CRYONICS: THE PROBABILITY OF RESCUE
By R.C.W. Ettinger
INTRODUCTION
Probability concepts often seem simple, and indeed not only experts but ordinary
people apply them usefully every day; they are indispensable to survival,
let alone successful living.
But at the same time, some of the most fundamental concepts are so subtle
and slippery that even the greatest names in mathematics have often been
confused.
When we try to apply probability to cryonics, we are combining two concepts,
both very poorly understood by almost everyone, including the
"experts"biologists and mathematicians.
The following is an attempt to summarize and clarify the evidence for eventual
reversibility of freezing and other damage to cryonic suspension patients,
and to put it in a probabilitytheory framework. Laymen and scientists face
two different problems in understanding this approach.
For the laymen, the greatest obstacle is just their own humility; they have
often been trained to disbelieve in their own powers of analysis and judgment.
It is useful to remember that the CommanderinChief of the U.S. armed forces
is usually a layman (the President), and that the CEOs of major corporations
selling technology are seldom engineers; yet they assess and understand technical
advice (in broad outline) and make the ultimate decisions. Nonscientists
reading the material below will find some of it too technical, but most of
it understandable, if not easy. In particular, they will learn why their
intuitions are often correct.
Scientists will face a harder jobviz., breaking the habit of preferring
precise but irrelevant numbers over imprecise but pertinent data. What this
means will become apparent as we proceed.
The first part of the discussion, laying the groundwork, may be the most
difficult for both laymen and scientists (for different reasons). I hope
the former will not be put off by the mathematics, which they can skip, and
that the latter will pay attention.
Some scientistsusually honesthave forgotten their ethics when discussing
cryonics, and made totally irresponsible and unfounded statements about the
alleged "low probability" of successwithout ever making a probability
calculation, or offering any basis for such a calculation! I hope some of
them will reconsider.
FOUNDATIONS OF PROBABILITY THEORY
(These notes are adapted from original work I did about forty five years
ago; the work was reviewed and stamped kosher by experts in the field.)
Approach A: von Mises
One of the best known "frequency" theories of probability is that advocated
by Richard von Mises. [1] According to him, the probability of' an event,
with reference to an experiment, is defined as the limiting relative frequency
(in the sense of measure theory) of occurrence of the event in an infinite
sequence of identical experiments; the sequence has the randomness property
that the results are "indifferent to place selection." (Lay readers, please
don't get nervous; what this means will gradually become clear, at least
in outline.)
That neither infinite sequences nor identical experiments are ever encountered
in the real world does not disturb von Mises; he points out that theories
generally apply idealized mathematical models to nonideal situations, and
that his theory meets the tests of' useful description and prediction.
He considers that the technical term probability can apply only to mass phenomena
or repetitive events, and never to "isolated" events or to intensity of belief
as met in common parlance. "...if one speaks of the probability that the
two poems known as the Iliad and the Odyssey have the same author, no reference
to an infinite sequence of cases is possible and it makes no sense to assign
a numerical value to such a 'possibility.'" [2]
Remarks:
a) The experiments cannot be identical, even for idealized sequences, since
if the experiments were truly identical so would be the results, giving rise
to certainty rather than probability. (This is on the classical level, not
the quantum level.) What is actually required for the idealized sequence
is that the experiments be identical on the operational level; but it is
also necessary that the operator's control be imperfect.
b) One of the points slurred over in the exposition by von Mises is that
one can only estimate, but never know, any probability. For example, even
if a perfectly symmetrical coin were available, no finite sequence of trials
could do more than suggest that the probability p for "heads" is in the
neighborhood of 1/2. To leap from this estimate to the conclusion that "p
actually is exactly 1/2" would require an additional postulate of an ad hoc
character.
c) Leibnitz is said to have objected to frequency theories on the ground
that one could never exactly prescribe the conditions of the experiment.The
answer of the frequentists seems to be that the application of a mathematical
model is generally inexact. But von Mises' theory is extraordinarily vague
with respect to application; in fact, this vagueness seems severely to restrict
its usefulness; and it has led to the contention by Jeffreys [7], Koopman
[4], and many others that the theory really relies on intuition for its
application, so that it is not a true frequency theory at all.
In particular, von Mises does not anywhere answer the important question:
when is "reference to an infinite sequence of cases" possible? In other words,
by what criterion is one to distinguish "repetitive" from "isolated" events?
This point may be worth elaborating with a couple of examples:
The toss of a die is specifically admitted by von Mises as a typical repetitive
event; yet by what criterion can it be so regarded? The criterion cannot
be the fact that dice have often been tossed: for it is clear that the toss
ofe.g.a 12sided die would be equally a "repetitive" event, although
such a thing may never have existed before and might be destroyed after one
toss.
The criterion cannot be merely the range of one's imagination. Von Mises
would regard the question of life on Mars as outside the theory; yet it may
be as easy to imagine a population of planets Marsor of Milky Way galaxies
for that matteras to imagine a sequence of tosses of 12sided dice.
Since frequentists are generally definite enough about whether a particular
event is within the scope of the theory, they are certainly applying some
criterion, although they do not seem to know what it is. In the synthesis
later on, I claim to make this criterion explicit, and thereby to extend
the scope of the theory.
The key lies in this question: just how does one obtain information (as in
the case of the 12sided die) from a hypothetical experiment?
Approach B: Doob
An approach somewhat different from that of v. Mises may be found in what
is sometimes called the "classical frequency" theory of probability; names
associated with this are Doob, Neyman, Feller, and Cramer.
A mathematical model is set up on the basis of measure theory. Following
Cramer [3], whose ideas for our purpose are similar enough to those of Doob,
we may paraphrase the chief axiom of the theory thus:
AXIOM: "To any random variable X in the ndimensional space Rn there corresponds
a set function P(S) uniquely defined for all Borel sets S in Rn, such that
P(S) represents the probability of the event X S; the function P(S) is a
nonnegative and additive set function such that
P(Rn) = 1."
O.K., not many readers are at home with Borel sets. Again, don't get nervous;
we'll just say that pursuing this approach gives us an interpretation of
the "limit" to which a relative frequency is said to tend.
But it turns out that the application of this theory also is a matter of
judgment and guesswork. In the case of a die, for example, the axiom tells
us that certain probabilities exist such that
P(ace) + P(deuce) + P(three) + P(four) + P(five) + P(six) = 1,
but what these numbers may be, or whether they are all the same, the theory
cannot say.
