Effects of neutral selection on the evolution of molecular species

Abstract
We introduce a new model of evolution on a fitness landscape possessing a tunable degree of neutrality. The model allows us to study the general properties of molecular species undergoing neutral evolution. We find that a number of phenomena seen in RNA sequence-structure maps are present also in our general model. Examples are the occurrence of "common" structures which occupy a fraction of the genotype space which tends to unity as the length of the genotype increases, and the formation of percolating neutral networks which cover the genotype space in such a way that a member of such a network can be found within a small radius of any point in the space. We also describe a number of new phenomena which appear to be general properties of neutrally evolving systems. In particular, we show that the maximum fitness attained during the adaptive walk of a population evolving on such a fitness landscape increases with increasing degree of neutrality, and is directly related to the fitness of the most fit percolating network.

arXiv:adap-org/9712005v1 15 Dec 1997
Eﬀects of neutral selection on
the evolution of molecular species
M. E. J. Newman
Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501. U. S.A.
Robin Engelhardt
Center for Chaos And Turbulence Studies, Dept. of Chemistry,
University of Copenhagen, Universitetsparken 5, Copenhagen Ø, Denmark
Abstract
We introduce a new model of evolution on a ﬁtness landscape po ssessing a tunable
degree of neutrality. The model allows us to study the genera l properties of molecular
species undergoing neutral evolution. We ﬁnd that a number o f phenomena seen in
RNA sequence-structure maps are present also in our general model. Examples
are the occurrence of “common” structures which occupy a fra ction of the genotype
space which tends to unity as the length of the genotype incre ases, and the formation
of percolating neutral networks which cover the genotype sp ace in such a way that
a member of such a network can be found within a small radius of any point in the
space. We also describe a number of new phenomena which appea r to be general
properties of neutrally evolving systems. In particular, w e show that the maximum
ﬁtness attained during the adaptive walk of a population evo lving on such a ﬁtness
landscape increases with increasing degree of neutrality, and is directly related to
the ﬁtness of the most ﬁt percolating network.
1 Introduction
Biological molecules such as proteins and RNAs undergo evolution jus t as
organisms do, selected for their ability to perform certain function s by the
reproductive success which that ability imparts on their hosts. It is believed
that many mutations of a molecule are evolutionarily neutral in the se nse
that they do not change the ﬁtness of the molecule to perform the function
for which it has been selected. We have many examples of proteins wh ich
Preprint submitted to Proc. R. Soc. London B 20 November 2018

appear to possess approximately the same conformation and to pe rform the
same function in diﬀerent species, but which have diﬀerent sequenc es. Such
proteins may diﬀer only by a single amino acid or may have whole regions
which have been substituted or inserted, or they may even be so diﬀ erent as
to appear completely unrelated. A mutation is said to be neutral if it c hanges
a molecule into one of these functional equivalents, leaving the viabilit y of its
host unchanged. This idea was ﬁrst explored in detail by Kimura [1,2].
In fact, as Ohta has pointed out [3], it is not necessary that the ﬁtn ess of a
molecule remain precisely the same under a given mutation for that mu tation
to be considered neutral. A mutation which produces a change in ﬁtn ess will
cause the population sizes of the original and mutant strains to dive rge expo-
nentially from one another over time. However, the time constant o f this ex-
ponential is inversely proportional to the change in ﬁtness. Thus if the change
in ﬁtness is small, its eﬀects will be felt only on very long time-scales. If these
time-scales are signiﬁcantly longer than the time-scale on which muta tions
occur (the inverse of the mutation rate), then the change in ﬁtne ss will never
be felt. In eﬀect, the mutation rate places a limit on the resolution wit h which
selection can detect changes of ﬁtness, so that small ﬁtness cha nges are eﬀec-
tively, if not precisely, neutral.
It is possible that the concept of neutral selection can also be applie d to
the evolution of entire organisms. Certainly there are changes pos sible in an
organism’s genome which have no immediate eﬀect on its reproductive success,
or which produce an eﬀect suﬃciently small that selection cannot de tect it on
the available time-scales. In this paper we will primarily use the languag e
of molecular evolution, but the reader should bear in mind that the ide as
described may have wider applicability.
Despite the long history of the idea, many aspects of neutral evolu tion are
still not well understood. In particular, we have very little idea of th e general
behaviours that can be expected of systems (molecules or organis ms) with a
signiﬁcant degree of evolutionary neutrality. The primary reason f or this gap
in our understanding is that, despite many decades of hard work, w e still have
a rather poor idea of the way in which genomic sequences map onto mo lecular
structures and hence onto a ﬁtness measure. In the case of ent ire organisms
the equivalent problem is that of calculating the genotype-phenoty pe map-
ping, which is even less well understood. One simple case in which neutr al
evolution has been investigated in some detail is that of RNA structu re [4–7],
although calculations so far are limited to secondary structures, a nd even these
cannot be calculated with any reliability, so that these studies should be taken
more as a qualitative guide to the behaviour of systems undergoing n eutral

selection than an accurate representation of the real world. The trouble with
this approach however is that RNAs are not a suﬃciently general mo del that
the results gained from their study can be applied to other systems , such as
protein evolution or the evolution of whole organisms.
At the other extreme, studies have been performed of extremely simple mathe-
matical models of neutral evolution in the context of genetic algorit hms [8,9].
An example is the “Royal Road” genetic algorithm studied by van Nimwe -
gen et al. [10,11]. These models possess highly artiﬁcial ﬁtness functions cho-
sen speciﬁcally to show a high degree of neutrality whilst at the same t ime
being simple enough to yield to analytic methods. Like RNA secondary s truc-
tures, these models have given us some insight into the type of eﬀec ts we may
expect neutral evolution to produce, but, like RNAs, they are not suﬃciently
general to be sure that these insights apply to other systems as w ell.
In this paper, therefore, we propose a new mathematical model o f neutral
evolution. This model is an abstract model of a genotype to ﬁtness map in the
spirit of the Royal Road model. This approach allows us to sidestep th e prob-
lems of incorporating the chemistry of real molecules in our calculatio ns and
to investigate the properties of the system more quickly and in grea ter detail
than is possible with, for example, RNA structure calculations. In ad dition,
the model is more general than either the Royal Road ﬁtness func tion or the
RNA sequence-structure maps of Refs. [5,6]. In fact, it possesse s regimes in
which it mimics the behaviour of both of these systems, as well as pro tein- and
organism-like regimes. Because the behaviours of our model cover such a wide
range of possibilities, it seems reasonable to conjecture that gene ric features
of the model which span all of these regimes may be common to most s ystems
undergoing neutral selection. This is the power of our model, and th ese general
results are the results that we will concentrate on in this paper; we believe
that the generic behaviours of our model should be visible in the evolu tion of
real systems such as proteins which are, as yet, beyond our abilitie s to study
directly.
In Section 2 we introduce our landscape model of neutral evolution . In Sec-
tion 3 we discuss its properties and compare these with previous res ults for
other systems undergoing neutral evolution. In Section 4 we discu ss the im-
plications of our results for evolving molecular species. In Section 5 w e give
our conclusions.

2 The model
Selective neutrality arises as a result of the many-to-one nature o f the sequence-
structure or genotype-phenotype maps found in biological syste ms. Many pro-
tein sequences, for example, map onto the same tertiary structu re, and since
the ﬁtness is primarily a function of the structure, such sequence s possess (at
least approximately) the same ﬁtness. We wish to construct a mode l of this
phenomenon without resorting to actual calculations of the struc ture of any
particular class of molecules. Only in this way can we hope to create a m odel
which is general enough to represent the behaviours of many diﬀer ent such
classes. Our approach is to employ a “ﬁtness landscape” model of t he type
ﬁrst proposed by Wright [12,13] which maps sequence (or genotype ) directly
to ﬁtness. Structures (or phenotypes) appear in our model as c ontiguous sets
or “neutral networks” of sequences possessing the same ﬁtnes s.
Our model is a generalization of the N K model proposed by Kauﬀman [14,15],
which is itself a generalization of the spin glass models of statistical ph ysics [16].
Consider a sequence of N loci, which correspond to the nucleotides in an RNA
or to amino acids in the case of a protein. At each locus i we have a value
xi drawn from an appropriate alphabet, such as {A,C,G,U} for RNAs, or the
set of 20 amino acids in the case of proteins. We denote the size of th e al-
phabet by A. Each locus interacts with a number K of other “neighbour”
loci, which may be chosen at random or in any other way we wish. (Kauﬀ man
refers to these interactions as epistatic interactions, though th is nomenclature
is strictly only appropriate to the case where we are modelling the ﬁtn ess of
whole organisms.) In the case of RNAs, bases most often interact w ith one
other base to form either a Watson-Crick or a G-U pair. Some bases have
both pairing and tertiary interactions. Some, in the single stranded regions,
have very little interaction with any others. Thus a value of K = 1 might
be approximately correct for RNA. 1 For proteins, which have more complex
types of interactions, a higher value of K may be appropriate.
Each locus i makes a contribution wi to the ﬁtness of the sequence, whose
magnitude depends on the value xi at that locus and also on the values at
each of the K neighbouring loci. There are AK+1 possible sets of values for
the K + 1 loci in this neighbourhood, and hence AK+1 possible values of wi.
1 It is possible to generalize the model to allow diﬀerent loci t o interact with dif-
ferent numbers of neighbours, which gives a behaviour more r epresentative of true
RNAs and proteins. In this paper however, we will conﬁne ours elves to the case of
constant K for simplicity. As we will see, even with this constraint the model is still
able to duplicate the behaviours seen in real systems.

