the ways in which risk assessments can be
flawed.
3.3 Historical examples of Model and Theory Failure
A dramatic example of a model failure was the Castle Bravo nuclear test on March 1
1954. The device achieved 15 megatons of yield instead of the predicted 4-8
megatons. Fallout affected parts of the Marshal Islands and irradiated a Japanese
fishing boat so badly that one fisherman died, causing an international incident
(Nuclear Weapon Archive 2006). Though the designers at Los Alamos National
Laboratories understood the involved theory of alpha decay, their model of the
reactions involved in the explosion was too narrow, for it neglected the decay of one
of the involved particles (lithium-7), which turned out to contribute the bulk of the
7
Additional theories and models can also be deliberately introduced in order to lower the
probability of argument failure, and in section 4, we shall see how this has been done for the
safety assessment of the LHC.
10
explosion’s energy. The Castle Bravo test is also notable for being an example of
model failure in a very serious experiment conducted in the hard sciences and with
known high stakes.
The history of science contains numerous examples of how generally accepted
theories have been overturned by new evidence or understanding, as well as a
plethora of minor theories that persisted for a surprising length of time before being
disproven. Classic examples for the former include the Ptolemaic system, phlogiston
theory and caloric theory; an example for the latter is human chromosome number,
which was systematically miscounted as 48 (rather than 46) and this error persisted
for more than 30 years (Gartler 2006).
As a final example, consider Lord Kelvin’s estimates of the age of the Earth
(Burchfield 1975). They were based on information about the earth’s temperature
and heat conduction, estimating an age of the Earth of between 20 and 40 million
years. These estimates did not take into account radioactive heating, for radioactive
decay was unknown at the time. Once it was shown to generate additional heat the
models were quickly updated. While neglecting radioactivity today would count as a
model failure, in Lord Kelvin’s day it represented a largely unsuspected weakness in
the physical understanding of the Earth and thus amounted to theory failure. This
example makes it clear that the probabilities for the adequacy of model and theory
are not independent of each other, and thus in the most general case we cannot
further decompose equation (3).
4. Applying our analysis to the risks from particle physics research
Particle physics is the study of the elementary constituents of matter and radiation,
and the interactions between them. A major experimental method in particle physics
involves the use of particle accelerators such as the RHIC and LHC to bring beams
of particles to near the speed of light and then collide them together. This focuses a
large amount of energy in a very small region and breaks the particles down into
their components, which are then detected. As particle accelerators have become
larger, the energy densities achieved have become more extreme, prompting some
concern about their safety. These safety concerns have focused on three possibilities:
the formation of ‘true vacuum’, the transformation of the earth into ‘strange matter’,
and the destruction of the earth through the creation of a black hole.
4.1 True vacuum and strange matter formation
The type of vacuum that exists in our universe might not be the lowest possible
vacuum energy state. In this case, the vacuum could decay to the lowest energy state,
either spontaneously, or if triggered by a sufficient disturbance. This would produce
a bubble of ‘true vacuum’ expanding outwards at the speed of light, converting the
universe into different state apparently inhospitable for any kind of life (Turner and
Wilczek 1982).
Our ordinary matter is composed of electrons and two types of quarks: up quarks
and down quarks. Strange matter also contains a third type of quark: the ‘strange’
quark. It has been hypothesized that strange matter might be more stable than
11
normal matter, and able to convert atomic nuclei into more strange matter (Witten
1984). It has also been hypothesized that particle accelerators could produce small
negatively charged clumps of strange matter, known as strangelets. If both these
hypotheses were correct and the strangelet also had a high enough chance of
interacting with normal matter, it would grow inside the Earth, attracting nuclei at
an ever higher rate until the entire planet was converted to strange matter —
destroying all life in the process. Unfortunately strange matter is complex and little
understood, giving models with widely divergent predictions about its stability,
charge and other properties (Jaffe, Busza et al. 2000).
One way of bounding the risk from these sources is the cosmic ray argument: the same
kind of high-energy particle collisions occur all the time in Earth’s atmosphere, on
the surface on the Moon and elsewhere in the universe. The fact that the Moon or
observable stars have not been destroyed despite a vast number of past collisions
(many at much higher energies than can be achieved in human experiments) suggest
that the threat is negligible. This argument was first used against the possibility of
vacuum decay (Hut and Rees 1983) but is quite general.