Hence some of the same criticisms made of the v. Mises theory apply here
also. We have a new notion of limit, and a new and more powerful calculus
of probability, but with respect to applicationfinding the underlying
probabilitieswe find the same vagueness.
Approach C: Laplace
That definition of probability most commonly called "classical" is associated
chiefly with the names of Pascal, Fermat, and Laplace. It states that if
an experiment can have any of n "equipossible" and mutually exclusive outcomes,
and if m of these outcomes are "favorable" to the event in question, then
the probability of the event, referred to the experiment, is the ratio m/n.
This definiton does not seem "circular", as claimed by v. Mises, because
the notion of "equally likely" is less general, and more primitive, than
the Laplacian notion of numerical probability. The definition is certainly
incomplete, however, since it leaves "equally likely" as an undefined term.
In philosophical language, the definition has only a "constitutive" aspect,
and no "epistemic" aspect.
The definition is also said to be weak in that it cannot apply to those cases
(e.g. a loaded die) where no breakdown into equally likely cases seems possible.
However, the criticism seems to me ill founded. What we essentially have
here is simply another starting point for a probability calculus, a viewpoint
slightly different from that of v. Mises or Doob; with respect to application,
this definition seems hardly weaker than the latter two; there are ways in
which the Laplacian notion can be doctored up to include such experiments
as tosses of a biased die. (One might say, e.g., that the die was equally
likely to be in any of several specified positions as it approached the ground;
this specification would favor the heavy side.)
The three theories so far considered all recognize that there must be some
sort of correspondence between probabilities and relative frequencies, but
none of them gives a criterion for actually assigning a probability to a
physical eventnot even the theory of v. Mises, although he claims to give
the physical events first consideration.
The Laplacian definition is also said to be ambiguous when the possible outcomes
are infinitely numerous. Mood e.gr. points out that the probability that
an integer drawn at random from all the integers will be even depends on
the ordering of the integers. However, this objection at most seems academic
rather than practical. After all, there does not seem to be any way to draw
an integer at random from all the integers! (Even if you could, in the sense
of measure theory there would be a zero probability of drawing one small
enough to write down!)
In short, the Laplacian definition seems to me to rest on essentially equal
footing with the two previously mentioned theories with regard to logical
adequacy. From the practical standpoint, there are many instances where the
application of the Laplace rule is the quickest and easiest resort.
Approach D: Koopman
There is a (minority) school, mostly in England, holding that the frequency
theory of probability is inadequate; some names which have been associated
with this thesis are Keynes, Koopman, Good, Jeffreys, Kendall, and de Finetti.
B.O. Koopman [4] e.g,. has produced a detailed axiomatic theory, based on
intuition, which seems to be in many respects as practical as v. Mises' frequency
theory and yet broader.
His axioms are based on what he calls "laws of thought," which he claims
are "not subject to experimental verification." Using two axioms, together
with a "body of beliefs," he is able to construct probability statements
about all kinds of events. In general, these statements are nonstatistical
and nonnumerical, but for a certain class of problems Koopman deduces a
numerical theory which, for many important applications, is in agreement
with frequency theory.
On the epistemic side, Koopman's position is esentially as follows: "The
intuitive thesis in probability holds that both in its meaning and in the
laws which it obeys, probability derives directly from the intuition, and
is prior to objective experience; ... and it holds that all the socalled
objective definitions of probability depend for their effective application
to concrete cases upon their translation into the terms of intuitive
probability."
What he means is closely related to the remarks made in connection with the
v. Mises theory. In the case of a die, for example, the frequency theory
does not give any explicit criterion which allows us to regard the toss as
a to regard the toss an a repetitive event; neither does it give us any
justification for taking the probability of a deuce e.g. as exactly 1/6.
In practice, we rely on intuition, any frequency correspondence we may be
able to exhibit being only an approximate one.
From one point of view, therefore, the difference between the Koopman "intuitive"
theory and the v. Mises "frequency" theory is not so great after all. They
are built on different formal bases; but in each case, in applications one
must start with given probabilities, neither theory giving any explicit way
to assign a value to an initial probability. The v. Mises theory supposedly
restricts the events considered to outcomes of sequences of experiments;
but, as pointed out by Feller, these are only conceptual experimentsand
we are not told just what is and what is not "conceivable".
Further remarks:
Kendall [5] has said that the intuitive attitude toward probability is one
which ". . . takes probability as 'a degree of rational belief'…and
does not attempt to analyze it into simpler ideas," and this is true of the
Koopman view. The question is, can intuition be analyzed into simpler ideas,
or is intuition indeed "prior to objective experience"?
To maintain that intuition cannot be analyzed seems philosophically and
psychologically naive; and in fact many studies are available which show
just how certain kinds of "intuition" are molded by experience. Intuition
varies from one person to another, from time to time in a given individual,
and from age to age with respect to mankind as a whole. Yesterday's obvious
truth becomes today's fallacy, and today's difficult lesson becomes tomorrow's
truism. In short, it seems consistent with modern ideas to assert, in flat
contradiction to Koopman, that intuition derives from experience, and can
in principle be explicitly analyzed in terms of experience. In fact, I claim
below to show how probability intuitions arise. (This really isn't very
mysterious, or shouldn't be.)
SYNTHESIS
Reconciliation of "Frequency" Probability and "Personal" Probability
We seek to show that the dominant frequency theory of probability can be
applied in an extremely simple and natural way so as to include "single events"
and "subjective" probability. It will then be possible to assign a definite
(although not necessarily precise) numerical probability to any event of
whatever kind: this probability will be at once objective (frequency) and
subjective (dependent on state of knowledge).
(We are considering the epistemic or operational definition of probability,
and not the calculus of probability or axioms of combination.)
We start with the v. Mises definition of probability of an event with reference
to an experiment: the probability is the postulated limiting relative frequency
of occurrence of the event in an infinite sequence of "identical" experiments
whose results are indifferent to place selection. (The experiments are
"identical" on the operational level.) Now let us note the important features
of the application of this definition.
1. The postulated limiting frequency is never known exactly. Unless one adopts
the dubious alternative of special postulates, such as ergodic principles,
one cannot do more than estimate the probability to a finite number of decimal
places on the basis of experience.