Following Kauﬀman and Johnsen [14] we choose this set of values at ra ndom.
However, Kauﬀman and Johnsen chose the values to be random rea l numbers
in the interval 0 ≤ wi < 1. We by contrast choose them to be integers in
the range 0 ≤ wi < F . Thus if F = 2 for example, each contribution wi is
either zero or one. Now we deﬁne the ﬁtness W of the entire sequence to be
proportional to the sum of the contributions at each locus:
W = 1
N(F − 1)
∑
i
wi. (1)
The ﬁtness of all sequences thus falls in the range from zero to one , and there
are N F − N + 1 possible ﬁtness values in this range.
In the limit in which F → ∞ the probability that two sequences will possess
the same ﬁtness becomes vanishingly small, and our model therefor e possesses
no neutrality and is in fact exactly equivalent to the N K model. However,
when F is ﬁnite the probability of two sequences possessing the same ﬁtnes s
is ﬁnite, so that the model possesses neutrality to a degree which in creases
as F decreases. Neutrality is greatest when F takes the smallest possible
value of two. Two sequences with the same ﬁtness may be equivalent either
to molecules which fold into the same conformation and perform the s ame
function, or they may be equivalent to molecules with diﬀerent confo rmations
but approximately the same contribution to the reproductive succ ess of the
host organism. The ruggedness of the landscape is controlled by th e parameter
K, and is largest when K takes the maximal value of N − 1 [14,17]. In the
next section, we investigate the properties of the landscapes gen erated by our
model, and show that with the right choice of parameters they can b e used to
mimic real biological systems, such as RNAs.
3 Evolution on neutral landscapes
The topology of a ﬁtness landscape depends on the types of mutat ion al-
lowed to molecules evolving on it. In biological evolution, point mutation s—
mutations of the value at a single locus—are the most common. In this case,
a neutral network is deﬁned to be a set of sequences which all poss ess the
same ﬁtness and which are connected together via such point muta tions. In
the molecular case, we assume that closely similar sequences have th e same
ﬁtness because they fold into the same conformation, so that the se neutral
networks correspond to (tertiary) structures. In the organis mal case they cor-

respond to phenotypes. 2
The model described in the last section possesses neutral networ ks of exactly
this type. The total ﬁtness W in the model ranges from zero to one, but the
greatest number of sequences have ﬁtness close to W = 0 .5. (In the extreme
case where K = N − 1 the distribution of W is binomial. When K < N − 1 it is
approximately but not exactly so. Examples of these distributions a re shown
in Figure 1 (a).) We would therefore expect the largest neutral net works to be
those with ﬁtness close to W = 0.5 and this is indeed what we ﬁnd in practice.
Typically there are a large number of small neutral networks and a s mall
number of large ones. In Figure 1 we show histograms of the sizes of the
neutral networks for N = 20 and various values of K. For the RNA-like case
K = 1, the histogram appears to be convex, indicating a distribution wh ich
falls oﬀ faster than a power law. The same behaviour has been in seen in RNA
studies by Gr¨ uneret al. [6]. As K increases the distribution ﬂattens, and by
the time we reach K = 5 it is markedly concave. Thus the behaviour seen in
RNAs is not in this case generic. For some intermediate value of K close to
K = 2, the distribution appears to be power-law in form, perhaps indica ting
the divergence of some scale parameter governing the distribution , in a manner
familiar from the study of critical phenomena [18].
We ﬁnd that the total number of neutral networks SN grows exponentially as
aN with sequence length. In Figure 1 (b) we show the number of networ ks in
our model for K = 1, both for two-letter {G,C} alphabets, and for a four-
letter {A,C,G,U} alphabet. We ﬁnd that a ≈ 1.5 for the A = 2 case and
a ≈ 2.3 for the A = 4 case. Interestingly, Stadler and co-workers [6,7,20] have
performed the same calculations for RNA sequences using the full s econdary-
structure calculation, and they also ﬁnd an exponential increase in the number
of structures with sequence length with values of a = 1.6 and a = 2.35 for the
two- and four-letter cases respectively. This suggests to us tha t this behaviour
is more general than the speciﬁc secondary-structure map emplo yed in the
Stadler calculations.
The largest neutral networks on our landscapes percolate, which is to say, they
ﬁll the sequence space roughly uniformly, in such a way that no sequ ence is
2 One might argue that the individual ﬁtness levels should cor respond to structures,
not the neutral networks. However, there are presumably man y structures which
are not similar enough to be easily accessible from one anoth er by point mutations,
and yet which possess similar ﬁtnesses, at least to the mutat ion-limited accuracy
with which selection can distinguish. Thus it seems more app ropriate to draw a
correspondence between networks and structures in the case of this model.

1 10 100 1000 10000 100000
size of network
−4
−3
−2
−1
frequency of occurrence
5 10 15
fitness W
frequency
5 10 15 20
length N
neutral networks
(a)
(b)
Fig. 1. Histogram of the frequency of occurrence of neutral n etworks as a function
of size for a particular realization of a landscape with N = 20, A = 2, F = 2, and
(circles) K = 1, (squares) K = 2, (diamonds) K = 5. Inset (a): the frequency of
occurrence of sequences as a function of ﬁtness for N = 20, A = 2, F = 2, and
(solid line) K = 19, (dotted line) K = 0. Inset (b): the number of neutral networks
as a function of N for K = 1, F = 2, and (circles) A = 2, (squares) A = 4.
more than a certain distance away from a member of the percolating network.
Determining which networks are percolating is not an easy task. We h ave
developed two diﬀerent measures to help identify percolating netwo rks, and
we describe them further in Ref. [19]. Gr¨ uner et al. [6] introduced instead the
idea of a “common” network, which is one which contains greater tha n the
average number of sequences. We can employ this deﬁnition with our model
too. We ﬁnd that the common networks in the model form a small fra ction
of the total number of networks, that fraction decreasing expo nentially as
N increases, as shown in the inset to Figure 2. The same result is found in
RNAs [6].
In the main frame of Figure 2 we show the fraction of sequences whic h fall in
the common networks as a function of N. As the ﬁgure shows, our numerical
studies indicate that this fraction increases with sequence length, tending to
one in the limit of large N. Even though the common networks form a smaller
and smaller fraction of all networks as N becomes large, they nonetheless cover
more and more of the sequence space. These results have interes ting evolu-

8 10 12 14 16 18 20
length of sequence N
0.6
0.7
0.8
0.9
1.0
fraction of sequences in common networks
8 12 16 20
length N
−2
−1
common networks
Fig. 2. The fraction of all sequences which fall in common net works as a function of
N for a particular realization of the model with A = 2, F = 2, and (circles) K = 2,
(squares) K = 4, (diamonds) K = 6. Inset: the number of common networks as a
fraction of the total number of networks for the same landsca pes.
tionary implications: they imply that as sequences become longer, a la rger
and larger majority of structures (the small networks) are vanis hingly un-
likely to occur through natural selection. Evolution can only ﬁnd the smaller
and smaller fraction of “common” structures. The same conclusion s have been
reached in the case of RNAs. However, the results presented her e indicate that
these conclusions are not speciﬁc to RNAs, and probably apply to mo st sys-
tems undergoing neutral evolution.
Next we have examined the dynamics of populations evolving on our lan d-
scapes. These studies have yielded some of the most interesting re sults of this
work. In their studies with N K landscapes, Kauﬀman and Johnsen [14] made
a useful approximation in representing evolving populations by their single
dominant sequence. This approximation is only valid in the case in which t he
time-scale for mutation is much longer than the time-scale on which se lection
acts. For the moment we will assume this to be the case. A “random h ill-
climber” is a population of this type, represented by a single dominant strain,
which tries mutations—point mutations in the present case—until it ﬁ nds one
with higher ﬁtness than the current strain. In this way the hill-climbe r per-

0 4 8 12 16
number of fitness levels F
0.6
0.7
0.8
0.9
1.0mean fitness
0 20 40 60 80
time t
0.0
0.2
0.4
0.6
0.8
fitness W
Fig. 3. The maximum ﬁtness attained by a random hill-climber averaged over ten
simulations with N = 20, K = 4 and A = 2, as a function of the neutrality param-
eter F (circles). The lower curve (squares) is the ﬁtness of the mos t ﬁt percolating
neutral network averaged over the same ten runs. Inset: the ﬁ tness of one of the
hill-climbers in the simulation as a function of time.
forms an adaptive walk through sequences of ever-increasing ﬁtn ess until it
reaches a local ﬁtness optimum. To study neutral landscapes we m odify this
strategy so that the hill-climber samples adjacent sequences at ra ndom until it
ﬁnds one of ﬁtness greater than or the same as itself. Such a climber will move
at random on a neutral network until it ﬁnds a mutation which takes it to a
network of higher ﬁtness. In the upper curve of Figure 3 we show t he average
ﬁtness attained by such a walker over ten simulations on our landsca pes as a
function of the neutrality parameter F . Recall that neutrality increases with
decreasing F . As the ﬁgure shows, the hill-climber on average ﬁnds higher ﬁt-
ness maxima for higher degrees of neutrality. In other words, neu trality helps
the population to attain a greater ﬁtness. This is certainly an idea wh ich has
been entertained before in the literature, but it is lent a new convict ion when
we see it emerge in the behaviour of a general model such as this.
The lower curve on Figure 3 shows the ﬁtness of the most ﬁt percola ting net-
work averaged over the same ten landscapes. The curve follows qu ite closely
the form of the curve for the ﬁtness of the local maxima found by t he hill-

climber. Our explanation of this result is as follows. The climber moves d iﬀu-
sively on a neutral network until it ﬁnds a one-mutant neighbour wh ich belongs
to a network of greater ﬁtness, at which point it shifts to that net work. This
process continues until it reaches a non-percolating network, at which point it
is conﬁned to the region occupied by the network and can only get as high as
the local maximum within that region. Thus the highest ﬁtness attain able on
a landscape with neutrality depends directly on the highest ﬁtness a t which
there are percolating networks. Since the landscapes with the gre atest degree
of neutrality also have more and ﬁtter percolating networks, this e xplains why
higher ﬁtnesses are attained on landscapes with lower values of F .
The inset to Figure 3 shows the ﬁtness of one of our hill-climbers as a f unction
of time, and we can clearly see the steps in this function where the clim ber ﬁnds
its way onto a neutral network of higher ﬁtness. Similar steps have been seen,
for example, in laboratory experiments on the evolution of bacteria [21,22].
Although it appears in the ﬁgure that nothing happens in the periods between
these jumps, it is at these times that the climber diﬀuses around its n etwork,
testing new mutations to ﬁnd one of higher ﬁtness. It is this diﬀusive motion
which allows us to ﬁnd higher ﬁtness sequences on landscapes with hig her
degrees of neutrality. Van Nimwegen et al. [10] have dubbed these periods of
apparent stasis “epochs”. They also bear some similarity to the pala eontologi-
cal “punctuated equilibria” described by Eldredge and Gould [23,24], a lthough
there are many other possible explanations for the periods of stas is seen in
fossil evolution.
To investigate the epochs in more detail, we have performed simulatio ns of
true populations evolving on our landscape. In these simulations we t ake a
population of M sequences which at each generation reproduce with proba-
bility proportional to their ﬁtness, in such a way that the total pop ulation
size remains constant. Reproduction is also subject to mutation at some rate
q per locus: with probability q the value at any locus in a sequence changes
to a new randomly chosen one when the sequence is reproduced.
In Figure 4 we show the results of such a simulation for a population of
M = 1000 sequences with N = 20 and a mutation rate of q = 5 × 10− 6. For this
value of q the mutation rate is low enough that the hill-climber approximation
used above is reasonable and, as the ﬁgure shows, the epochs visib le in the
case of the hill-climber appear also in the average ﬁtness measured in the
population simulations (solid line). For the simple case of the Royal Roa d
genetic algorithm, van Nimwegen et al. [11] have studied the epochs extensively
and given a number of analytic and numerical results which may gener alize to