An influential analysis of the risk from strange matter was carried out in (Dar, De
Rujula et al. 1999) and formed a key part of the safety report for the RHIC. This
analysis took into account the issue that any dangerous remnants from cosmic rays
striking matter at rest would be moving at high relative velocity (and hence much
less likely to interact) while head-on collisions in accelerators could produce
remnants moving much at much slower speeds. They used the rate of collisions of
cosmic rays in free space to estimate strangelet production. These strangelets would
then be slowed by galactic magnetic fields and eventually be absorbed during star
formation. When combined with estimates of the supernova rate, this can be used to
bound the probability of producing a dangerous strangelet in a particle accelerator.
The resulting probability estimate was < 2 ! 10
-9
per year of RHIC operation.
8
While using empirical bounds and experimentally tested physics reduces the
probability of a theory error, the paper needs around 30 steps to reach its conclusion.
For example, even if there was just a 10
-4
chance of a calculation or modelling error
per step this would give a total P(¬A)!"!0.3%. This would easily overshadow the risk
estimate. Indeed, even if just one step had a 10
-4
chance of error, this would
overshadow the estimate.
A subtle complication in the cosmic ray argument was noted in (Tegmark and
Bostrom 2005). The Earth’s survival so far is not sufficient as evidence for safety,
since we do not know if we live in a universe with ‘safe’ natural laws or a universe
where planetary implosions or vacuum decay do occur but we have just been
exceedingly lucky so far. While this latter possibility might sound very unlikely, all
observers in such a universe would find themselves to be in the rare cases where
8
(Kent 2004) points out some mistakes in stating the risk probabilities in different versions of
the paper, as well as for the Brookhaven report. Even if these are purely typesetting mistakes,
it shows that the probability of erroneous risk estimates is nonzero.
12
their planets and stars had survived, and would thus have much the same evidence
as we do. Tegmark and Bostrom had thus found that in ignoring these anthropic
effects, the previous model had been overly narrow. They corrected for this
anthropic bias and, using analysis from (Jaffe, Busza et al. 2000), concluded that the
risk from accelerators was less than 10
-12
per year.
This is an example of a demonstrated flaw in an important physics risk argument
(one that was pivotal in the safety assessment of the RHIC). Moreover, it is
significant that the RHIC had been running for five years on the strength of a flawed
safety report, before Tegmark and Bostrom noticed and fixed this gap in the
argument. Although this flaw was corrected immediately after being found, we
should also note that the correction is dependent on both anthropic reasoning and on
a complex model of the planetary formation rate (Lineweaver, Fenner et al. 2004). If
either of these, or the basic Brookhaven analysis is flawed, the risk estimate is
flawed.
4.2 Black hole formation
The Large Hadron Collider experiment at CERN was designed to explore the
validity and limitations of the Standard Model of particle physics by colliding beams
of high energy protons. This will be the most energetic particle collision experiment
ever done, which has made it the focus of a recent flurry of concerns. Due to the
perceived strength of the previous arguments on vacuum decay and strangelet
production, most of the concern about the LHC has focused on black hole
production.
None of the theory papers we have found appears to have considered the black holes
to be a safety hazard, mainly because they all presuppose that any black holes would
immediately evaporate due to Hawking radiation. However, it was suggested by
(Dimopoulos and Landsberg 2001) that if black holes form, particle accelerators
could be used to test the theory of Hawking radiation. Thus critics also began
questioning whether we could simply assume that black holes would evaporate
harmlessly.
A new risk analysis of LHC black-hole production (Giddings and Mangano 2008)
provides a good example of how risks can be more effectively bounded through
multiple sub-arguments. While never attempting to give a probability of disaster
(rather concluding "there is no risk of any significance whatsoever from such black
holes") it uses a multiple bounds argument. It first shows that rapid black hole decay
is a robust consequence of several different physical theories (A
1
). Second it discusses
the likely incompatibility between non-evaporating black holes and mechanisms for
neutralising black holes: in order for cosmic!ray–produced stable black holes to be
innocuous but accelerator-produced black holes to be dangerous, they have to be
able to shed excess charge rapidly (A
2
). Our current understanding of physics
suggests both that black holes decay and that even if they didn’t, they would be
unable to discharge themselves. Only if this understanding is flawed will the next
section come into play.
13
The third part, which is the bulk of the paper, models how multidimensional and
ordinary black holes would interact with matter. This leads to the conclusion that if
the size scale of multidimensional gravity is smaller than about 20 nm, then the time
required for the black hole to consume the Earth would be larger than the natural
lifetime of the planet. For scenarios where rapid Earth accretion is possible, the
accretion time inside white dwarves and neutron stars would also be very short, yet
production and capture of black holes from impinging cosmic rays would be so high
that the lifespan of the stars would be far shorter than the observed lifespan (and
would contradict white dwarf cooling rates) (A
3
).