(Even when tossing a perfect coin, one cannot assume that the nervous system
of the person doing the tossing yields perfectly random results; but experience
indicates that the magnitude of any shove delivered by human muscles, below
a certain level of accuracy, is random or nearly so.)
It is important to note here that the accuracy with which experience allows
us to estimate a probability is of no theoretical importance: a probability
of 0.5 +/ 0.4 is just as much in the scope of the theory as a probability
of 0.5 +/ 0.0000001.
2. In practice, the experiments are not identical even on the operational
level. Coins are asymmetrical; mortality tables are subject to revision;
etc.
It is important to note therefore that any sequence of experiments remains
in the scope of the theory so long as the experiments are sufficiently nearly
identical to allow useful calculation.
3. Although frequentists allege that "isolated events" are outside the theory,
there are at least two kinds of such events which they save from isolation:
a) Suppose I make a new toy, the 12sided die mentioned earlier, toss it
once and then destroy it. The event that on this toss a particular face will
land up is in a sense an isolated event, since experience contains no sequence
of such tosses. We therefore seek a larger class of experiments to save the
event from isolation; in fact we take cognizance of our experience with
symmetrical bodies and random shoves, and this does the trick.
b) "Derivative" probabilities are also recognized. E.g., the event "Phogbound
will be reelected senator" is an isolated event; but it is essentially equivalent
to the event "A majority of voters favor Phogbound," and the probability
of this latter event can be estimated by sampling.
4) We emphasize again what we stated in the definition: a probability refers
not only to an event but also to an experiment in a sequence. Since specifying
a particular experiment and sequence is equivalent to specifying a state
of knowledge, we see that ordinary frequency probability has a subjective
aspect. For examples see below.
Next, a short discussion of certain contentions by frequentists and
subjectivists will clear the way for our general approach.
Frequentists say that the vague common parlance notion of "probability",
which applies to single events, hypotheses, etc., is outside the theory.
They seem to overlook the fact that this common parlance notion has enormous
practical success every day for everyone, and that this success should be
capable of analysis. In fact, I assert that this success is due to the
(nonexplicit) application of just the theory here presented.
Subjectivists say that intuition is at the root of probability, and that
intuition is prior to experience. This, as noted, seems to be a psychologically
naive position: intuition is varied, intuition is educable, and in fact intuition
derives mainly (although unconsciously) from experience.
An example will illustrate our method, and should clarify almost everything
for those who have been uneasy with the stilted language of mathematics.
Suppose that next year an interconference football game is scheduled between
Wayne State University and Michigan State University. We shall find three
different "probabilities" that Wayne will wineach objective and with a
frequency interpretation, but of course posited on different states of knowledge
or different experiments.
Bettor A from Alabama knows nothing about Michigan teams, but he does know
that the Associated Press poll has picked Michigan State (by a margin not
specified), and that in several years this poll has consistently picked about
80% winners, give or take. For A, therefore, the probability that Wayne will
win is about 20% or 0.20 or 1/5, and refers to the following experiment:
pick the team chosen by the A.P. poll, and in the long run you will be right
about 80% of the time.
Bettor B is a visiting Bantu with no knowledge of American football or polls.
For him, the probability that Wayne will win is 1/2, and refers to the following
experiment: pick a team by some arbitrary system, perhaps the team wearing
a color you like better; in the long run, you will pick about 50% winners.
Bettor C is a coach who "rates" Michigan State four touchdowns better than
Wayne; he further knows that, over a period of years, only 5% of fourtouchdown
underdogs, so rated by him, have won. For him, then, the probability that
Wayne will win is 0.05 or 1/20, referred to the experiment indicated.
We see that by suitably examining experience one can find several probabilities
for the same event;
probabilities for the same event; it is of obvious importance to choose that
one which is based on the most appropriate experience (the most appropriate
sequence of experiments).
As another example we choose an extreme instance specifically cited by v.
Mises as outside the theory, and show that there is at least one way in which
it can be treated on exactly the same basis as any other event, viz., the
event "The Iliad and the Odyssey have the same author."
We might make a list of eminent literary historians, tabulate their performance
records with reference to disputes, as similar to this as possible, which
have finally been settled, and then canvass their opinions. Thus we might
arrive at a probability p for the event "the prevailing opinion is right,"
referred to the following experiment: when a question of literary history
arises, consult this group of experts and adopt the consensus opinion; in
the long run, you will be right 100 p percent of the time.
Note that, in examining experience to construct the Kollektiv (v. Mises'
word for the set of events or sequence or experiments), we must look for
instances as "similar as possible" to the one in question, and it might be
thought that often one would be unable to find enough sufficiently similar
instances. This is not the case. By considering ever broader classes of
experiments, one can always find a suitable sequence of recorded experiments
wbich are "identical" with respect to some sufficiently loose operational
criterion.
In practice, one will often be confronted with the following dilemma: whether
to use a sequence of experiments which are only roughly similar with respect
to some relatively strict criterion; or to use a sequence of experiments
which are very nearly identical, but with respect to a much looser criterion.
In the football example above, bettor C, the coach, might have gradually
changed his rating system over the years, so that the indicated recorded
sequence contains experiments by no means identical; while bettor B, the
Bantu, refers to an ideal sequence of indistinguishable experimentsand
yet the Kollektiv constructed by C is clearly superior.
Scientists tend to love precisely quantified information, and with good reason;
but sometimes they forget that the relevance of the information is more important
than its precision.
Note, finally, that there is no difference at all, in principle, between
constructing a Kollektiv for an historical hypothesis or, for example, for
a combination at cardseven though such great minds as that of von Mises
have been confused on this point. The difference is merely that in the latter
case (cards) it is more obvious which Kollektiv is most appropriate, and
it is one with minimal difference in the experiments; relevance and precision
here happily coincide, but in most problems of real life this is not the
case.
Deciding which body of experience is most appropriate is not difficult in
principle: that body of recorded experiments is most appropriate which bears
the most detailed resemblance to the experiment at hand. In the football
example above, the Coach's experience was obviously the most suitable (if
available) because it most nearly approximated a sequence of games
Waynevs.Michigan State1953. The Bantu's sequence, on the other hand, was
random teamvs.random team. The Bantu's probability number is much less
uncertain (indeed it is ideally precise), but also much less appropriate,
corresponding to a much less relevant body of knowledge.
