Sufficiency on the Stock Market

0.1 MB·

by  Peter Harremoës

Information and Sufficiency on the Stock Market
Peter Harremoës
Copenhagen Business College
Copenhagen
Denmark
Email: [email protected]
Abstract—Itiswell-knownthatthereareanumberofrelations leads to Bregman divergences and sufficiency lead to local
6 betweentheoreticalfinancetheoryandinformationtheory.Some proper scoring rules. The basic result is that any strictly local
1 of these relations are exact and some are approximate. In this proper scoring rule is proportional to logaritmic score. The
paper we will explore some of these relations and determine
0 linkbetweeninformationtheoryandstatisticsisnowverywell
under which conditions the relations are exact. It turns out
2
that portfolio theory always leads to Bregman divergences. established [6].

n The Bregman divergence is only proportional to information Convex optimization also appear in thermodynamics and
a divergence in situations that are essentially equal to the type statistical mechanics where the goal is to extract as much
J of gambling studied by Kelly. This can be related an abstract
energy as possible from some physical system. The notion
7 sufficiency condition.
of entropy obviously play an important role in both theories,
2
I. INTRODUCTION butthebestinterpretationhasbeendebatedeversinceShannon

] decidedtocallhisquantityentropy.Sinceallthesetheoriesare
T
The relation between gambling and information theory has related we also get a link between finance theory and physics
.I been known since Kelly [1]. Later Kelly’s theory has been sothereisawholetopiccalledeconophysicswhereideasfrom
s
extendedtotradingofassets,butthelinktoinformationtheory physics are applied to economic systems. We hope that the
c
[ is weaker than in the case of gambling [2]. In both gambling presentpaperwillhelptounderstandtowhatextendquantities

theory and more general portfolio theory logarithmic terms in finance are really proportional to quantities in information
1
v appear because we are interested in the exponential growth theory, statistics, or physics.
3 rate. In this paper we shall demonstrate that portfolio theory The general idea of using Bregman divergencesfor convex
9 consist of two parts. The general part is related to Bregman optimization was presented in [7]. In the present paper we
5 divergences and this part is shared with a number of other will develop the theory further. Therefore there will be some
7
convex optimization problems. If a sufficiency condition is overlap between then the presentation in [7] and the present
0
. imposedonthegeneraltheorywearriveata theorywherethe paper.Thesecondgoalofthispaperisapplythegeneraltheory
1 Bregman divergence reduces to information divergence. The to portfolio theory.
0
sufficiency is essentially equal to Kelly’s theory of gambling.
6
II. OPTIMIZATION
The general theory of convex optimization and Bregman
1
: divergences has a number of important applications. In each Assume thatour knowledgeofa system can be represented
v
of the applications we get a strong link to information theory by an element in a convex set S that we will call the state
i
X if a sufficiency condition is imposed. Therefore sufficiency space. The simplest case of a state space is the simplex of
r conditions will lead to strong relations between the different probability measures on a set. In quantum information theory
a
applications. thestatespaceisthesetofdensitymatricesonaHilbertspace.
In information theory an important goal is to compress. For states s0 and s1 and t ∈ [0,1] the convex combination
As long as we restrict to uniquely decodable codes we get a (1−t)·s0+t·s1 is identified with the mixed state where s0
Bregmandivergence.Thesufficiencyconditioncorrespondsto is taken with probability 1−t and the state s1 is taken with
allowingcodewordsrealvaluedlengthwhichisrelevantwhen probability t. The pure states are the extreme points of the
weallowblockcodeswithnoupperlimitontheblocklength. state space.For simplicity we will assume thatthe state space
Thisleadstothewidespreaduseofinformationdivergencein is a finite dimensional convex compact set.
information theory. The link between information divergence Let A denote a subset of the feasible measurements such
and the notion of sufficency was emphazied already by Kull- thata∈A mapsS into a distributionon the real numbersi.e.