0 400 800 1200 1600
time t
0.0
0.2
0.4
0.6
0.8
1.0
mean population fitness
0 400 800 1200 1600
time t
0.0
1.0
2.0
3.0
4.0
5.0
population entropy
Fig. 4. Solid line: the average ﬁtness of a population of M = 1000 sequences evolving
on a landscape with N = 20, K = 4, A = 2 and F = 2. The mutation rate is
q = 5 × 10− 6. Dotted line: the entropy of the population, which is a measu re of the
diversity of sequences present.
present case. 3
The length of the epochs in Figures 3 and 4 increase, on average, wit h in-
creasing ﬁtness. This behaviour was also seen in the Royal Road, an d occurs
because as the ﬁtness increases the number of structures with h igher ﬁtness
still dwindles. 4 The length of the epochs in fact depends on the rate of diﬀu-
sion of the population across the neutral network, which in turn de pends on q,
and on the density of “points of contact” between the network an d other net-
works of higher ﬁtness [2]. Another interesting feature of the epo chs, also seen
in the Royal Road, is that their average ﬁtness does not correspo nd exactly
to the ﬁtness of any of the networks. Typically the average ﬁtnes s is a little
lower than the ﬁtness of the dominant structure in the population b ecause
3 In fact, the Royal Road model can be reproduced as a special ca se of the present
model in which the epistatic interactions between loci take a particular structure,
being collected in blocks, rather than placed randomly betw een pairs of loci.
4 It would do this under any circumstances, but the problem is m ade particularly
severe by the binomial distribution of ﬁtness values mentio ned above, which has an
exponentially decreasing tail for high ﬁtness.

deleterious mutants are constantly arising. Even though these mu tants are
selected against, there are at any time enough of them in the popula tion to
make a noticeable diﬀerence to the average ﬁtness. Since the numb er of pos-
sible mutants with lower ﬁtness than the dominant sequence increas es with
increasing ﬁtness, it is also possible to get error threshold eﬀects w ith increas-
ing ﬁtness [25,26]. As the ﬁtness increases, there may come a point w here the
rate at which deleterious mutants arise in the population exceeds th e rate at
which they are suppressed by selection, and at this point further im provement
in ﬁtness becomes impossible. Thus there may be a dynamical limit on th e
ﬁtness of populations, independent of the limit imposed by the struc ture of
the landscape discussed above. (This is true of landscapes without neutral
evolution too, though the eﬀect is much more prominent in the neutr al case.)
In Figure 4 we also show the entropy of the population as a function o f time
(dotted line). The entropy is deﬁned as
S = −
∑
i
pi log pi, (2)
where pi is the average probability of ﬁnding a particular member of the
population with sequence i. 5 In this case, we see that the entropy is low
during the epochs, indicating that the population is compact, and he nce that
the strong-selection approximation made in the case of the random hill-climber
applies. Only during the selection events themselves, when the popu lation
moves to a new neutral network does the entropy increase.
Simulations similar in spirit to ours have been performed for population s of
tRNAs by Fontana and co-workers [5,27]. In these simulations the au thors
chose a “target” structure which was artiﬁcially selected for, and they also
observed epochs in the evolution as the population passed through a succession
of increasingly ﬁt structures on its way to the target.
4 Discussion
The aim of this work is to study a model of neutral evolution which is ge neral
enough to encompass behaviours typical of other more speciﬁc mo dels which
have been employed in the past. In this way we can reproduce in a gen eral
5 In fact, calculating this entropy for a population of ﬁnite s ize in a large sequence
space is not a trivial task. The technical details of the calc ulation will be addressed
in more detail in another paper [19].

context the results which have been observed as special cases an d hence in-
vestigate the extent to which these results are general propert ies of ﬁtness
landscapes possessing neutrality, or particular to the systems in w hich they
were ﬁrst observed. In this spirit, we put forward the following con jectures
about the ﬁtness landscapes on which biological molecules evolve, ba sed on
the results of the investigations outlined in this paper.
(1) The total number of possible structures increases exponent ially with se-
quence length. The exponential constant of this increase appear s to be
approximately numerically equal in both the general model and the o nly
speciﬁc case in which it has been studied, that of RNA secondary str uc-
ture.
(2) There are a large number of structures which correspond to a small num-
ber of sequences, and a small number of structures which corres pond to a
large number of sequences. The exact form of the histogram of st ructure
frequency, shown in Figure 1, varies depending on the parameters of our
model. However, for certain values of the parameters it has a form similar
to that seen in RNA studies, whilst for others it appears to follow a po wer
law.
(3) The “common” structures—ones which correspond to a large n umber of
sequences—constitute an exponentially decreasing fraction of th e total
number as sequence length increases. Conversely, however, the y cover a
fraction of the sequence space which tends to unity for long seque nces.
(4) The evolution of populations, at least on short time-scales, is do minated
by the presence of neutrality. Neutrality helps the population to ﬁn d
structures of high ﬁtness without having to cross ﬁtness barrier s. The
highest ﬁtness which can be found in this way is closely related to the
ﬁtness of the highest percolating neutral network, which itself de pends on
the amount of neutrality. For landscapes with a higher degree of neutrality
therefore, the population typically reaches a higher ﬁtness.
(5) The ﬁtness may be limited by error threshold eﬀects, which are p artic-
ularly severe for landscapes of this type, because the size of the n eutral
networks (and hence the ratio of numbers of beneﬁcial and harmf ul mu-
tants) falls exponentially with increasing ﬁtness.
(6) Evolution proceeds in jumps separated by “epochs” in which the ﬁtness
appears to change very little. In fact, the population uses these e pochs to
diﬀuse across the current neutral network, allowing it to search a larger
portion of sequence space for beneﬁcial mutations.

5 Conclusions
To conclude, we believe that by studying a simple and general model o f a neu-
tral landscape, we should be able to distinguish properties of speciﬁ c systems
undergoing neutral selection from properties common to all such s ystems. We
have found a number of potential candidates for inclusion in a list of s uch
common properties. There are many interesting lines of investigatio n which
we have not been able to pursue in this short work, including details of the
structure and size of the neutral networks such as percolation m easures and
covering radii, details of population dynamics on these networks inclu ding
entropy and other statistical measures of the structure of suc h populations,
calculations of the length of epochs, of the maximum ﬁtness obtaina ble on
these landscapes, of the eﬀects of error threshold eﬀects on ma ximum ﬁtness,
and many eﬀects of the variation of the parameters of the model, p articu-
larly the eﬀects of the variation of the level of epistasis K and the neutrality
parameter F . Some of these questions will be addressed in a forthcoming work.
Acknowledgements
The authors would like to thank James Crutchﬁeld, Melanie Mitchell, Er ik
van Nimwegen and Paolo Sibani for interesting discussions. This work was
supported in part by the Santa Fe Institute and DARPA under gran t number
ONR N00014–95–1–0975.
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Adaptive Competition, Market Efficiency, Phase Transitions and Spin-Glasses

Abstract
We analyze a simple model of adaptive competition which captures essential features of a variety of adaptive competitive systems in the social and biological sciences. Each of N agents, at each time step of a game, joins one of two groups. The agents in the minority group are awarded a point, while the agents in the majority group get nothing. Each agent has a fixed set of strategies drawn at the beginning of the game from a common pool, and chooses his current best-performing strategy to determine which group to join. For a fixed N, the system exhibits a phase change as a function of the size of the common strategy pool from which the agents initially draw their strategies. For small pool sizes, the system is in an efficient market phase. All information that can be used by the agents' strategies is traded away, no agent can accumulate more points than would an agent making random guesses, and thus the commons suffer,since relatively few points are awarded to the agents in total. For large initial strategy pool sizes, the system is in an inefficient market phase, in which there is predictive information available to the agents' strategies, and some agents can do better than random at accumulating points. In this phase, the total number of points awarded to the agents is greater than in a game in which all agents guess randomly, and so the commons do relatively well. At a critical size of the strategy pool marking the cross-over from the efficient market to the inefficient market phases, the commons do best. This critical size of the pool grows monotonically with N. The behavior of this system has some features reminiscent of a spin-glass.

Figure 1
Past Prediction
0 0 0 1
0 0 1 1
0 1 0 0
0 1 1 1
1 0 0 0
1 0 1 0
1 1 0 0
1 1 1 1

Adaptive Competition, Market Efficiency, Phase Transitions and
Spin-Glasses
Robert Savit, Radu Manuca and Rick Riolo
Program for the Study of Complex Systems and Physics Department
University of Michigan
Ann Arbor, MI 48109
Abstract
In this paper we analyze a simple model of adaptive competition which captures essential
features of a variety of adaptive competitive systems in the social and biological sciences.
In this model, each of N agents, at each time step of a game, joins one of two groups.
The agents in the minority group are awarded a point, while the agents in the majority
group get nothing. Each agent has a fixed set of strategies drawn at the beginning of the
game from a common pool, and chooses his current best-performing strategy to
determine which group to join. We find that for a fixed N, the system exhibits a phase
change as a function of the size of the common strategy pool from which the agents
initially draw their strategies. For small pool sizes, the system is in an efficient market
phase in which all information that can be used by the agents' strategies is traded away,
and no agent can accumulate more points than would an agent making random guesses.
In addition, in this phase the commons suffer, and relatively few points are awarded to
the agents in total. For large initial strategy pool sizes, the system is in an inefficient
market phase, in which there is predictive information available to the agents' strategies,
and some agents can do better than random at accumulating points. In addition, in this
phase, the total number of points awarded to the agents is greater than in a game in which
all agents guess randomly, and so the commons do relatively well. At a critical size of
the strategy pool marking the cross-over from the efficient market to the inefficient
market phases, the commons do best. This critical size of the pool grows monotonically,
and in a very simple way with N. The behavior of this system has features reminiscent
of a spin-glass in statistical physics, with the small pool size phase being, in a certain
sense, more glassy than the large pool phase. We argue that the structure we have
elucidated has important implications for a range of phenomena in the social and
biological sciences, as well as for the general study of adaptive, competitive systems.
12/3/97 a