While each of these arguments have weaknesses the force of the total argument
(A
1
,A
2
,A
3
) is significantly stronger by the combination of them. Essentially the paper
acts as three sequential arguments, each partly filling in the grey area (see figure 1)
left by the previous. If the theories surrounding black hole decay fail, the argument
about discharge comes into play, and if against all expectation black holes are stable
and neutral the third argument shows that astrophysics constrains them to a low
accretion rate.
4.3 Implications for the safety of the LHC
What are the implications of our analysis for the safety assessment of the LHC? First,
let us consider the stakes in question. If one of the proposed disasters were to occur,
it would mean the destruction of the earth. This would involve the complete
destruction of the environment, 6.5 billion human deaths and the loss of all future
generations. It is worth noting that this loss of all future generations (and with it, all
of humanity’s potential) may well be the greatest of the three, but a comprehensive
assessment of these stakes is outside the scope of this paper. For the present
purposes, it suffices to observe that the destruction of the earth is at least as bad as 6.5
billion human deaths.
There is some controversy as to how one should combine probabilities and stakes
into an overall assessment of a risk. Some hold that the simple approach of expected
utility is the best, while others hold some form of risk aversion. However, we can
sidestep this dispute by noting that in either case, the risk of some harm is at least as
bad as the expected loss. Thus, a risk with probability p of causing a loss at least as
bad as 6.5 billion deaths is at least as bad as a certain 6.5!!!10
9
p deaths.
Now let us turn to the best estimate we can make of the probability of one of the
above disasters occurring during the operation of the LHC. While the arguments for
the safety of the LHC are commendable for their thoroughness, they are not
infallible. Although the report considered several possible physical theories, it is
eminently possible that these are all inadequate representations of the underlying
physical reality. It is also possible that the models of processes in the LHC or the
astronomical processes appealed to in the cosmic ray argument are flawed in an
important way. Finally, it is possible that there is a calculation error in the report.
Recall equation (1):
(1) P(X)!=!P(X|A)!P(A)!+!P(X|¬A)!P(¬A)
14
P(X) is formed from two terms. The second of these represents the additional
probability of disaster due to the argument being unsound. It is the product of the
probability of argument failure and the probability of disaster given such a failure.
Both terms are very difficult to estimate, but we can gain insight by showing the
ranges they would have to lie within, for the risk presented by the LHC to be
acceptable.
From (1), we obtain that:
(4) P(X) # P(X|¬A)!P(¬A) .
If we let l denote the acceptable limit of expected deaths from the operation of the
LHC, we get: 6.5!!!10
9
P(X) $ l. Combining this with equation (4), we obtain:
(5) P(X|¬A)!P(¬A) $ 1.5!!!10
-10
l .
This inequality puts a severe bound on the acceptable values for these probabilities.
Since it is much easier to grasp this with an example, we shall provide some numbers
for the purposes of illustration. Suppose, for example, that the limit were set at 1000
expected deaths, then P(X|¬A)!P(¬A) would have to be below 1.5!!!10
-7
for the risk to
be worth bearing. This requires very low values for these probabilities. We have seen
that for many arguments, P(¬A) is above 10
-3
. We have also seen that the argument
for the safety of the RHIC turned out to have a significant flaw, which was unnoticed
by the experts at the time. It would thus be very bold to suppose that the argument
for the safety of the LHC was much lower than 10
-3
, but for the sake of argument, let
us grant that it is as low as 10
-4
— that out of a sample of 10,000 independent
arguments of similar apparent merit, only one would have any serious error.
Even with the value of P(¬A) were as low as 10
-4
, P(X|¬A)!would have to be below
0.15% for the risk to be worth taking. P(X|¬A) is the probability of disaster given
that the arguments of the safety report are flawed!and is the most difficult
component of equation (1) to estimate. Indeed, few would dispute that we really
have very little idea of what value to put on P(X|¬A). It would thus seem overly
bold to set this below 0.15% without some substantive argument. Perhaps such an
argument could be provided, but until it is, such a low value for P(X|¬A) seems
unwarranted.
We stress that the above combination of numbers was purely for illustrative
purposes, but we cannot find any plausible combination of the three numbers which
meets the bound and which would not require significant argument to explain either
the levels of confidence or the disregard for expected deaths. We would also like to
stress that we are open to the possibility that additional supporting arguments and
independent verification of the models and calculations could significantly reduce
the current chance of a flaw in the argument.