Briefly to recapitulate thus far:
We define probability with reference to a v. Mises Kollektiv (except that
ours is always finite, while v. Mises' is infinite). To apply the definition
in a particular case, we must find a recorded sequence of experiments which
are sufficiently similar by some suitable criterion. This can be done in
many ways (corresponding to different bodies of knowledge), but usually one
can find a most appropriate way; in principle, one always can. The recorded
relative frequency of occurrence of the event in this sequence gives the
value of p, and no better value can be found without introducing new postulates.
(It should be carefully noticed that the rough, experimental value p is NOT
an approximation to some precisely defined but unknown "real" value of p.
There is no such "real" value, just as there are no infinite sequences of
experiments conducted under precisely defined conditionsleaving out of
account the question of physical statistics at the level of elementary particles.
The value of p is almost always, necessarily, and inherently vague to some
extent .... In estimating population means by sampling, it is true that a
population mean can have a well defined value in certain cases, namely cases
where the population is unchangingbut this is a rarity in the real world
and of very little interest.)
Thus we treat frequency probability and personal probability on a single
objective basis, such that we can assign a definite (although imprecise)
numerical probability to any event of whatever kind. But "whatever kind"
still needs some clarification, as does the question of the uncertainty in
the number p, and certain other old questions of doctrine.
Some Probability Notions Clarified
In this section, arguments are abridged even more drastically, and sometimes
only conclusions are stated. Those interested in the full discussion may
write the author.
A random experimentfollowing Cramer [3]is just one whose outcome cannot
be surely predicted. Hence randomness is a quality of the observer as well
as of the experiment.
Single experiments: Since we define a probability as a relative frequency,
what in meant by the probability of the outcome of an isolated experiment?
Von Mises says that although actuarial statistics may show that only about
1% of Americans die in their fortieth year, it is meaningless to speak about
the probability of a particular individual dying during the year. He will
live or he will die; he will not live 99% (at least according to precryonics
ideas). Yet it is obvious that if we select such a person at random, we have
reason for considerable confidence that he will live. In fact, in contrast
to that of v. Mises, our formulation above for the probability of an event
permits it to be unique in some respects, yet also demands that it be considered
part of a set.
Past Events & States of Nature. A great deal has been written about the
distinction between "random variables" and "states of nature," and whether
it is possible to make "true probability statements" about the latter. Mood
[9] and Neyman [10] for example claim a distinction between "random variables"
and "unknown constants" or states of nature. For example, past events (the
notion goes) cannot have probabilities; they either occurred or they did
not.
In my view there is no such proper distinction. The outcome of every random
experiment is equally a "random variable" in that it is not known in advance;
likewise, every such outcome is equally a "state of nature," in that (from
a deterministic viewpoint, on the classical physics level) it is unalterably
fixed in the structure of the world. To keep the discussion brief, I'll cite
just one example, which should be convincing.
Everyone agrees that cointossing is within the scope of the theory: the
future fall of a coin is a random variable. But what about a coin already
tossed but not yet inspected? A coin already tossed represents a past event
and a state of natureyet from the standpoint of the observer, there is
no difference whatever between a future toss and a past (but still unknown)
toss. The observer's bet will be the same in either case.
Similar remarks apply to sample means, point estimation, confidence intervals,
and to all questions of a priori versus a posteriori probabilities. All are
treated on the same basis.
The Principle of Insufficient Reason. Disagreements persist about
the "principle of indifference," "principle of cogent reason," "equal
distribution of ignorance," etc. My own view is one I have not seen expressed
by any single writer, although its elements are scattered through the literature.
The idea has much practical importance.
Thomas Bayes published his famous essay [6] in 1764, containing a theorem
about the probability
of an hypothesis as a function of the outcome of an experiment or test. One
element of the formula is the a priori probability of the hypothesis.
(After a bit we'll see what this means, with an example.) Many writers consider
this a priori probability to be, in general, impossible to ascertain
or even meaningless, hence useless, and they avoid this very powerful theorem.
Often those who shy away from a Bayes estimate of an unknown parameter will
use, instead, a "maximum likelihood" estimate due to R.A. Fisherbut it
turns out that this is the same as the estimate obtained by maximizing the
Bayes a posteriori probability, IF THE A PRIORI DISTRIBUTION IS UNIFORM,
and this is an implicit application of the "principle of insufficient
reason"i.e., assuming all outcomes equally likely, in the absence of any
information. In confidence interval estimation, also, since confidence intervals
usually have maximum likelihood estimates as their centers, anyone who uses
them is ACTING just AS THOUGH he believed in the principle of insufficient
reason.
Finally we note, with Jeffreys [7], that in practice one cannot avoid using
the principle of insufficient reason, in the following sense. Whenever one
seeks a probability, one looks for relevant information, and he bases his
estimate exclusively on this. But this is the same as saying that one begins
with the principle of insufficient reason, and then modifies his judgment
according to the information at hand!
This is more than a play on words. Consider a probability that is "well known"
on the basis of a very long series of observationsperhaps the probability
of heads on a coin toss, which we tried to show was based on total human
experience with giving random shoves to symmetrical bodies. In the strict
sense, this "well known probability" is merely a sample mean, and we should
estimate the "true" probability by use of Bayes' formula. But we cannot do
this, because there does not exist any larger background of experience in
terms of which we might define an a priori probability. We therefore
fall back on the principle of indifference in the form of a maximum likelihood
estimate, and this estimate equates the sample mean with the population mean,
fixing the probability (and leaving it slightly blurred).
In a manner of speaking, then, what we dobecause there is nothing else
to dois this: we assume that the portion of the world that we know is a
representative sample of the whole. The interpretation of this principle,
however, can be very tricky.
II
APPLICATIONS
In the period of more than forty five years since I did this work, it apparently
has remained customary for statistical studies to favor the "maximum likelihood"
approach, and many biologists and sociologists, for example, do not know
any other. This leads to many absurdities and even economic waste.
For example, consider the claim of Dr. Bobaloo, the shaman, who says his
chants can bring rain in the desertand suppose that, in a test observation,
rain does indeed follow the chant, although the U.S. weather service assessed
a very small chance of rain. How are we to judge the probability that the
medicine man can influence the weather?
Using the usual maximum likelihood approach, it might be calculated, say,
that the observed result had less than a 1% chance of occurrence on a random
basis. Hence, a statistician might say, the claim is "validated, at the 0.99
significance level"even though common sense tells us this is balderdash.