Introduction
Most systems in the biological and social sciences involve a number of interacting
agents, each making behavioral choices in the context of an environment that is formed,
in large part, by the collective action of the agents themselves, and with no centralized
controller acting to coordinate agent behavior. In the most interesting and difficult to
analyze cases, the agents have heterogeneous strategies, expectations and beliefs [Arthur,
1994]. In some cases the agents' strategies may be self-validating, at least for a limited
time. For example, in the financial markets a wide-spread belief that a commodity will
rise in price may perforce result in a price rise for that commodity [Arthur et al, 1996].
But unless there are fundamental reasons for the price rise, such bubbles eventually tend
to burst, so that widely-shared strategies are often self-defeating in the long run. Thus, in
many systems, and most clearly in those in which agents compete for scarce resources,
successful agents will employ strategies that differentiate them from their competitors, so
that the agents will place themselves in groups in which target resources are not over-
utilized. Examples of such systems include animals foraging for food, firms searching
for profitable technological innovations, packets looking for paths through the internet,
or people trying to go to (overly) popular events or places. Furthermore, from the point
of view of overall system performance, the best strategy sets are those that result in
coordinated resource utilization so that average agent experience is relatively good, and
the scarce resource is consumed near its limiting rate. Examples of systems of competing
agents in which such coordinated allocation of resources is critical include ecological
communities [Cody and Diamond, 1975], routers trying to send packets over the Internet
[Kahin and Keller, 1997], and humans trying to decide on which night to go to a popular
bar [Arthur, 1994].
These systems are enormously complicated and their detailed dynamics may depend on
particular characteristics of the agents and their interactions. Nevertheless, there also are
fundamental properties which are shared by all these systems. If we have any hope of
ever understanding these kinds of collective adaptive systems, or even of understanding
the terms in which we should analyze them, we must first understand the dynamics
imposed by their most basic shared properties.
In this paper we analyze a simple model that incorporates the basic adaptive and
feedback features of systems of agents competing for a scarce resource [Challet and
Zhang, 1997]. In this model, each agent chooses to be in one of two groups at each time

step, and those agents in the minority group are awarded a point. Each agent has his own
set of strategies, chosen initially from a pool of available strategies, from which he uses
the currently best-rated strategy (based on past performance) to select a group to join.
Each strategy uses information about which group was in the minority during (a few)
previous time steps to predict which will be the minority group during the current time
step. We find that as a function of the size of the strategy pool available to the agents,
there is a transition which separates an efficient market phase from an inefficient market
phase. If the number of available strategies is not sufficiently large, then the system is in
a phase, in which the agents are "frustrated" in their attempt to find minority positions,
and many features of the aggregate behavior of the system in this phase are analogous to
those of the glassy phase of a spin-glass [Mezard, Parisi and Virasoro, 1987; Fischer and
Hertz, 1991]. In this phase the market is efficient, and it can be shown that all the
relevant information accessible to the agents' strategies has been traded away by their
competition [Malkiel, 1985; Fama, 1970; Fama, 1991]. Thus, no agent can ever exceed a
50% rate of success. (Remarkably, however, there is still information in the record of
which group was the minority group as a function of time, but that information cannot be
used by the agents' strategies.) In addition, the collective behavior of the agents is
generally substantially worse than in a random market in which the agents join one of the
two groups independently and randomly with equal probability. Above a critical size of
the strategy pool, the system is in an inefficient market phase. Here there is predictive
information available to the agents' strategies and some agents can achieve better than a
50% success rate. In addition, the collective experience of the agents is much improved
and is substantially better than in a random market, indicating an emergent coordination
among the agents. As a function of the size of the strategy pool, the best collective
performance occurs near the critical size of the strategy pool which marks the cross-over
from the efficient market phase to the inefficient market phase. The critical size of the
strategy pool scales with N, the number of agents playing the game, in a very simple
way. These unexpected emergent properties have profound implications for the study
and epistemology of competitive markets in the biological and social sciences.
We first describe the model, then present the results of our analysis and finally discuss
the implications of our work for the study of adaptive, competitive social and biological
systems.

The Model
The simple model of competition we discuss here consists of N agents playing a game
[Challet and Zhang, 1997]. The rules of the game are as follows: At each time step of
the game, each of the N agents playing the game joins one of two groups, labeled 0 or 1.
Each agent that is in the minority group at that time step is awarded a point, while each
agent belonging to the majority group gets nothing. An agent chooses which group to
join at a given time step based on the prediction of a strategy. The strategy uses
information from the historical record of which group was the minority group as a
function of time. A strategy of memory m is a table of 2 columns and 2m rows. An
example of an m=3 strategy is shown in Fig. 1. The left column contains all the eight
possible combinations of three 0's and 1's. To use this strategy, an agent observes which
group was the minority group during the last three time steps, and finds that string of 0's
and 1's in the left hand column of the table. The corresponding entry in the right hand
column contains that strategy's determination of which group (0 or 1) the agent should
join during the current time step.
In each of the games discussed here, all strategies used by all the agents have the same
value of m. At the beginning of the game each agent is randomly assigned s (>1) of the
22m possible strategies, with replacement.1 For his current play the agent chooses his
strategy that would have had the best performance over the history of the game up to
that time. Ties between strategies are decided by a coin toss. Following each round of
decisions, the cumulative performance of each of the agent's strategies is updated by
comparing each strategy's latest prediction with the current minority group. Because the
agents each have more than one strategy, the game is adaptive in that agents can choose
to play different strategies at different moments of the game in response to changes in
their environment; that is, in response to new entries in the time series of minority groups
as the game proceeds. Because the environment (i.e. the time series of minority groups)
is created by the collective action of the agents themselves, this system has very strong
feedback, reminiscent of, for example, the financial markets.

Results
In what follows we will report and interpret the results of this game for a range of values
of N (odd), m and s=2. The qualitative results also hold for other values of s that are not
extremely large.2 To achieve stable results, the game must be run for a long enough
time. What is long enough depends on N and m. For our runs, in which N ranged
between 11 and 1001, we found that 10,000 time steps were generally sufficient, except
for the largest values of N, which required runs of 100,000 time steps. To start the game,
we also create a short (of order m) random history of 0's and 1's, so that the strategies can
be initially evaluated. The asymptotic statistical results of any run do not materially
depend on what this random string is.
To begin to understand the behavior of this system, consider the time series of the
number of agents belonging to group 1 (≡L1). (This information is not available to the
agents but it is available to the researchers.) The mean of this series is generally close to
50% for all values of N, m and s (we shall return to this point below), and so the standard
deviation, σ, of this time series is a measure of how well the commons do: The smaller
σ, the more total points are awarded to all agents combined. That is, if there are typically
many fewer than 50% of the agents in the minority, then σ will be large and there will be
few total points awarded, while if σ is small, then most of the time the minority group
will consist of only slightly fewer than half of the agents, and more total points will be
awarded.
The behavior of σ is quite remarkable. In Fig. 2, we plot σ for these time series as a
function of m for N=101 and s=2. For each value of m, 32 independent runs were
performed. The horizontal dashed line in this graph is at the value of σ for the random
game, i.e. for the game in which the agents randomly choose 0 or 1, independently and
with equal probability at each time step.
Note the following features:
1. For small m, the average value of σ is very large (much larger than in the random
case). In addition, for m<6 there is a large spread in the σ 's for different runs
with different (random) initial distributions of strategies to the agents, but with
the same m.

2. There is a minimum in σ at m=6 at which σ is less than the standard deviation of
the random game. We shall refer to the value of m at which the σ vs. m curve
(for fixed N) has its minimum as mc.3 Thus, in Fig. 2, mc=6. Also, for m≥mc,
the spread in the σ's appears to be small relative to the spread for m<mc.
3. As m increases beyond 6, σ slowly increases, and for large m approaches the
value for the random game.
The system clearly behaves in a qualitatively different way for small and large m. To
further study the dynamics in these two regions, we consider the time series of minority
groups, (≡G) the data publicly available to the agents. We want to study the information
content of strings of consecutive elements of this time series of various lengths (including
strings of length m) for different values of m and N. To do this, we consider the
conditional probability P(1|uk). This is the conditional probability to have a 1
immediately following some specific string, uk, of k elements of G. For example,
P(1|0100) is the probability that 1 will be the minority group at some time, given that
minority groups for the four previous times were 0,1,0 and 0, in that order. Recall that in
a game played with memory m, the strategies use only the information encoded in strings
of length m to make their choices. In Fig. 3, we plot P(1|uk) for G, the time series of
minority groups generated by a game with m=4, N=101 and s=2. Fig. 3a shows the
histogram for k=m=4 and Fig. 3b shows the histogram for k=5. Note that the histogram
is flat at 0.5 in Fig. 3a, but, remarkably, is not flat in Fig. 3b. Thus, any agent using
strategies with memory (less than or) equal to 4, will find that those strings of minority
groups contain no predictive information about which group will be the minority at the
next time step. But recall that this time-series was itself generated by players playing
strategies with m=4. Therefore, in this sense, the market is efficient and no strategy
playing with memory (less than or) equal to 4 can, over the long run, accumulate more
points than would be accumulated randomly. (In principle, an agent switching strategies
adaptively can do better than random, but we have explicitly verified that this does not
happen in this phase. The expicit reason for this is described in the results section and in
more detail in [Manuca, Riolo and Savit, 1998].) But note also that the time series of
minority groups is not a random (IID)4 string. There is information in this string, as
indicated by the fact that the histogram in Fig. 3b is not flat. However, that information
is not available to the agents playing the m=4 game who collectively generated that string
in the first place.

We can repeat this analysis for m≥6 (N=101, s=2). For this range of m, the
corresponding histogram for k=m is not flat, as we see in Fig. 4 for the m=6 game. In
this case, there is significant information available to agents playing the game with
memory m and the market is not efficient. Indeed, some individual agents can
accumulate more points than they would by simply making random guesses. We have
verified these statements by explicitly calculating the points won by individual agents in
games with different values of m. For m<6 (N=101, s=2), no agent ever achieves results
statistically greater than 50%, while for m≥6 some agents do win more than 50% of the
time (statistically significantly). It is noteworthy that even for m=5, the histogram of
conditional probabilities with k=m=5 is flat, even though σ is less than in the random
game.
How does the system behavior change as we change the number of agents? One can
repeat the calculation of σ for different N. One finds, plotting σ vs. m for fixed N, that
in all cases one obtains a graph similar to that in Fig. 2, but in which the position of the
minimum, mc, is proportional to lnN. In addition, σ and the spread in σ behave in very
simple ways with changes in N which differ depending on whether m is greater than or
less than mc. In Fig. 5 we study the behavior of σ as a function of N for m=3 and m=16.
For the range of values of N used in these figures, m=3 is to the left of the minimum in
the curve of σ vs. m (i.e. 3<mc for all these values of N) and m=16 is to the right of the
minimum (16>mc). In Fig. 5a we plot σ vs. N on a log-log scale. We see that for m=3 σ
is proportional to N, while for m=16, σ is proportional to N1/2. This is typical: for fixed
m, and m<mc, σ is proportional to N, while for fixed m and m>mc σ is proportional to
N1/2. In Fig. 5b, we plot, again for m=3 and 16, the spread in σ, i.e., the standard
deviation of the σ's (≡∆σ ) as a function of N, on a log-log scale. Here we also see power
law behavior: for m=3 ∆σ is proportional to N, while for m=16, ∆σ is proportional to
N1/2. As before, this behavior is representative of the two behaviors seen for values of
m<mc and values of m>mc, respectively.
The transition between these very different behaviors is at mc~lnN. We have found,
using scaling arguments that, to a first approximation, σ2/N is a function only of 2m
N
≡z.
To see this explicitly, we plot in Fig. 6 σ2/N as a function of z on a log-log scale for
various N and m (with s=2). We see first that all the data fall on a nearly universal
curve. The minimum of this curve is near 2mc/N=zc≈0.5, and separates the two different
phases. The slope for z<zc approaches -1 for small z, while the slope for z>zc