back and Leibler in 1951 in the paper entitled “Information a random variable. The elements of A may represent actions
and Sufficiency” [3]. (decisions) that lead to a payoff like the score of a statistical
In statistics the idea of scoring rules has its roots in the decision,theenergyextractedbyacertaininteractionwiththe
1920’s in the Dutch book theorem by Ramsay and de Finetti. system, (minus)the lengthof a codewordofthe nextencoded
McCathy [4] studied scoring rules in a more systematic way inputletterusingaspecificcodebook,ortherevenueofusing
and Dawid, Lauritzen and Parry [5] have recently extended acertainportfolio.Iftheactionaisappliedtothestate sthen
the notion of proper local scoring rules. Proper scoring rules we get a random variable a(s) that we will allow to take
values in R ∪ {−∞}. For each s ∈ S we define F (s) = Theorem 4. If (t1,t2,...,tn) is a probability vector on the
supa∈AE[a(s)]. Without loss of generality we may assume statess1,s2,...,sn with s¯= ti·si and aopt is the optimal
thatthe setof actionsA isclosed so thatwe mayassume that action for s¯then
P
thereexistsa∈AsuchthatF (s)=E[a(s)] andin thiscase
C ≥inf t ·D (s ,a)+D (s¯,a ).
we say that a is optimal for s. We note that F is convex but F a i F i F opt
F need not be strictly convex. If a is an action anXd s is optimal then
opt
Definition 1. If F (s) is finite the regret of the action a is
supD (s ,a)≥C +D (s ,a).
F i F F opt
defined by i
DF (s,a)=F (s)−E[a(s)] (1) III. SUFFICIENCY
Proposition 2. The regret DF has the following properties: Let(sθ)θ denoteafamilyofstatesandletΦdenoteanaffine
transformation S → T where S and T denote state spaces.
• DF (s,a)≥0 with equality if a is optimal for s. ThenΦissaidtobesufficientfor(s ) ifthereexistsanaffine
• If aˆ is optimal for the state sˆ = ti · si where transformation Ψ:T →S such thaθt Ψθ (Φ(s ))=s .
θ θ
(t1,t2,...,tℓ) is a probability vector thPen We define a transformation Φ to be an isomixture if Φ
ti·DF (si,a)= ti·DF (si,aˆ)+DF (sˆ,a). has the form Φ = ki=1pi · Φi where (p1,p2,··· ,pk) is
a probability vector and Φ is a isometry, i.e. a bijective
i
• Xti·DF (si,a) is minXimal if ais optimal for ti·si. transformation of thePstate into itself. We say that the regret
IfthePstateiss1 butoneactsasifthestateweres2 onPesuffers DF on the state space S satisfies the iso-sufficiency property
a regret that equals the difference between what one achieves if
and what could have been achieved. DF (Φ(s1),Φ(s2))=DF (s1,s2) (5)
Definition 3. If F (s1) is finite the regret is defined by for any isomixture S → S that is sufficient for (s1,s2).
The notion of sufficiency as a property of divergences was
DF (s1,s2)=inafDF (s,an) (2) introduced in [9]. The crucial idea of restricting the attention
wheretheinfimumistakenoveractionsa thatareoptimalfor totransformationsofthestate spaceintoitself wasintroduced
s2. in [10].
The center of a convex set S is the set of point in S that
If there exists a uniqueaction a such that F (s)=E[a(s)]
are invariant under isometries of S. Note that the center is
then F is differentiable which implies that the regret can be
convexand non-empty[11]. If the center of the state space is
written as a Bregman divergence in the following form
not a point there are many Bregman divergences that satisfy
DF (s1,s2)=F (s1)−(F (s2)+hs1−s2,∇F (s2)i). (3) the sufficiency condition.
In the context of forecasting and statistical scoring rules the Proposition 5. Let G denote the set of isometries of a state
use of Bregman divergences dates back to [8]. spaceS andletµ denotethe Haarprobabilitymeasureon G.