However, our analysis implies that the current safety report should not be the final
word in the safety assessment of the LHC. To proceed with the LHC on the
arguments of the most recent safety report alone, we would require further work on
estimating P(¬A), P(X|¬A), the acceptable expected death toll, and the value of
15
future generations and other life on earth. Such work would require expertise
beyond theoretical physics, and an interdisciplinary group would be essential. If the
stakes were lower, then it might make sense for pragmatic concerns to sweep aside
this extra level of risk analysis, but the stakes are astronomically large, and so further
analysis is critical. Even if the LHC goes ahead without any further analysis, as is
very likely, these lessons must be applied to the assessment of other high-stakes low-
probability risks.
5. Conclusions
When estimating threat probabilities, it is not enough to make conservative estimates
(using the most extreme values or model assumptions compatible with known data).
Rather, we need robust estimates that can handle theory, model and calculation
errors. The need for this becomes considerably more pronounced for low-probability
high-stake events, though we do not say that low probabilities cannot be treated
systematically. Indeed, as pointed out by (Yudkowsky 2008), if we could not
correctly predict probabilities lower than 10
-6
,
!we could not run lotteries.
Some people have raised the concern that our argument might be too powerful: for it
is impossible to disprove the risk of even something as trivial as dropping a pencil,
then our argument might amount to prohibiting everything. It is true that we cannot
completely rule out any probability that apparently inconsequential actions might
have disastrous effects, but there are a number of reasons why we do not need to
worry about universal prohibition. A major reason is that for events like the
dropping of a pencil which have no plausible mechanism for destroying the world, it
seems just as likely that the world would be destroyed by not dropping the pencil.
The expected losses would thus balance out. It is also worth noting that our
argument is simply an appeal to a weak form of decision theory to address an
unusual concern: for our method to lead to incorrect conclusions, it would require a
flaw in decision theory itself, which would be very big news.
It will have occurred to some readers that our argument is fully applicable to this
very paper: there is a chance that we have made an error in our own arguments. We
entirely agree, but note that this possibility does not change our conclusions very
much. Suppose, very pessimistically, that there is a 90% chance that our argument is
sufficiently flawed that the correct approach is to take safety reports’ probability
estimates at face value. Even then, our argument would make a large difference to
how we treat such values. Recall the example from section 2, where a report
concludes a probability of 10
-9
and we revise this to 10
-6
. If there is even a 10% chance
that we are correct in doing so, then the overall probability estimate would be
revised to 0.9!!!10
-9
+ 0.1!!!10
-6
" 10
-7
, which is still a very significant change from the
report’s own estimate. In short, even serious doubt about our methods should not
move one’s probability estimates more than an order of magnitude away from those
our method produces. More modest doubts would have a effect.
The basic message of our paper is that any scientific risk assessment is only able to
give us the probability of a hazard occurring conditioned on the correctness of its
main argument. The need to evaluate! the reliability of the given! argument ! in order
16
to adequately address the risk was shown to be of particular relevance in low-
probability high-stake events. We drew a three-fold distinction!between theory,
model and calculation, and showed how this can be more useful than the common
dichotomy in risk assessment between model and parameter uncertainties. By
providing historic examples for errors in the three fields, we clarified the three-fold
distinction and showed where flaws in a risk assessment might occur. Our analysis
was applied to the recent assessment of risks that might arise from experiments
within particle physics. To conclude this paper, we now provide some very general
remarks on how to avoid argument flaws when assessing risks with high stakes.
Firstly, the testability of predictions can help discern flawed arguments. If a risk
estimate produces a probability distribution for smaller, more common disasters this
can be used to judge whether the observed incidences are compatible with the
theory. Secondly, reproducibility appears to be the most effective way of removing
many of these errors. By having other people replicate the results of calculations
independently our confidence in them can be dramatically increased. By having
other theories and models independently predict the same risk probability our
confidence in them can again be increased, as even if one of the arguments is wrong
the others will remain. Finally, we can reduce the possibility of unconscious bias in
risk assessment through the simple expedient of splitting the assessment into a ‘blue’
team of experts attempting to make an objective risk assessment and a ‘red’ team of
devil’s advocates attempting to demonstrate a risk, followed by repeated turns of
mutual criticism and updates of the models and estimates (Calogero 2000).
Application of such methods could in many cases reduce the probability of error by
several orders of magnitude.
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