Common sense is savedand your intuition vindicatedby throwing out the
NeymanPearson approach with its "Type I and Type II errors" and appealing
to Bayes, even though the Bayes formula requires an estimate of the a priori
probability that the shaman is a rainmaker. It's true the this probability
is not accurately known, but we do know that it is very smallcertainly
much smaller than 0.01. (We know this because, in all the history of the
world, despite many claims, there is no proven case of chanting or dancing
affecting the weather.) Using some appropriately small number for the a priori
probabilityperhaps the number 0.000000000001the a posteriori probability,
even after the one positive observation, turns out very small, and we dismiss
the claim. (Actually, we wouldn't even have bothered watching the demonstration.)
... It is true, of course, that a long succession of successes would make
us take the shaman seriously, but one or two or a few will not.
Something similar happens in studies of "extrasensory perception," ESP.
If a "success" occurs that would have been unlikely on a random basis, the
ESP researcher may assert that the "faculty" (of telepathy, psychokinesis,
etc.) is established, at a "significance level" of 0.99, or 0.999, or 0.9999,
or whatever.
Again, the approach was inappropriate; a Bayes approach should have been
used, with some extremely small number for the a priori probability that
the faculty exists. (Again, in the whole history of the world, despite
innumerable citations, there is not a single proven instance of any such
faculty, as far as I have been able to determine.)
Actually, those who use the NeymanPearson or similar "significance level"
methods are only kidding themselves, because they are left with the problem
of deciding the appropriate leveland this decision, if correctly made,
is equivalent to making an estimate of the a priori probability. One might
as well do it explicitly, and get the advantage of the Bayes method.
In the world of commerce, it still appears common to use the NeymanPearson
approach, which I have proven to be inferior. In quality testing, for example,
a simpleminded "significance level" test after sampling might show quality
to be too low. But if the manufacturer has a good reputation, and there are
other favorable circumstances such as details of production design, a favorable
a priori estimate might yield a Bayes calculation showing satisfactory quality.
People tend to be afraid of a priori estimates when they are only rough guesses.
But a rough guess, properly used, is much more useful than an exact datum
inappropriately applied. (Think again about the football example: Bettor
B's number was the most precise, but Bettor C's was by far the most useful.)
A Suggested Estimate for the Exponential Life Parameter
In 1953 I applied these ideas to a practical technical and commercial problem,
that of estimating the mean.life of vacuum tubes by sampling in ordered
observations. While vacuum tubes have mostly gone down the tubes, the
"exponential life parameter" still has its uses.
The following little problem is not meant for the average reader, but is
intended to abash partlyeducated biologists. (The average reader is invited
back to the next section, however.)
The expressions that follow will look a bit unfamiliar and clumsy to
mathematicians. The main reason for using this unconventional notation is
to reduce downloading time. Any reader who wants to take the trouble can,
of course, convert to conventional notation by using the following equivalences:
h = theta; Int [f] = integral of f from 0 to infinity; sigma 1, r
[x_{i}] = sum from 1 to r of x_{i}. So:
If the underlying distribution is
(1)
f(x; h) = 1/h [exp (k/h)] (x > 0, h > 0)
then the maximum likelihood estimate for h based on the first r out of n
observations is [8]
(2)
HF = N/r, where N = {sigma 1, r [
x_{i}] + (n  r) x_{r}}/r
Although this estimate is unbiased, minimum variance, efficient, and sufficient,
there is an important sense in which it is not the best possible estimator.
The maximum likelihood estimate is that estimate which maximizes the probability
density for
(h; x_{1},…..,x_{r}) on the assumption that all values
of h are equally likely a priori. But this assumption is very unrealistic;
e.g. in tube testing it is certainly reasonable to assume that a moderate
value for the mean life is more likely than a very large or a very small
value, and the estimate should be weighted accordingly.
I shall therefore suggest a plausible a priori distribution, and a new estimate
based thereon, indicate some of its properties, and show in what way it is
"better" than the maximum likelihood estimate.
The following assumed a priori distribution for h is certainly a better
approximation to the "true" a priori distribution than is the uniform; it
also has a plausible look and contains room for adjustment depending on
experience in the industry, and it is simple to handle:
(3)
g(h) = [a/hb] exp[c/h]
(a > 0, c > 0, b > 1)
where b and c are location and scale parameters respectively and a is the
normalization factor.
Maximizing the probability density
(4)
f_{1} (h;
x_{1},….,x_{r}) = {f_{2
}[x_{1},...,x_{r};h] g(h)}/ { Int f_{2
}(x_{1},….,x_{r}; h) g(h) dh}
gives the new estimate
(5) H_{B = }{sigma 1,r [x_{i}] + (n  r) x_{r
}+ c}/(r + b) = (H_{F} + c/r) [r/(r + b)]
with a p.d.f. which can be shown to be
(6) h (H_{B}) = {[r + b]/[ r(r  1)!]}
(r/h)^{r} [H_{B}(r + b)/r  c/r]^{r  1} exp
{r/h[H_{B}(r + b)/r  c/r]
The mean of this estimate turns out to be
(7)
E(H_{B}) = [r/(r + b)] [h + c/r]
so that while H_{B }is not unbiased in general for small samples,
it becomes unbiased for large samples; we see also that when h = c/b,
H_{B} is unbiased for samples of any size, and h = c/b maximizes
g(h).
The variance of H_{B} is easily shown to be
(8)
Var H_{B} = [r^{2}/(r + b)^{2}]
[h^{2}/r] = [r^{2}/(r + b)^{2}] Var H_{F}
which is smaller than the variance of H_{F}!
The important feature of H_{B} is that it does not throw away the
information that moderate values of h are more likely a priori than extreme
values.
Let us review once more what the above means:
If tubes are produced in batches, and mean life varies from batch to batch,
and a steady buyer wants to estimate as well as possible on the average,
then the Modernized Bayes approach is indicated; a less satisfactory substitute
would be a confidence interval approach using maximum likelihood estimation.
If however some important decision, such as the letting of a contract or
installation of military equipment, rests (for some reason) on a single sampling,
then the situation is essentially different. Here the required probability
must apply not to overall repeated sampling, but just to that subsequence
of samples yielding exactly those observations actually obtained! Hence the
Classical Bayes approach is required, and any other is a poor substitute.
In such a case one must not be deterred by the fact that the a priori
distribution of is known only roughly; one must not throw away important
information merely because that information is less exact than one might
wish!