Bregman divergences satisfy the Bregman identity Let Φ denote the projection s → g(s) dµ. Let F denote a
concavefunctiononthecenterofS.ThenDF (Φ(s1),Φ(s2))
ti·DF (si,s˜)= ti·DF (si,sˆ)+DF (sˆ,s˜) defines a Bregman divergence onR S that satisfies the iso-
but ifXF is not differentiabXle this identity can be violated. If sufficiency condition.
the state s2 has the unique optimal action a2 then Proposition 6. Assume that S is a state space. If the diver-
genceD satisfiestheiso-sufficiencypropertythenthereexists
F(s1)=DF (s1,s2)+E[a2(s1)] (4) a F˜ suchFthat
sothefunctionF canbereconstructedfromD exceptforan
F
affinefunctionofs1. SimilarlythedivergenceDF isuniquely DF˜(s1,s2)=DF (s1,s2)
determined by the function F. and F˜(Φ(s))=F˜(s) .
Consider the case where the state is not know exactly but
If the state space is a one dimensional simplex then the
weknowthats∈S forsomesetofstates.Theminimaxregret
only sufficient transformation is the reflection and the above
of the set S is defined as
conditiononF issufficienttoconcludethatEquation5holds.
C =infsupD (s ,a).
F F i
a i Proposition 7. If the state space has the shape of a ball then
Using general minimax results we get anyfunctionF ontheballthatisconcaveandinvariantunder
rotations satisfies the iso-sufficiency condition.
C =supinf t ·D (s ,a)
F i F i Proof: Assume that the isomixture Φ is sufficient for
~t a Xi {s0,s1}. Then Φ is also sufficient for any affine conbination
where the supremum is taken over all probability vectors ~t ofs0 ands1. Inparticularwe mayreplaces0 ands1 byaffine
supported on S. This result can improved. combinationsfor the form sti =(1−ti)·s0+ti·s1 that are
extreme points in S. Since Φ is assumed to be sufficient it Example 11. A specialasset is the safe asset wherethe price
maps s into an extreme points. Hence Φ acts as a rotation relative is 1 for any feasible price relative vector. Investingin
ti
on the intersection of the state space and the affine span this asset correspondsto place the moneyat a safe place with
of s1,s2 and U. Since F is invariant under rotations the interest rate equal to 0 % .
divergence D is also invariant under rotations implying that
F A portfolio is an asset given by a probability vector~b =
DFTh(Φes(ism1)pl,eΦst(csa2s)e)=oDfaFb(asl1l,iss2a)n.interval,whichcorresponds (b1,b2,...,bk) where for instance b5 =0.3 means that 30 %
ofthemoneyisinvestedinassetno.5.Thetotalpricerelative
to the probabilitymeasureson a binaryalphabet. This special
case was discussed in [10]. The balls in dimensions 2, 3, and isX1·b1+X2·b2+···+Xk·bk = X~,~b .Ifanassethasthe
property that the price relative is only positive for one of the
5 correspond to density matrices of a 2 dimensional Hilbert D E
feasible price relative vectors,then we may call it a gambling
space over the real numbers, over the complex numbers, and
asset. For any set of possible assets we may extend the set
over the quarternions.
of assets by a numberof idealgambling assets so that any of
We say that the states s0 and s1 are orthogonal and write
the possible assets can be written as a portfolio of the ideal
s1⊥s2 if there exists an affine function φ : S → [0,1] such
gambling assets. This can be done without changing the set
φ(s0) = 0 and φ(s1) = 1. The following theorem can be
of feasible pricerelative vectors.Thereforethe set of possible
proved by the same technique as [7, Thm. 4] except that we
portfolios may be considered as a convex subset of a set of
will make sufficientprojectionsby takingthe mean actionsof
portfolios of some ideal gambling assets.
a groups equipped with the Haar probability measure.