approaches zero for large z, consistent with the results of Fig. 5.5 Because σ2/N depends
only on z, it is clear that for fixed z σ is proportional to N1/2 for any fixed z, both above
and below zc. In addition, it can be shown that, for fixed z ∆σ is approximately
independent of N, approaching a z-dependent constant as N→∞ . The N→∞ limit of ∆σ
is large for small values of z and decreases monotonically with increasing z. It is unclear
whether or not ∆σ is non analytic at zc.
Discussion
In this section, we will first present some qualitative arguments that explain the different
behaviors for small and large m, and the scaling results with N. More detailed
explanations along with results of corroborating simulations will appear in a forthcoming
publication [Manuca, Riolo, and Savit]. Following that, we will discuss some
implications of this study for a wide variety of social and biological systems.
Consider first the small m region (m<mc). In this region, the time series of the number
of agents belonging to group 1 (≡L1) has a very unusual structure. Consider the time
series of minority groups, and focus on one particular binary string, χ, of length m.
Now, from L1 form the time series of the number of agents which choose group 1 in
response to the first, third, fifth and other odd occurrences of χ. Now form the time
series of the number of agents which choose group 1 in response to all the even
occurrences of χ. It turns out that the standard deviation of the series of responses to the
odd occurrences of χ is of order N1/2, while the standard deviation of the time series of
responses to the even occurrences of χ is of order N. This is true for all possible strings
of length m in the time series of minority groups. Thus, in the whole time series of the
number of agents belonging to group 1, there is a bursty structure with large (order N)
excursions from the mean separated by smaller excursions of order N1/2. For large N,
the contributions of the large, order N excursions to the standard deviation dominate, and
so σ ∝ N in this region.
This behavior can be understood as follows: Consider the pool of all 22m strategies. For
s=2, each agent chooses two strategies randomly from the pool. For very large N, the
strategies played by the agents are a good approximation to the population of the pool.
Now, divide the strategies into two groups according to their response to some string, say
χ. The first time χ appears in the time series of minority groups, the agents will choose
to join group 0 or 1, and which strategies they play will be a more or less random sample
of the strategies in the pool. Consequently, we expect that the population of group 1 will

deviate from 50% by a number of order N1/2. Suppose that the minority group turns out
to be group 1. Then, in the pool of strategies, all those whose response to χ is 1 will get
an additional point. The next time the string χ appears in the series of minority groups,
those strategies whose response to χ is 1 will be preferentially chosen. Now, if N is
large, then approximately 75% of the agents will join group 1. (i.e., on average, only
those agents both of whose strategies respond to χ with a 0 will join group 0.)
Consequently, the minority group at this time step will be group 0, and all those
strategies which respond to χ with a 0 will get a point. Thus, after the second occurrence
of χ, all strategies will have gained one point due to their response to χ, some following
the first occurrence, and the others following the second occurrence. There will therefore
be no strong preference for the agents' responses to the third occurrence of χ (but see the
discussion in [Manuca, Riolo and Savit, 1998]), and deviations in the membership of
group 1 at the third occurrence of χ will again be of order N1/2. This period two process
repeats, for all strings and gives rise to the structure described above. A more detailed
description of this dynamics will appear in [Manuca, Riolo and Savit, 1998].
The precise distribution of strategies to the agents at the beginning of the game differs
from run to run. The existence of the period two oscillations described above does not,
for large N, depend on the details of that distribution. However, the coefficient in front
of N in the expression for the standard deviation in the responses to even occurrences of
a given string does depend on the distribution of initial strategies. Consequently, the size
of σ will vary from run to run, but, in leading order, will still be proportional to N.
Thus, the spread in the σ's is also proportional to N in leading order in the low m phase
(m<mc), as we saw in Fig. 5.
The existence of different runs with different σ's is reminiscent of the glass phase of a
spin-glass system. The analogy goes deeper: In the spin-glass case different samples of
the glassy material behave differently as a result of frozen in randomly distributed
impurities. From one sample to the next, the precise way in which the impurities are
distributed affects the thermodynamic behavior of that sample. A spin-glass order
parameter, which is non-zero in the glassy phase just expresses the fact that the
thermodynamics of different samples is different. In our case, the strategies are
distributed to the agents randomly, but the precise way in which that randomness is
realized in one game affects the size and distribution of the membership of groups 0 and
1, thus leading to different time-averaged behavior which is most pronounced in the low
m phase. Mathematically, ∆σ , the spread in the σ's, bears some resemblance to a spin-

glass order parameter. Furthermore, while the expectation value of the number of agents
belonging to group 1 is nearly 50% in the high m phase, it deviates more strongly from
50% in the low m phase, and differs from run to run. This is also analogous to the
behavior of a spin-glass, in that the magnetization in the glassy phase is non-zero and
differs from sample to sample. Although there is some sense in which the low m phase is
glassy, we do not mean to imply that the transition at mc is a transition from a glassy to a
normal phase. Both the high and low m phases partake of glassy behavior and have
elements of broken ergodicity due to the initial distribution of strategies, although the
glassiness is more pronounced in the low m phase. This analogy will be more fully
treated in a forthcoming publication [Manuca, Riolo and Savit].
In contrast to the behavior in the low m phase, for fixed m
≥mc σ, the standard deviation
of L1, scales with N1/2. In this high m phase, for a given N, the strategy space is
sufficiently large so that the strategies assigned to the N agents are not a representative
sample of the entire strategy space. Consequently, the arguments used in describing the
low m phase do not hold, and there are not the same very strong temporal correlations in
the response of the system to successive occurrences of a given string. The scaling of σ
with N1/2 just represents the variations in the membership of N agents whose decisions
are not very tightly coupled. However, there are still strong dependencies among the
decisions of the agents. In fact, it is the remarkable emergent coordination among the
agents' decisions that accounts for the fact that σ is below the random result in this phase.
The precise way in which this comes about will be discussed in detail elsewhere
[Manuca, Riolo and Savit, 1998], but the issue can be thought of in the following way:
Strategies that would have been most effective in the past at predicting minority groups
are more likely to be used by the agents. Those strategies have specific predictions for
each string of length m. Consequently, there is an induced dependence among the
decisions of the agents which can be expressed as nontrivial conditional probabilities.
That is, the probability that at some time agent i chooses group 1, changes depending on
the choices of the other agents at that time.6 It can be shown that, for fixed m, such
nontrivial conditional probabilities cause deviations in σ from that of the random game.
In the high m phase, this term is negative, giving rise to a lower σ.
Note that the region of greatest coordination (smallest σ) is when z=2m/N is of order
one. We can understand this in the following way: Dependencies among the strategies
are induced by the agent's selection of the best strategies, as described above. But each
chosen strategy dictates a response to 2m different strings of length m. Thus, as m

increases, for fixed N, it becomes increasingly difficult for strategies to coordinate all of
their entries. (I.e., at any moment there are N strategies in play, and they must minimize
fluctuations over 2m choices.) Consequently, for fixed N, maximal coordination will be
achieved for some finite m (=mc), whose value is a monotonically increasing function of
N. As m increases for fixed N, coordination becomes less effective, and σ approaches
the result of the random game from below. It can be shown that the correction term is
proportional to 1/2m [Manuca, Riolo and Savit].7
It is quite remarkable that σ2/N lies on a universal curve as a function of the scaling
variable z=2m/N, as shown in Fig. 6. One immediate consequence of this is that the
critical value of m, mc is proportional to lnN as we have found. An intriguing result here
is that for maximum coordination N should be roughly the same size as the dimension of
the strategy space. Thus, it is the dimension of the strategy space that seems to be the
relevant measure of the diversity of strategies available to the agents.
Implications
This work points up a number of very important and general features that must be
confronted in any analysis of adaptive competition among N players. First, it is clear that
the size of the strategy space available to the agents and the number of agents playing the
game are crucial parameters in determining the qualitative behavior of the system.
Moreover, their ratio, z, seems to be the predominant parameter for many of the most
important features of the game. In particular, if, for a given number of agents, the
strategy space is not large enough, then the game will be efficient in a well-defined
sense, but the commons will suffer. There appears to be a critical size of the strategy
space that gives rise to the greatest common good, at least as measured by σ, and
everywhere in the inefficient market phase the common good is enhanced over a strictly
random game by an implicit emergent coordination. Note that in this game there is no
central controlling authority. The improvement of the common good in the inefficient
phase is a purely emergent effect. This general structure, an efficient, but poorly
performing market when the strategy space is small, a market whose average
performance is close to random when the strategy space is large (but which, nonetheless,
may produce wealthy agents [Manuca, Riolo and Savit, 1998]), and good performance,
both for the commons and for individuals when the dimension of the strategy space is
matched to the number of players is likely to be a characteristic that transcends the
specific model discussed here. These observations have clear relevance to a number of
important issues. For example, there may be significant public policy implications for

the design of markets and other regulatory issues, as well as implications for the study of
a wide range of competitive systems including ecological systems.
The phase structure of this model as a function of 2m/N is quite intriguing. In particular,
it is quite remarkable that there is such a clear transition between the efficient and
inefficient phases. Although some of the results of this system may be specific to the
model, the observation of the general phase structure and the features it shares with a
spin-glass has clear implications for the epistemology of adaptive systems. It suggests, at
least as a first step, a set of concepts and tools which are likely to be fruitful in thinking
about, and analyzing adaptive competitive systems.
One of our most noteworthy results is that the agents so efficiently and selectively trade
away the information accessible to them in the efficient phase (z<zc). The overly
competitive dynamics in this phase is remarkably successful at removing information
from the reach of the players, and thus removing any possibility that any agent could
perform well. What is even more remarkable is the selective way in which this
information is removed. There is still significant information contained in the history of
minority groups, as we see in Fig. 3b. But it is simply not available to the players whose
decisions produced that series in the first place. The precise way in which this
information is removed from the system will be discussed elsewhere [Manuca, Riolo and
Savit, 1998].
It is clear that the cost of over competition both to the individual and to the collective
may be much higher than one would a priori have thought. Not only can no agent
individually perform well, but the over-competition eliminates the possibility that the
common good could be well-served. This may be an important insight for a variety of
decision makers. For example, such insights could affect the cost-benefit analysis
associated with a company's decision about whether to invest in technological innovation
in an already crowded market. There is also an important methodological and conceptual
lesson from these observations: The way in which information is removed from access
to agents in an efficient or overcrowded market may be considerably more subtle than
one might have thought. This suggests a reevaluation of the concept of an efficient
market, and the tools and vocabulary used to describe it.
Although the behavior we have elucidated is very intriguing, it is important to remember
that there are many effects that may play a major role in specific systems and which