Aspects of Probability Concerning Cryonics
We have said that it is possible to calculate a probability (indeed, usually
many probabilities, corresponding to different states of knowledge) for any
event of whatever kind. But we have also pointed out that observed probabilities
(those taken directly from the most appropriate experience) always have some
uncertaintysometimes so much that the number has little value; and subsidiary
or derived probabilitiesthose calculated from the observed probabilities
by the rules of combinationhave even more uncertainty. Sometimes the end
result is simply p = 0.5 ± 0.5, which is the equivalent of a shrug.
But it must be emphasized, first, that an unknown probability (one which
has not yet been investigated), or a highly uncertain one, is not the same
as a small probability. Saying the odds are unknown is NOT the same as saying
the odds are adverse.
But we are not admitting even that the odds are unknown. Stick around.
We are talking, of course, about the chance of eventual rescue of patients
frozen today, or in the near future, by relatively crude methods.
There are several aspects to the estimate. First, we disregard the chance
of global nuclear war or other worldwide catastrophe, which might destroy
both the dead and the living: in most respects we do not plan our individual
lives with this in mind, after all. (But the Cryonics Institute, for example,
does make plans intended to minimize risk in limited nuclear war.)
Second, we disregard the business risks of any particular organization (although,
again, the Cryonics Institute for example strives for extreme prudence, with
never any debt. ) In almost any industry, some will dive and some will thrive.
Third, we disregard the risks of politics, assuming that in the U.S.A. at
least there will persist a sufficient degree of personal freedom to allow
our movement to exist.
In all three of the aforesaid risk areas, it should be again noted, we are
dealing not with fixed probabilities, but with conditions in flux and subject
to feedbacks: we are dealing with MOTIVATED BEHAVIOR, and in this area no
one has yet developed adequate statistical tools. In other words, when things
start to look bad, usually somebody does something about it, and pretty soon
things look better again.
The remaining area of interest is the scientific chance of rescuethe chance
of development of skills sufficient to reverse freezing damage and senile
debility as well as other deficits in the frozen patient, allowing restoration
to youthful good health with memory and personality largely intact.
To persuade a lay reader, gradually, that (s)he has or can acquire the competence
to assess questions of technical feasibilitythis is not easy, but I think
not impossible. Remember the CommanderinChief and the CROs. It may help
to remind ourselves of the astonishing feats we accomplish every day.
Intuition and Probability
While intuition is certainly fallible, it is also educable, and in many areas
all of us routinely rely on it for lifeanddeath estimates.
Think about crossing the street. Small children, and dogs and cats, may not
think about it, or forget abut it, and die. But most of us have learned to
gauge our chances in traffic very quickly and very efficientlyand we don't
need the slightest knowledge of mathematics in any explicit or conscious
sense.
In a sense, gauging traffic is an awesome calculation. We have to estimate
the speeds and trajectories sometimes of several vehicles, as well as our
own speed and agility: a computer directing antitank fire and evasive maneuvers
in a similar situation has only been available in recent yearsand in some
respects the human brain (even in unconscious behavior!) is still far ahead
of the computer's capabilities.
Some types of mental analog computation involve much more abstract thought
than the computation of traffic patterns. Think for a moment about the prediction
by Goddard and Tsiolkovsky, in the early part of this century, that there
would be moon rockets. Why did these rocket experts know there would be moon
rockets, while most other expertseven at a much later datedoubted or
denied it?
The answer is that they looked at the central issue and at the sweep of history,
ignoring some of the troublesome details, including the expense.
It is highschool simple (once a Goddard has shown you how) to prove
mathematically that a rocket can reach the moon: all you need is enough explosive
and a way of controlled release. But in the early part of the century many
ingredients of practical importance were missinghigh quality refractories
and electronic instrumentation, for example, as well as the economic resources.
(It took a national effort by the world's wealthiest country to accomplish
it.) But anyone with a a ense of history had to know that these details were
only a matter of time and determination. There was always a question about
the political feasibility of a moon rocket; there was never a question of
its scient.ific or engineering feasibility.
In assuming that the petty details would in time be worked out, Goddard and
Tsiolkovsky relied on intuition; they did not make explicit calculations
of probability. But their view of history entailed an implicit calculation
whichhowever roughwas relevant. They had a better grip on reality than
the fusspot experts whose aberrant intuition focused on little, temporary
obstacles of no longterm importance.
For another example of educated intuition, consider Leonardo da Vinci,
whose "inventions" included the airplane, in a manner of speaking.
Leonardo did not really design a working flying machine, of course. He only
dabbled in concepts, and in any case the materials and technologies were
not available to implement t hem. But he saw to the center of the problem.
Could a flying machine be built? Of course, since flying machines (birds
for example) already existed; the air obviously provides support to a suitably
designed frame, and the rest is detail. Given time, Leonardo would have worked
out the details too, including innovations in materials and power supply.
Most experts' failure of intuition occurs because they are unconsciously
tied to a short time frame: if we can't do it soon (in the framework of a
professional career), we can't do it. But frozen patients have a longer time
frame; they can wait.
Unlimited WealthNot "Probable" but Certain
Many people admit the goals of cryonics are possible of achievement, but
they put a low estimate on the probability or feasibilityoften on the basis
of economics.
John W. Campbell, late editor of Astounding science fiction magazine, said
he could not conceive of repair after rupture by freezing of every single
cell in the body. (This does not occur, but his ignorance is not the point
here.) He didn't think cell repair physically impossible; he just considered
the repair job too monumentally difficulti.e., expensive. But the fact is
that (in the indefinite future) expense is no object. Factnot conjecture.
Eric Drexler, Conrad Schneiker, and many others have given rather detailed
arguments to show that automated repair mechanisms will become available
at some datemachines that are selfreplicating, selfimproving, and
"intelligent" to any necessary degree. This implies not only the physical
capacity for microscopic repair, but also the economic capacity, since such
machines will represent unlimited wealth. (Raw materials abound; it is
organization of matter and energy that is critical.)
But unlimited wealth does not depend on the invention of any particular new
devices. Its basis already exists! Specifically, the exponential growth of
wealth is already with us; some people will recognize it under the name of
compound interest.
Obscuring the fact, unfortunately, is the fog of destructive tendencies in
human society, which can hide or even destroy any amount of productivity.