We now consider a situation where the assets are traded
Theorem 8. Assume that the state space S satisfies the once every day. For a sequence of price relative vectors
following properties: X~1,X~2,...X~n and a constant re-balancing portfolio ~b the
1.Forandtwopurestatess1ands2thereexistsanisometry wealth after n days is
of S such that Φ(s1)=s2. n
2. For any three pure states s1,s2, and s3such that s1⊥s3 Sn = X~i,~b (6)
ands2⊥s3 thereexistsanisometryofS suchthatΦ(s1)=s2 iY=1D E
and Φ(s3)=s3. n
3.Thestatespacehasatleastthreeorthogonalpurestates. = exp log X~i,~b (7)
4. Anystate can be written as amixture oforthogonalpure Xi=1 (cid:16)D E(cid:17)!
states. = exp n·E log X~,~b (8)
If the regret D satisfies the iso-sufficiency property given
F where the expectation is t(cid:16)aken whith rDespectEtio(cid:17)the empirical
by Equation 5, then D is uniquely determined except for a
F
multiplicative factor. distributionofthe pricerelativevectors.Here E log X~,~b
Remark 9. Condition 4 seems to be redundant, but we have is proportionalto the doubling rate and is denothed WD ~b,PEi
not been able to prove this. where P indicates the probability distribution of X~. Ou(cid:16)r goa(cid:17)l
When the state space is a simplex the uniquely determined istomaximizeW ~b,P bychoosinganappropriateportfolio
divergenceisinformationdivergenceandwhenthestatespace ~b. In [2] and [7] it(cid:16)was(cid:17)tacitly assumed that a unique optimal
isdensitymatricesonacomplexHilbertspacewegetquantum
portfolio exists but this is not always the case. Here we will
relative entropy.
not assume uniqqueness.
Lemma 10. Assmue that the state space satisfies the condi- Definition 12. Let~b1 and~b2 denote two portfolios. We say
tionsin Theorem8. Ifs0⊥s1 thenanyoptimalactiona fors1 that ~b1 dominates ~b2 if X~j,~b1 ≥ X~j,~b2 for any j =
satisfies E[a(s0)]=−∞.
1,2,...,n.Wesaythat~b1DstrictlyEdomiDnates~b2Eif X~j,~b1 >
Proof:Since s0 ands1 are orthogonalandthe conditions
in the previoustheoremis fulfilled the we have thatthe regret X~j,~b2 for any j =1,2,...,n. D E
prersotpriocrtteidontaoltthoeinlifnoermseagtimonendtiv{etrg∈en[0c,e1,]b|u(t1in−fotr)msa0ti+ontsd1iv}eirs- D ForaEvector~v =(v1,v2,...,vk)∈Rk thesupportsupp(~v)
is the set of indices i such that v > 0. We note that
m∞geun.mcHeieesnqctuaeakleisnn∞faov(feForr(aoscr0tt)iho−ongsEotnh[aaalt(dasir0set)r]io)bpu=ttiimo∞nasl fswoorhDesr1Fe.(Ttshh0ee,rsien1f)foo=rie- iif∈~b1sustprpictl~by2 dosmucinhattehsat~b~b21isftraicntdlyoidnolymiinfattehser~eei ewxhisetrse a~eni
E[a(s0)]=−∞ for any action a that is optimal for s1. denotesthe(cid:16)i’t(cid:17)hbasis vector.The consequenceis that we may
removeassetsnumberiif~e isstrictlydominatedbecauseone
i
IV. PORTFOLIO THEORY willneverputanymoneyonthatparticularasset.Similarly,~b1
Let X1,X2,...,Xk denote price relatives for a list of k dominates~b2 ifandonlyifthereexistsani∈supp ~b2 such
assets.ForinstanceX5 =1.04meansthatassetno.5increases that~b1dominates~ei.Wedonotdecreasethemaxima(cid:16)ldo(cid:17)ubling
its value by 4 %. rate by removing assets that are dominated, but sometimes
assets that are dominated but not strictly dominated may lead measuresare projected (reverse informationprojection)into a
to non-uniquenessof the optimal portfolio. convex set. Here we should note that information divergence
is convex but not strictly convex in the second argument.