could alter the emergent structure, fundamentally. For example, while this model is
adaptive, it is not evolutionary. There is no discovery of new strategies by the agents, no
mutation, no recombination, no sex. Evolutionary dynamics may drastically alter the
phase structure of the system. In fact, we believe that some kinds of evolutionary
dynamics may remove the efficient, over competitive phase and drive the system to an
effective strategy space corresponding to the region around mc. Nevertheless, any
analysis of specific systems which share the competitive dynamics we have discussed
here, must take account of the structure we have described, for it is the ground upon
which the description of more specific systems, possibly with more complex dynamics, is
based.
Finally, and perhaps most importantly, our work raises the question of what really are the
fundamental terms in which we ought to think about N-agent adaptive competitive
systems. For example, the fact that in the efficient phase (m<mc), σ is so strongly
dependent on the initial distribution of strategies and, is proportional to N for fixed m,
raises the question of how such simple quantities should be interpreted. Clearly, the
variance of this time series does not carry the information one might suppose, and at the
very least must be supplemented with a specification of the spread in the variance. As
another example, the fact that the market is efficient in this phase, but that the time series
of minority groups is not random suggests the need for a more sophisticated approach to
characterizing publicly available information. The fact is that there is no well developed
epistemology for complex adaptive systems, and we are still quite unsure of what the
important issues are or what are the most robust ways of characterizing the dynamics of
such systems. But the study of simple models, and the elucidation of the variety of
behaviors which they manifest can lead us toward a deeper understanding of how to
properly frame the questions that we can sensibly ask, and sensibly answer, for complex
systems.

Footnotes
1. As we shall see, the dynamics of adaptivity are crucial to our results. It is, therefore,
essential that s>1 so that the agents have more than one strategy with which to play. For
s=1 the game devolves into a game with a trivial periodic structure. It is also worth
noting, parenthetically, that the majority game, in which each agent in the majority group
gets a point has trivial periodic structure.
2. The dependence of the results of the game on s are interesting and will be discussed in
detail elsewhere. However, the qualitative picture we present here obtains for s<<22m
3. As we shall see below, the value of m at the minimum of the curve, mc, increases
with increasing N.
4. IID stands for independent and identically distributed, and indicates a sequence whose
entries are chosen independently, from some probability distribution that does not
change. In our case, IID would mean that the 0's and 1's were all chosen independently,
and that the probability to choose 1 did not change over time. This is the simplest, most
intuitive meaning of "random".
5. For different values of s there are systematic changes in the shape of this scaling
curve. These will be discussed elsewhere [Manuca, Riolo and Savit, 1998].
6. Another way to say this is that, as a result of the competition among each agent's
strategies, the joint probability for any set of agents to make specific choices is not equal
to the product of the individual probabilities for each agent's choice.
7. There is another finite-size effect that contributes to a σ that is lower than would be
obtained in the random game. Independent of induced coordination, for a given initial
distribution of strategies, the decisions of the N agents following a given string of m 0's
and 1's in the time series of minority groups will be constrained. Therefore, for finite N,
the probability distribution of the number of agents choosing group 1 following a specific
m-string will not, in general, be symmetric. It can be shown that skewed distributions
lower the standard deviation of a random process. However, as m increases, the agents
will, over time, respond to a larger number of different m-strings, each with a different
probability distribution of, say 1's. Consequently, as m increases, the effect of the
skewness in each of the responses to a specific string will be averaged away.

Figure captions
Fig. 1. An example of an m=3 strategy.
Fig. 2. σ as a function of m for N=101 and s=2. 32 independent runs of 10,000 time
steps were performed for each value of m. σ for each run is indicated by a dot. The
horizontal dashed line is at the value of σ for the random game described in the text.
Note the broad spread in the values of σ for m<6.
Fig 3a. A histogram of the conditional probability P(1|uk) with k=4 for the game played
with m=4. There are 16 bins corresponding to the 16 possible combinations of 4 0's and
1's. The bin numbers, when written in binary form yield the strings, u k.
Fig. 3b A histogram of the conditional probability P(1|uk) with k=5 for the game played
with m=4. There are 32 bins corresponding to the 32 possible combinations of 5 0's and
1's. The bin numbers, when written in binary form yield the strings, u k.
Fig. 4 A histogram of the conditional probability P(1|uk) with k=6 for the game played
with m=6. There are 64 bins corresponding to the 64 possible combinations of 6 0's and
1's. The bin numbers, when written in binary form yield the strings, u k.
fig. 5a. σ as a function of N for fixed m, for two values of m (3 and 16), in the two
phases of the system, on a log-log scale. Note that for m=3, σ∝Ν, while for m=16,
σ∝Ν 1/2.
Fig. 5b. The spread in σ, ∆σ , as a function of N for fixed m, for two values of m (3 and
16), in the two phases of the system, on a log-log scale. Note that for m=3, ∆σ∝Ν, while
for m=16, ∆σ∝Ν 1/2.
Fig 6. σ2/N as a function of z≡2m
N
for various values of N, on a log-log scale.

References
[Arthur, 1994] W. Brian Arthur, "Complexity in Economic Theory: Inductive Reasoning
and Bounded Rationality", Amer. Econ. Assoc. Papers and Proc 84: 406-411.
[Arthur et al, 1996] W. B. Arthur, J. Holland, B. LeBaron, R. Palmer, and P. Tayler,
"Asset Pricing Under Endogenous Expectations in an Artificial Stock Market", Santa Fe
Institute Working Paper 96-12-093.
[Challet and Zhang, 1997] D. Challet and Y.-C. Zhang, "Emergence of Cooperation and
Organization in an Evolutionary Game", preprint, University of Fribourg.
[Cody and Diamond, 1975] Ecology and Evolution of Communities, M. L. Cody and J.
M. Diamond (eds.), (Harvard Univ Press, Cambridge MA).
[Fama, 1970] E. F. Fama, "Efficient Capital Markets: A Review of Theory and
Empirical Work", Journal of Finance; 25(2), 383-417.
[Fama, 1991] E. F. Fama, "Efficient Capital Markets: II", Journal of Finance, 46(5),
1575-1617.
[Fischer and Hertz, 1991] Spin Glasses, K. H. Fischer and J. A. Hertz (Cambridge
University Press, Cambridge)
[Kahin and Keller, 1997] Coordination of the Internet, B. Kahin and J. Keller (eds),
(MIT Press, Cambridge).
[Malkiel, 1985], A Random Walk Down Wall Street, 4th Edition, B. G. Malkiel, (W. W.
Norton, New York)
[Manuca, Riolo and Savit, 1998] R. Manuca, R. Riolo and R. Savit, in preparation.
[Mezard, Parisi and Virasoro, 1987] Spin Glass Theory and Beyond, M. Mezard, G.
Parisi and M. A. Virasoro (World Scientific, Singapore).

0 2 4 6 8 10 12 14 16
Figure 2: N=101 S=2
m
St. dev.

0 2 4 6 8 10 12 14 16
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
m−bit past
P(1|m−bit past)
Figure 3a: m = 4
0 5 10 15 20 25 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(m+1)−bit past
P(1|(m+1)−bit past)
Figure 3b: m = 4

0 10 20 30 40 50 60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
m−bit past
P(1|m−bit past)
Figure 4: m = 6

m=3
slope=1
slope=1/2
Figure 5a: s=2
N
St. Dev.
m=3
slope=1
slope=1/2
−2
−1
Figure 5b: s=2
N
St. Dev. Spread

N=11
slope=−1
−3
−2
−1
−2
−1
Figure 6: s=2
z
Stdev^2/N


Geometric statistical inference

Geometric statistical inference

Abstract
Finite sample size corrections to the reparametrization-invariant solution of the inverse problem of probability are computed, and shown to converge uniformly to the correct distribution.

arXiv:adap-org/9801001v1 31 Dec 1997
Geometric statistical inference
Vipul Periwal
Department of Physics, Princeton University, Princeton, N ew Jersey 08544
Finite sample size corrections to the reparametrization-i nvariant solution of the inverse problem
of probability are computed, and shown to converge uniforml y to the correct distribution.
I. INTRODUCTION
A basic problem in statistics, with applications in divers ﬁelds, is the det ermination of the probability distribution
that underlies a ﬁnite set of experimental results involving continuo us variables. From the practical point of view,
one usually has a deﬁnite ﬁnite-parameter model for the probability distribution on theoretical grounds, and the
parameters are ﬁxed by ﬁtting to the data set. As is obvious, this p arametric inference introduces a theoretical bias,
since the true distribution may not even lie in the parameter set chos en. It is therefore of interest to attempt a direct
determination of the probability distribution without resorting to ﬁn ite-dimensional approximations. Of course, for
any ﬁnite data set, one obtains only a probabilistic description of the probability distribution, but the spread of
possible probability distributions decreases as the size of the data s et is increased [1].
This set-up is fundamental in pattern theory, a context in which re parametrization invariance is of great importance.
In visual information processing algorithms, for example, if the infe rred pattern depends on, say, the orientation of the
object, then the object recognition capabilities of the algorithm ar e not likely to be eﬃcient. In speech recognition,
variations in the tempo of speech should, again, be treated as repa rametrizations of the same underlying pattern, as
should frequency variations.
In the Bayesian approach to the problem, one starts by assuming a space of probability distributions, {w}, within
which the true distribution lies, and given a set of observations, f , one obtains a probabilistic description of the true
distribution by using
prob(w|f ) = prob(f |w) · prob(w)
prob(f ) ; (1)
in words, the probability of the distribution w being the true distribution, given f, is the probability of f given
the distribution w, multiplied by the probability of w occuring in the space of all distributions, normalized by the
probability of f occuring in any of the distributions in the set {w}. The maximum likelihood (ML) estimate of w is
then that distribution w that maximizes prob( f |w) · prob(w). To go further in this direction, one needs to ﬁgure out
what the a priori probability prob( w) should be, and how to compute the ML estimate. Clearly, to avoid bia s, one
wants the space of distributions to be as general as possible, cons istent with computability.
In one dimension, Bialek, Callan and Strong [2] have recently given an e legant formulation of this problem for
continuous distributions in one dimension. They used Bayes’ rule to w rite the probability of the probability distribution
Q, given the data {xi}, as
P [Q(x)|x1, x2, ..., xN ] = P [x1, x2, ..., xN |Q(x)]P [Q(x)]
P (x1, x2, ..., xN ) = Q(x1)Q(x2) · · ·Q(xN )P [Q(x)]∫
[dQ(x)]Q(x1)Q(x2) · · · Q(xN )P [Q(x)] , (2)
where the factors Q(xi) arise because each xi is chosen independently from the distribution Q(x), and P [Q] encodes
the a priori hypotheses about the form of Q. The optimal least-square estimate of Q, Q est(x, {xi}), is then
Qest(x; {xi}) = ⟨Q(x)Q(x1)Q(x2) · · ·Q(xN )⟩(0)
⟨Q(x1)Q(x2) · · ·Q(xN )⟩(0) , (3)
where ⟨· · ·⟩(0) denotes an expectation value with respect to the a priori distribution P [Q(x)].
In this ﬁeld-theoretic setting, Ref. [2] assumed that the prior dist ribution P [Q] should penalize large gradients, so
written in terms of an unconstrained variable φ ≡ ln(ℓQ) ∈ (−∞, +∞), they assumed
Pℓ[φ(x)] = 1
Z exp
[
− ℓ
∫
dx(∂xφ)2
]
× δ
[
1 − 1
ℓ
∫
dxeφ(x)
]
, (4)
with ℓ a global length parameter that they later ﬁxed by means of other consider ations. This form for the prior
distribution is very simple, and quite minimal in terms of underlying assu mptions, so is almost an ideal example of
the Bayesian set-up.