Wars, heedless breeding, and assorted insanities have kept the standard of
living orders of magnitude below what technology has made possible. But the
exponential growth machine already exists.
(The existence of the money machine is obscured not only by outright ignorance
and recklessness, but also by milder forms of imprudencein capitalist
societies, for example, by mindless frivolity. There is staggering waste
in the constant churn of fashion and proliferation of fads and choices: do
we really need a new model auto every year, let alone dozens of them? But
this will have to be cured by education and maturation; the socialist planned
economy is worse.)
We NOW have the technology and natural resources (even on a worldwide basis)
to give everyone a decent living (with family planning), and a good deal
left over for investment. Much of this investment should take the form of
research and development. This alone would GUARANTEE an exponential upward
spiral, wealth growing without bound. Again, the underlying assumption is
that political disasters will not ruin everything; we are concerned with
the scientific probability of success. In the U.S., real income per capita
does grow every year, albeit unevenly. The compound interest djinn exists
even without robots. Any amount of money will eventually be available for
the repair, revival, rejuvenation, and rehabilitation of the frozen patients.
This money growth will take two forms: first, the dollar growth in the trust
funds or organizational funds of the patients through compound interest over
the years (assuming that organizations such as the Cryonics Institute are
successful in keeping the ravages of occasional inflation at bay); second,
and more important, the exponential growth of productivity will make everything
cheaper relative to income. (How many people could afford an air conditioner
in 1936?)
This is without counting on selfreplicating machines. With such machines,
the curve of exponential growth becomes nearly vertical.
Summing Up
"Probability" Derives from "Experience"
The main thrust of our argument has been that, in attempting to estimate
any probability, one must abstract or summarize as much as possible of the
most relevant experience. This is scientific, not the myopic obsession with
precise but irrelevant data one sees so often.
For a summary of experience bearing on cryonics, please review the six pages
following (reprinted in many issues of The Immortalist). (On the Web site,
go to Contents, then Principles of Experience.)
We can recapitulate in slightly different terms:
1. In the modern era, not a single goal of science, so far as I know, has
been shown impossible (although some have proven more difficult than expected,
and others become irrelevant). Oddson for success.
2. Many cells survive even uncontrolled freezing, and there have been partial
successes with freezing mammalian brains: oddson that, even in freezedamaged
brains, injury is limited and reparable.
3. The Precedent Principle, the Feinberg Principle, and the prospect of
nanotechnology assure us that atombyatom manipulation of tissue (frozen
or not) will allow construction or reconstruction, in finest detail, of any
human configuration known, designable, or capable of inference. (For a discussion
of prospective molecular repair technology, see the Drexler citation on the
last page.) Oddson that you can be restored.
In still other words, from the whole sweep of history, and our best understanding
of the way the world works, we conclude the "gamble" of cryonics is oddson
in your favor: the probability of success (from the standpoint of technical
feasibility) is much closer to one (certainty) than to zero. The number may
be imprecise, but it is the best and most scientific estimate available.
Congratulations! You have just won a ticket to forever, transfers included.
P.S. If "forever" is a little too long, focus instead on just the next century
or two, then reconsider.
ADDENDUM
Goals of TechnologyThe Record
This is added to clarify and expand the application of probability to
the problem of repair of cryoinjured patients.
To estimate a probability, we need a recorded sequence of experiments "similar"
to the one at hand. By broadening our definition of "similarity" as much
as necessary, we can always do this. The broader the criteria of similarity,
the less precise the estimate will be, to be surebut even very broad criteria
can still yield useful information. (See the football example previous.)
Now we want to estimate the probability of success in repairing and reviving
frozen cryonics patients. Nothing very similar has ever been attempted, so
we loosen criteria until we get to "attempts to achieve difficult new
technology." There have been many such, especially in the last few centuries.
Can we compile statistics on how many have succeeded and how many have failed
and how many are still open?
My own sense of history tells me this explicit calculation is unnecessary,
but we can at least look at a few indicators.
First, let's try to think of attempts which have failed, using reasonable
definitions. An "attempt" means a serious effort of competent people in their
area of competence. "Attempt" is also defined with reference to ends, not
means. "Failed" means abandoned by all serious people.
One place to start might be the patent office record of patents that were
refused because the examiners were convinced the gadget didn't work or could
never work.
Gadgets that could never work? A prime and recurrent example might be a
"perpetual motion machine"something that violates the first or second law
of thermodynamics, yielding "free" energy. But gadgets like this are NOT
submitted by serious, competent people; they don't count.
Gadgets that don't work as submitted? Ornithopters might be an examplemachines
that fly by flapping their wings. The ones actually built either didn't work
at all, or else worked only for a very short time and then crashed. But these
don't count either, for two reasons.
First, a practical flappingwing machine may yet be built. Birds and bats
and insects and (formerly) pterosaurs fly by flapping their wings; and with
future materials and power plants and stabilizing systems, larger machines
may also.
Second, we must look at the ends and not just the means. Before there were
any heavierthanair flying machines, there were several possibilities. These
included ornithopters, planes with airscrews, jets, and rockets. All were
once considered impossible or forever impractical, but the last three have
been realized. The endflyinghas been achieved. Only one of the
meansornithoptershas not been successful, and that one still may be one
day.
Hold onwhat about Lysenko? The Russian biologistwho claimed acquired
traits could be inheritedwas regarded as "serious" only in the Soviet Union,
but that was a large constituency. Shouldn't that count as a failed goal
of technology? Well, aside from the "iffy" characterization of him as serious,
at worst only the means failed, not the ends. The ends were to breed better
plants, and that is being done apaceeven if Lysenko's methods have been
discarded and disredited.
Maybe I haven't tried hard enough, but I have not thought of a single example
of a serious goal of technology that has failed and remains without serious
advocates.
On the other side, what about goals of technology that were once thought
by most people to be impossible or forever impractical, but that in fact
were achieved? or that were previously not even IMAGINED, and yet were
achieved? They are many and notorious, including:
Abrasives. Air conditioning and twoway heat pump. Air to air missiles. Algebras.
Alloys. Alphabet. Anaesthesia. Analgesics. Analog computers. Anatomies.
Antibiotics. Antidepressants. Armies. Artificial insemination. Asepsis in
surgery. Assembly line. Auger. Automated factories. Automatic gene sequencing.
Automobiles. Adze. Axe.