Definition 13. A set A of assets is said to dominate the set
Therefore the reversed information may be non-unique.
of assets B if anyasset in B is dominatedby a by a portfolio
of assets in A.
V. SUFFICIENT PORTFOLIOS
Proposition 14. If~b0 is optimal for the distribution δ~v then Lemma 16. Assume that there are only two price relative
the support of~b is a subset of the support of ~v.
vectorsandthatthesetofassetsisminimaldominating.Ifthe
Proof:IfP =δ thenE log X~,~b =log ~v,~b .The Bregman divergence
~v
portfolio~b isaprobabilitydisthributDionovEeirstocksDsoifEwelet W ~b ,P −W ~b ,P (11)
P Q
~b∗ denote the conditional distribution of~b on the support of
(cid:16) (cid:17) (cid:16) (cid:17)
~v. Then isproportionaltoinformationdivergenceD(PkQ)thenthere
log ~v,~b0 ≤log ~v,~b∗ are only two gambling assets.
D E D E
with equality if and only if the supportof~b is a subset of the Proof: Let
sisupapsourbtsoeft~vo.f Tthheerseufpoproer~bt o=f ~~vb.0 implies that the support of~b X~ = (X1,X2,...,Xk)
Let~b denote a portfolio that is optimal for P. The regret Y~ = (Y1, Y2, ..., Yk)
P
of choosing a portfolio according to Q when the distribution
denote the two price relative vectors. If P = (s,t) then the
is P is given by the Bregman divergence
vector~b=(b1,b2,...,bn) is log-optimal if and only if
W ~b ,P −W ~b ,P .
P Q
X Y
i i
(cid:16) (cid:17) (cid:16) (cid:17) s + t ≤ 1
If~bQ is not uniquelydetermined we take a minimum over all b1X1+···+bkXk b1Y1+···+bkYk
~b that are optimal for Q. for all i ∈ {1,2,...,k} with equality if b > 0. Since we
i
have assumed that none of the assets are dominated by other
Example 15. If the assets are orthogonalgamblingassets we
portfoliosonlytwooftheseinequalitiescanholdwithequality.
get the type of gambling described by Kelly. There will be
one-to-onecorrespondencebetween price relative vectors and Therefore we may assume that only b1 and b2 are positive.
Hence we may assume that there are only two assets.
assets. For a probability disttribution P over price relative
vectors the optimal portfolio ~b is a vector with the same Let δ1 denote the measure concentrated on X~ and let δ2
coordinates as the probability vePctor P. We have denotethe measureconcentratedonY~. Since the measuresδ1
W ~bP,P −W ~bQ,P =D(PkQ) (9) a−n∞dδ.2Naorwe orthogonalLemma 10we have thatW (cid:16)~bδj,δi(cid:17)=
(cid:16) (cid:17) (cid:16) (cid:17)
so the sufficiency condition is fulfilled in gambling. W ~b ,δ = E log X~,~b
δj i δi δj
If a set of possible assets it embedded as a subset C in a (cid:16) (cid:17) = log hX~ ,D~b Ei
set of ideal gambling assets then C may be identified with i δj
a convex set of probability distributions. Now maximizing D E
W ~b,P overpossibleportfolios~bisthesameasminimizing so that X~i,~bδj = 0. Since the support of ~bδi is a subset
the(cid:16)regre(cid:17)tgivenby(9)overQ∈C inthesetofportfoliosover of the suDpport ofEX~i we have that~bδi⊥~bδj. Therefore~bδ1 and
ideal gambling assets. Therefore~b may be identified with a ~b must be proportional to the basis vectors. Since~b and
Q δ2 δ1
reversed information projection of Q on C. ~b are vectorsin a 2-dimensionalspace and their coordinates
δ2
As proved in [2] the regret satisfies arenon-negativewe havethat~b mustproportionalto a basis
δi
vector. Since X~ ,~b = 0 for i 6= j we have that X~ is
W ~b ,P −W ~b ,P ≤D(PkQ). (10) i δj i
P Q
parallel with~bD. E
(cid:16) (cid:17) (cid:16) (cid:17) δi
In the set of portfolios over ideal assets there is a on-
Theorem 17. Assume that none of the assets are dominated
to-one correspondence between mixed states and portfolios.