There is, however, an important aspect in which this prior distributio n does not measure up—a probability distri-
bution is a density, and hence transforms in a speciﬁc manner under reparametrizations of the data, e.g. given a set
of N observations xi, we want to infer a distribution wx that is related in a very simple way to the distribution wexp
inferred from exp( xi) :
wx(z) = exp( z) · wexp(exp(z)). (5)
This reparametrization covariance of the estimated distribution tu rns out to be a strong constraint on the distribution
prob(w), and on the actual computation of the estimated distribution. In re cent work, I found a geometric modiﬁcation
of the approach of Ref. [2], which satisﬁes this desired reparametr ization covariance [4]. It is the details of this
solution, especially the corrections for ﬁnite sample size (which are o bviously of paramount importance for practical
applications), that are explained in the present paper.
This paper is organized as follows: Section 2 contains a short review o f Ref. [4], to keep the presentation here
relatively self-contained. In Section 3, the one-dimensional solutio n for ﬁnite sample sizes is given, along with a
discussion of corrections arising from ﬂuctuations about the sadd lepoint distribution. In section 4, I present some
conclusions, with comments on the solution of the inference problem in dimensions larger than 3.
II. REVIEW
Instead of the form (eq. 4) used in Ref. [2], I write Q(x) ≡
√
h(x) exp(φ(x)), with h(x) a metric in one dimension.
Hence
√
h(x) is a scalar density, and so is Q, which is crucial for a reparametrization covariant solution. An intuit ion
for the rˆ ole thath plays is that it is a local analogue of ℓ, determining binning intervals locally. Set
P [φ, h] ≡ 1
Z exp
[
− ℓ
∫
dx
√
h(x)
−1
(∂xφ)2
]
δ
[
1 − 1
ℓ
∫
dx
√
h(x)eφ(x)
]
, (6)
The inverse of the metric is just 1 /h(x) and the reparametrization invariant volume element is
√
h(x)dx, so P [φ, h]
is reparametrization invariant. Now, we want to evaluate
⟨Q(x1)Q(x2) · · · Q(xN )⟩(0)
=
∫
Dφ Dh
Dif f+
P [φ, h]
N∏
i=1
√
h exp[φ(xi)] (7)
= 1
Z
∫ dλ
2π
∫
Dφ Dh
Dif f+
exp [−S(φ, h; λ)] , (8)
where
S(φ, h; λ) = 1
∫
dx
√
h(x)
−1
(∂xφ)2 + iλ
∫
dx
√
h(x)eφ(x) −
N∑
i=1
[φ(xi) + 1
2 ln h(xi)] − iλ. (9)
Notice that the integral over all metrics has been divided by the volu me of the group of orientation preserving
diﬀeomorphisms. In one dimension, this division eliminates all but one glo bal degree of freedom from the metric.
There is no operational way to distinguish between the factor
√
h(xi) and exp( φ(xi))—these must occur together in
Q(xi). Taking the local symmetry into account, the number of local degre es of freedom is the same as in the approach
of Bialek et al. [2].
The equations of motion that follow from varying S are
− 1
2 (φ′)2 1
h + iλ exp(φ) −
∑
i
1√
h(xi)
δ(x − xi) = 0 , (10)
−
( 1√
h
φ′
) ′
+ iλ
√
h exp(φ) −
∑
i
δ(x − xi) = 0 , (11)
∫
dx
√
h exp(φ) = 1 , (12)
where I have used primes to denote d/dx. δ (x) denotes the scalar density such that
∫
dxδ(x) = 1 . Introduce a variable

y(x) =
∫ x√
h(s)ds, (13)
then it follows from eq. 10 and eq. 11 that
d2φ
dy2 = 1
( dφ
dy
) 2
. (14)
Note that the use of eq. 13 is limited by the non-singularity of
√
h.
It is now necessary to be careful about the limits of integration. Th is care in the boundary terms is unnecessary at
N = ∞, in accord with the fact that any ﬁnite data set will not indicate the tr ue limits of the probability distribution.
Suppose that x ranges from x− to x+, then integrating eq. 11 I ﬁnd
N − iλ +
∫ x+
x−
dx( 1√
h
φ′)′ = 0 , (15)
so, using eq. 12, it follows that
exp(−φ) = ( y+ − y−) (y − c)2
(y− − c)(y+ − c) (16)
where c is an arbitrary constant of integration. c is restricted only by c > y + > y −, or y+ > y − > c, since exp( φ)
must be positive. This ‘cyclic’ constraint on y+, y−, c arises because the algebra of inﬁnitesimal reparametrizations of
the line has a subalgebra isomorphic to sl(2, R).
In order to determine y(x), which satisﬁes
(y+ − c)(y − y−)
(y+ − y−)(y − c) =
∫ x
x−
[
N
∑
i
δ(x′ − xi)
]
dx′ , (17)
observe that the cross ratio on the left is projectively invariant, i.e., invariant under transformations of the form
z ↦→αz + β
γz + δ , (18)
with α, β, γ, δ real, αδ − βγ = 1 . This amounts to a three-parameter family of equivalent solutions y(x) determined
by the data. This projective invariance can be ﬁxed by setting c = ∞, y+ = 1 and y− = 0, which implies φ = 0.
Operationally, this means that the next measured data point will be o bserved with equal probability at any value
of y in the interval [0 , 1]. An important point to notice about this solution is that it is only valid at N = ∞. At ﬁnite
N, the metric determined by this solution is not smooth, being a sum over δ functions, hence we must take more care
in solving eq.’s 10,11,12 for ﬁnite N. Indeed, the action evaluated at this solution is inﬁnite for ﬁnite N.
III. FINITE N CORRECTIONS
Obviously, for any practical application of this theory, one needs a systematic scheme for computing ﬁnite sample
size corrections. To understand the theory at ﬁnite N, and to make contact with the higher-dimensional theory
commented on in Section 4, we rewrite eq. 9 as
S = 1
∫
dy(∂yφ)2 + iλ
( ∫
dyeφ − 1
)
+ N
[
−
∫
dy ˆPy(y)φ(y) −
∫
dx ˆPx ln
√
h(x)
]
, (19)
where y(x) is as in eq. 13, ˆPx(x) ≡ 1
N
∑ δ(x − xi), and ˆPy(y)y′ ≡ ˆPx(x). It is clear in this form that there are two
independent functional variations of S, one with respect to φ(y), and the other with respect to y(x) :
−∂2
y φ + iλ exp(φ) − N ˆPy(y) = 0 , (20)
∂x
[ ˆPx(x)
y′(x)
]
= 0 . (21)

For N large, suppose ˆPx(x) = P (x)+ 1√
N ρ(x), with P (x) a continuous density and ρ(x) : ⟨ρ(x)ρ(x′)⟩ = P (x)δ(x−x′),
following Ref. [2]. Now, a solution to eq.’s 20 and 21 at leading order in N is φ0 = 0 , and
√
h(x) ≡ ∂xy0(x) = P (x),
as reviewed in Sect. 2. Indeed, it would seem from eq. 17 that this so lution is exact, and no modiﬁcation for ﬁnite N
is necessary. This is not true, since for ﬁnite sample sizes eq. 17 deﬁnes a singular metric, an d hence is inconsistent
with our calculations which have assumed that the metric is non-singu lar implicitly.
We can, however, solve eq. 11 to get φ = φ
[ √
h, ˆPx
]
, and then use this solution in eq. 10 to determine h. We shall
ﬁnd that the leading correction at ﬁnite N is a correction to φ at O(N −1/2), with no correction to
√
h = P (x). This
ordering of the computational steps parallels the calculation in [2], wit h the determination of h here the analogue of
the determination of the length scale in [2]. What is diﬀerent in the pres ent approach, is that the formulæ derived
here are reparametrization-covariant, e.g., square roots of densities do not appear. To leading order, we ﬁnd iλ = N,
and
exp(φ0) = P (x)√
h
. (22)
Notice that eq. 22 has the correct reparametrization covariance by construction, with a function equated to a ratio of
densities.
At the next order, expanding eq. 11 about φ0, iλ = N, we ﬁnd
[
− 1√
h(x)
∂x
1√
h(x)
∂x + N eφ0
]
φ1 ≡
(
−∆ h + N eφ0
)
φ1 =
[ √
N ρ(x)√
h(x)
+ ∆ hφ0
]
. (23)
Eq. 23 is easily solved to obtain
φ1(x) =
∫
dx′√
h(x′)
[ √
N ρ(x′)√
h(x′)
+ ∆ hφ0(x′)
]
K(x, x′), (24)
with the leading term in K given by
K(x, x′) ≈ 1
√
N
[
eφ0(x)eφ0 (x′)
] −1/4
exp
(
−
√
N
∫ max(x,x′)
min(x,x′)
√
heφ0/2dx′′
)
. (25)
Note the reparametrization invariance of eq. 24.
We can now insert eq.’s 22,24 in eq. 9, and vary with respect to h to obtain h = h( ˆPx). This order of solving
the saddlepoint equations is somewhat simpler than solving the equat ions obtained by independent variations—the
results are, obviously, independent of this order of solution. We ﬁn d
Sred[h] = −N
∫
dx
[
P + 1√
N
ρ
]
ln P + 1
∫ (
N P φ1
[
ln
√
h − ln P − φ1
]
−
√
N ρ
[
ln
√
h + φ1 − ln P
])
, (26)
with φ1 = φ1(h). Now, it is important to note that while Sred appears to include terms with powers of N that should
be ignored at ﬁrst glance, the variation of φ1 with respect to h is O(1) even though φ1 = O(N −1/2). Thus, varying
eq. 26 with respect to h, we ﬁnd
2 N P δφ1
δ
√
h
(
ln
√
h − ln P
)
+ O(
√
N ) = 0
so to leading order in N, √h0 = P, φ0 = 0 so our old result is recovered. However, at the next order, we ﬁ nd no
correction to h of O(N −1/2). Thus the leading ﬁnite size correction is entirely encoded in eq. 24, wit h
√
h = P. At
higher orders in N −1/2, there are, of course, further corrections to φ, and to h.
Consider now the computation of the functional integral expande d about the saddlepoint solution found above.
The division by the volume of the group of diﬀeomorphisms eliminates th e integration over ﬂuctuations in the metric,
so we are left with computing the integral over the ﬂuctuations in φ. This integration gives the operator determinant
det−1/2(−∆ h + N eφ1). This determinant can be computed in the large N limit by the standard van Vleck technique,
explained in Ref. [5], for example. Thus
det
(
−∆ P 2 + N eφ1
) −1/2
∝ exp
(
−
√
N
∫
dxP (x)eφ1/2
)
, (27)