Babies from frozen embryos. Baking soda, baking powder. Ball bearings. Banking.
Bellows. Blacksmithing. Blood and urine analysis. Books. Bow and arrow. Brazing.
Bricks. Bridles & saddles. Bubble level. Butter. Buttons.
Calendar. Camera. Canals. Catalysts. Cathode ray tube. Central heating.
Centrifuge. Chain. Cheeses. Chess program that can defeat a chess champion.
Chunnel. Cities. Cloning. Clothing. Coders and decoders. Codes of law. Cogwheels.
Comb. Compass for drawing. Condoms. Contact lenses. Coolidge tube. Corporations.
Cosmetics. Cotton gin. Counting. Cryogenics. Cryopreservation of blood and
other tissue. Cryosurgery. CT and NMR scanners.
Dental amalgams. Dental floss. Dentist's drill. Deodorants. Depilatories.
Desalinization plants. Dietetics. Digital computers. Digital disk recorder.
Directories. Dirigibles. Distillation.Drawing. Drugs. Dynamite.
Earth satellites. Electric generator. Electric lights. Electric motor. Electric
shaver. Electrophoresis. Elevators. Endoscopy. Epoxy. Eyeglasses.
Facial tissues. Fake fat. Fake sugar. False teeth. Farming. Fax. Fertilizers.
Fiberoptics. File (paper). File (tool). Fingerprints. Fire making. Fishhook.
Fletching. Flint chipping. Flying machine. Flypaper. Forceps. Fork lift.
Bulldozer. Forks. Freeze drying. Furnace.
Genetic engineering. Genome mapping. Geometries. Glues & adhesives.
Goldsmithing. Growth factors. Gyroscopes.
Hair care. Hammer. Horse shoes. Host mothers. Hotels.
Insecticides. Insulin. Intercontinental missiles. Interferons. Internal
combustion engine. Internet databases. Invitro fertilization.
Jet propulsion.
Kevlar. Knife. Knitting. Knots.
Laminates. Language. Laparoscopy. Lathe. Lawnmowers. Laxatives. Letters of
credit. Logics. Lubricants. Lying.
Mace. Magnetic tape recorder. Magnets. Manuals of operation. Maps. Masers
& lasers & holograms. Metallurgy Microscope. Microtome. Mining. Mirror.
Money. Monoclonal antibodies. Moon rocket. Movies, including animation, special
effects, and 3D. Musical instruments.
Nails. Nanotechnology, just beginning. Nations. Navigation and location
by satellite.Navigation systems. Needle & thread. Newspapers. Nuclear
energy. Nuclear fusion, still in the works.
Oars & paddles. OCR. Offshore drilling rigs. Organ transplants. Oscilloscope.
Paint. Painting. Paper. Paving machines. Paving. Pendulum. Phonograph. Piping
& tubing. Planing tool. Planting machines. Plastics. Pliers. Pneumatics.
Pockets. Poetry. Polygraph. Potter's wheel. Pressure pump. Prostheses for
limbs. Protein sequencing. Pulley.
Quantum computers, just beginning. Quantum chemistry.
Radar. Radiation therapy and chemotherapy. Radio telescopes. Radio. Railroads.
Refractories. Rivets. Roads. Roller bearings. Rope. Ruler.
Safety razor. Sails. Sandpaper. Saw. Scanning tunneling microscope, atomic
force microscope, etc. Schools. Scintillation counters. Scissors. Screw machine.
Screws. Sculpture. Scythe. Seismic prospecting. Semaphore. Sewers.
Sextant. Shoes. Sickle. Skin grafts. Skyscrapers. Sledges. Smart bombs. Smelting.
Soap. Soldering. Solvents. Sonar. Sonograms. Soup. Spectroscope. Spinning
and weaving machines. Spoons. Stapling machines. Steam power. Stove. Stylus.
Submarines. Surveying. Sutures. Sword. Synthetic hormones. Syringe.
Tables of organization. Tattoos. Telegraph. Telephone. Telescope. Television.
Tempering steel. Thermometer. Tilling machines. Tissue typing. Toilet. Toilet
paper. Tongs. Tooth brush. Tooth paste. Transatlantic cable. Transgenesis.
Transistors. Transit. Traps & snares. Travois. Triodes. Turbines.
Typewriters.
Use of radioisotopes.
Vaccines. Vacuum pump. Virtual reality. Vise. Vitamins.
Wallpaper. Watches and clocks. Water pump. Welding. Welldigging. Wheel.
Wrenches. Writing.
Xray diffraction techniques. Xrays.
Yurt making.
Ziploc. Zipper.
If you have a suggestion to add to this list, please email us.
<cryonics@cryonics.org>
Sure, some of the above are arguably minor or redundant. Still, I think they
were all reasonably important in their time, and not obvious or easy ahead
of time. But a great many have doubtless been omitted. The important thing
is that there have been very many successful projects of technology, and
very few or none that have failed.
If you can think of a failure, that qualifies according to our criteria,
please let us know.
REFERENCES
1. v. Mises, Richard: Notes on the Mathematical Theory of Probability
and Statistics, Harvard U. Press, 1946.
2. v. Mises, Richard: "On the Foundations of Probability and Statistics,"
Annals of Math. Stat. 12 (1941), p. 191.
3. Cramer, Harald: Mathematical Models of Statistics, Princeton U.
Press, 1951.
4. Koopman, B.O.: "The Axioms and Algebra of Intuitive Probability," Annals
of Mathematics 41 (1940), p. 269.
5. Kendall, M.G.: "On the Reconciliation of Theories of Probability,"
Biometrika 36 (1949), p.101.
6. Bayes, Thomas: "An Essay toward Solving a Problem in the Doctrine of Chances,"
Philosophical Transactions, 1764.
7. Jeffreys, H.: Theory of Probability, Clarendon Press, Oxford, 1939.
8. Epstein, Benjamin and Sobel, Milton: "Some Tests Based on the First r
Ordered Observations Drawn from an Exponential Distribution," Wayne State
U. Technical Report No. 1, 1952.
9. Mood, Alexander McF.: Introduction to the Theory of Statistics,
McGrawHill, 1950.
10. Neyman, Jerzy: Lectures & Conferences on Mathematical
Statistics and Probability, U.S. Dept. of Agriculture, 1952.
11. Doob, J.L.: "Probability as Measure," Ann. Math. Stat. 12
(1941).