Therefore maximizing W ~b,P over~b in the original set of by a portfolio of the other assets. If the Bregman divergence
portfolioscorrespondsto m(cid:16)inim(cid:17)izingthe regretW ~b ,P − W ~b ,P −W ~b ,P (12)
P P Q
W ~b ,P over Q which again corresponds to m(cid:16)inimiz(cid:17)ing (cid:16) (cid:17) (cid:16) (cid:17)
Q is proportionalto information divergence D(PkQ) the mea-
D((cid:16)PkQ)u(cid:17)ndertheconditionthat~b ∈C inasetofportfolios sures P and Q are supported by k distinct price rela-
Q
on orthogonal gambling assets. The inequality (10) therefore tive vectors of the form (o1,0,0,...0), (0,o2,0,...0), until
states that information divergencedecreases when probability (0,0,...o ).
k
Proof: Assume that there exists a constant c > 0 such VI. CONCLUSION
that The link between portfolio theory and information theory
W ~b ,P −W ~b ,P =c·D(PkQ). (13) works on two levels. Parts of the theory can be stated and
P Q
proved on the level of convex optimization, where Bregman
(cid:16) (cid:17) (cid:16) (cid:17)
If~b =~b then divergences and related concepts play a central role. If we
P Q
further impose a sufficiency condition we have, essentially,
W ~b ,P −W ~b ,P =0 to restrict our attention to gambling as described by Kelly.
P Q
Adding certain assets that are dominated does not make any
(cid:16) (cid:17) (cid:16) (cid:17)
significant changes to the theory. In the case of gambling
and D(PkQ)=0 and P =Q. Therefore the mapping P →
~b isinjective.Thevectors~b formasimplexwithk extreme the correspondence between portfolio theory and information
P P
theory becomes perfect. Therefore the link between general
points.ThereforethesimplexofprobabilitymeasuresP hasat
portfolio theory and information theory is convayed by gam-
mostk extremepoints,soP issupportedonatmostk distinct
vectors that we will denote X~1,X~2,...,X~k. bling theory.
Assume that X~ and Y~ are two vectors of price relatives. Information divergence was introduced by Kullback and
Leiblerinthepaperentitled“OnInformationandSufficiency”.
Then Equation 13 holds for probability vectors restricted to
In the present paper we have made the notion of sufficiency
the set X~,Y~ . From Lemma 16 it follows that X~ and Y~
more explicit for portfolio theory. The introduction of ideal
are orthogonal. Therefore all the price relative vectors are
n o gambling assets paralellels the use of microscopic states as
orthogonal,andhavedisjointsupports.Sincethepricerelative
opposed to macroscopic states in physics. For microscopic
vectorshavedisjointsupport,anassetcanonlyhaveapositive
states we have reversibility and conservation of energy. Sim-
price relative for one of the price relative vectors. Therefore
ilarly, gambling corresponds to two-person zero sum games
each price relative vector has one asset that dominates any
where money is the conserved quantity. As we have seen
other asset in the support of the price relative vector. Since
these correspondencies are consequences of the sufficiency
we have assumed noneof the assets are dominatedeach price
condition.
relative vector is supported on a single asset.
If the price relative vectorsare as in Theorem 17 we are in ACKNOWLEDGEMENT
the situation of gambling introduced by Kelly [1]. ThaauthorwanttothankPrasadSanthanamforinvitingme
to Electical Engineering Department, University of Hawai’i,
Corollary 18. Assume that the Bregman divergence
where this paper was written.
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