which diﬀers from the corresponding expression in Ref. [2]. Eq. 27 is r eparametrization invariant. Putting everything
together, eq. 8 is given by
N∏
i=1
P (xi) exp
(
−
√
N
∫
dxP (x)eφ1/2 − 1
∫
dxP (x)−1(∂xφ1)2
)
. (28)
Eq. 28 can be used to evaluate the least-square estimate of the inf erred distribution.
Finally, we compute the magnitude of the correction φ1 :
⟨φ1(x)2⟩connected ∼ 1
√
N
, (29)
independent of x, implying uniform convergence to the solution reviewed in Sect. II. This uniform conv ergence is
of great practical importance, since it implies that our estimated dis tribution will make sense even in regions where
P is small. The uniformity can be directly traced to the reparametrizat ion invariance of our computations, which
enables the binning size to be adjusted locally enabling a more accurat e determination of the distribution. As this
uniformity is of great importance for practical applications, it is all t he more gratifying that a re-analysis motivated
by conceptual concerns regarding reparametrization covarianc e leads to a computational improvement.
IV. CONCLUSIONS
The statistical inference problem is of great interest in biophysics, for example, when x is a vector variable. Bialek,
Callan and Strong [2] gave an account of a higher dimensional variant of their theory, but I want to add a few remarks
here regarding a geometric version sketched in [4], taking into accou nt reparametrization invariance. In Sect.’s II
and III, we saw that a reparametrization-covariant Bayesian set -up could be obtained by introducing a metric, and a
scalar ﬁeld, dividing out by the group of diﬀeomorphisms. Since a metr ic in one dimension has only one global degree
of freedom, this left only one local scalar degree of freedom, which is what we expect to need to ﬁt the data, since the
data determines a scalar density on the interval [ x−, x+].
In higher dimensions, the aim is still to determine a scalar density, but the number of local degrees of freedom
in the metric in d dimensions is d(d + 1)/2 and reparametrizations form a d parameter local symmetry group. It is
clear then that a geometric formulation will require a reparametriza tion invariant constraint to reduce the number of
degrees of freedom in the metric, to the required d + 1 local degrees of freedom.
Reparametrization invariant constraints must be given by the vanis hing of tensor quantities, so in this case we have
to construct appropriate tensors from the metric. Investigatin g the constraints provided by the vanishing of various
curvature tensors, since we expect only a scalar local degree of f reedom to survive after imposing the constraint and
dividing out by reparametrizations, we are naturally led to consider c onformally Euclidean metrics, which can indeed
be characterized by the vanishing of a tensor constructed from t he curvature, the Weyl curvature, W δ
αβγ . W vanishes
identically in d ≤ 3, so the geometric theory will only be valid for d > 3.
The proposed generalization of eq. 6 to higher dimensions is now given by
P [hαβ] ∝ exp
(
−
∫
ddx
√
hL
)
δ
(
1 −
∫
ddx
√
h(x)
) ∏
x
δ
(
Wαβγ δ(x)
)
(30)
where L is a scalar constructed out of the metric hαβ, constrained to be a metric of vanishing Weyl curvature, W δ
αβγ ,
and
√
h ≡ det1/2(hαβ) as usual.
A variety of issues, some technical, have to be addressed in order t o deﬁne eq. 30 precisely: (i) The integration
measure for integrating over metrics is notoriously nonlinear; (ii) So lving the constraint W = 0 , and then extracting
the volume of the group of diﬀeomorphisms will leave a Jacobian; (iii) Wh at is an appropriate choice for L, given the
constraint W = 0?
Before tackling the technical issues, consider the logic of what we w ish to do. Metrics with vanishing Weyl curvature
are of the form
hαβ(x) = ∂f δ
∂xα
∂f ǫ
∂xβ e2φ(f (x)) δδǫ ≡ J δ
αJ ǫ
βe2φ(f (x)) δδǫ. (31)
For such metrics,
√
h = det(J)edφ. Now, following our steps in Sect. 2, we want to extremize (ignoring te chnicalities)

ˆS ≡
∫
ddx det(J)edφL[φ(f (x))] + iλ
( ∫
ddx det(J)edφ − 1
)
− N
∫
ddx ˆP (x) [ln det J + φ] , (32)
where N ˆP (x) ≡ ∑ δ(x − xi). Now, ˆP (x) is a density, so ddx ˆP (x) = ddf ˆP (f ), with ˆP (f ) det(J) = ˆP (x). We can
rewrite eq. 32 as
ˆS =
∫
ddf edφL[φ(f )] + iλ
( ∫
ddf edφ − 1
)
− N
∫
ddx ˆP (x) ln det J − N
∫
ddf ˆP (f )φ , (33)
It follows that there are two independent functional variations, o ne with respect to φ(f ), and the other with respect
to f (x), the latter giving
∂
∂xα
[
ˆP (x) ∂xα
∂f δ (x)
]
= 0. (34)
Eq. 34 is the analogue of eq. 21, and determines the co¨ ordinates f α in terms of the co¨ ordinates xα in which the data
is presented, and is reparametrization-invariant because P (x) is a density, not a function. Consider the special case
x = f : eq. 34 is the requirement that the density P in the co¨ ordinatesf is the constant density, exactly analogous to
our result in one dimension, with φ a constant.
It is amusing to explicitly construct f α co¨ ordinates which satisfy
det ∂f α
∂xβ = ˆP (x).
We construct these co¨ ordinates essentially iteratively as a one-d imensional distributions with parameters. If we
couldﬁnd f :
f 1(xα) = f 1(x1), f 2(xα) = f 2(x1, x2), . . .
then we would get
det ∂f α
∂xβ =
∏
α
∂f α
∂xα ,
so the problem is one of factoring ˆP appropriately. Deﬁne
P 1(x1) ≡
∫
ˆP (x1, x2, x3, . . .)
d∏
α=2
dxα,
P 2(x1, x2)P 1(x1) ≡
∫
ˆP (x1, x2, x3, . . .)
d∏
α=3
dxα,
.
.
.
P d(xα)
d−1∏
β=1
P β ≡ ˆP (xα), (35)
then
f 1(x1) ≡
∫ x1
P 1(z)dz,
f 2(x1, x2) ≡
∫ x2
P 2(x1, z)dz,
.
.
. (36)
is an explicit set of coordinates in which the given density ˆP is constant. Notice that each f α varies from 0 to 1, as
beﬁts integrals of one-dimensional normalized probability distributio ns, P α.

As explained in Ref.’s [2,4], we are interested in extracting a continuum d istribution from some set of binned data,
as independent as possible of the binning. In renormalization group t erminology, we are interested in the ‘infrared’
properties of the data, independent of the ‘cutoﬀ’. The terms mos t relevant in the infrared are precisely the terms
with the fewest derivatives. Thus, since Einstein-Hilbert action is a t otal derivative for conformally Euclidean metrics,
we consider L ≡ − 1
2 R∆ −1R, where ∆ is the Laplacian, and R is the Ricci scalar constructed from hαβ. This Lagrange
density is non-local for general metrics in dimensions larger than tw o, but is local for metrics with W = 0,
∫
ddx
√
h L = (d − 1)2
∫
ddf e(d−2)φ(f )δαδ ∂φ
∂f α
∂φ
∂f δ . (37)
The steps carried out in the previous section for determining the ﬁn ite N corrections can be carried out in an exactly
analogous fashion for this action for the leading correction, using t he Laplacian in d dimensions in the f co¨ ordinates
explicitly constructed above. At higher orders in N, the nonlinearity of L leads to a distinctly diﬀerent computation.
I hope to address these issues, along with the technical issues men tioned above, elsewhere.
In conclusion, I have computed the ﬁnite sample size corrections to the geometric solution of the statistical inference
problem I gave in earlier work. The uniform convergence of these co rrections should be important for applications so
it is of interest to test this theoretical construction in practice.
I am grateful to V. Balasubramanian and C. Callan for conversation s. This work was supported in part by NSF
grant PHY96-00258.
[1] J. Earman, Bayes or Bust?: A Critical Examination of Bayesian Conﬁrmat ion Theory (MIT Press, Cambridge, 1992); from a
physics perspective, the inference problem has been discus sed by V. Balasubramanian, “Statistical Inference, Occam’ s Razor
and Statistical Mechanics on the Space of Probability Distr ibutions”, Neural Computation (in press), cond-mat/96010 30;
“Occam’s Razor for Parametric Families and Prior on the Spac e of Probability Distributions”, in Maximum Entropy and
Bayesian Methods, K.M. Hanson and R.N. Silver (eds.), (Kluw er Academic, Amsterdam, 1996); “A Geometric Formulation
of Occam’s Razor for Inference of Parametric Distributions ”, adap-org/9601001; and references therein
[2] W. Bialek, C.G. Callan and S.P. Strong, Phys. Rev. Lett. 77 4693(1996)
[3] C. Callan explained to me the biophysical applications i n pattern recognition which motivated the investigation in Ref. [2].
See also D. Mumford, Pattern Theory: A Unifying Perspective , in Proceedings, First European Congress of Mathematicians ,
vol. 1, eds. A. Joseph, F. Mignot, B. Prum and R. Rentschler (B irkh¨ auser Verlag, Basel, 1992), and references therein.
[4] V. Periwal, Phys. Rev. Lett. 78 4671(1997)
[5] S. Coleman, Ch. 7, Aspects of symmetry (Cambridge University Press, Cambridge, 1975)