3. SITUACIÓN ACTUAL DE LA ACCIÓN POPULAR
3.4. REQUISITOS FORMALES
So far we have discussed investors in a (unconditional) horse race as an ide- alized setting for a decision maker who uses a probabilistic model for a single variable. In many practical situations, however, we are interested in condi- tional probabilities of a variableY given a variableX. In order to evaluate and build such probabilistic models, we introduce the notion of the conditional
86 Utility-Based Learning from Data
horse race. We shall see that most of the horse-race results from the previous sections can be generalized to the conditional horse race.
Throughout this section, we denote joint probability measures bypX,Y or
qX,Y, conditional probability measures by pY|X or qY|X, and the marginal
X-probability measures by pX orqX. Later in this book, when the meaning
is clear from the context, we shall drop the upper indices.
Conditional horse race
We generalize Definition 3.1 as follows.
Definition 3.7 (Conditional horse race) A conditional horse race is charac- terized by the discrete random variableY with possible states in the finite set
Y and the discrete random variableX with possible states in the finite setX; we identify each element ofY with a horse and each element ofX with a par- ticular piece of side information. An investor can place a bet that Y =y∈ Y
after learning that X =x∈ X, which pays the odds ratio (payoff) Oy|x>0
for each dollar wagered ifY =y, and0, otherwise.
This conditional horse race (see, Friedman and Sandow (2003b)) is slightly more general than the horse race with side information from Cover and Thomas (1991), Chapter 6; the latter is restricted to payoffs that are in- dependent ofX, but is otherwise the same as the conditional horse race.
We also generalize the notions of a bank account, introduce a worst (overx∈
X) bank account, and introduce the homogeneous expected return measure.
Definition 3.8 (Conditional bank account) Given that X =x, the riskless conditional bank account payoff, Bx, is
Bx= P 1
y∈Y
1
Oy|x
. (3.21)
Definition 3.9 (Worst conditional bank account) The worst conditional bank account has payoff
B= inf
x∈XBx. (3.22)
Definition 3.10 The conditional homogeneous expected return measure is given by pY|X(h)= pYy||xX(h)= Bx Oy|x , y∈ Y, x∈ X . (3.23)
The above pY|X(h) has the required properties of a conditional probability
measure, i.e.,pyY||xX(h)>0, ∀x∈ X, y∈ Y, andP
y∈Yp
Y|X(h)
y|x = 1, ∀x∈ X.
The conditional expected return, given X = x, under pY|X(h) is B x; this
return depends on the value ofX, which is known before bets are placed, but is independent of the value ofY.
The Horse Race 87
Conditional investor
We generalize Definition 3.4 as follows.
Definition 3.11 (Conditional investor) A conditional investor is a gambler who invests$1 in a horse race, i.e., a gambler who, after having learned that
X =x, allocates by|xto the eventY =y, where
X
y∈Y
by|x= 1, ∀x∈ X . (3.24)
We denote the conditional investor’s allocation by
b=
by|x, x∈ X, y∈ Y . (3.25)
Below we shall often refer to the conditional investor simply as investor, unless we need to make a distinction between a conditional and an unconditional investor.
Expected conditional wealth growth rate
We generalize Definition 3.5 as follows.
Definition 3.12 The expected conditional wealth growth rate corresponding to a probability measurepX,Y and a betting strategyb is given by
WY|X b, pX,Y
=EpX,Y [log (b,O)] =
X
y∈Y,x∈X
pX,Yx,y log(by|xOy|x). (3.26)
Below we shall often refer to the expected conditional wealth growth rate simply as expected wealth growth rate.
We generalize Lemma 3.1, which states that, asymptotically, the gambler’s wealth grows exponentially withWY|X(b, p) as growth rate.
Lemma 3.2 The wealth, Wn, of an investor after n independent successive
bets in a conditional horse race, where the horses win with probabilities given by pX,Y, is related to the expected wealth growth rate as
WY|X b, pX,Y = lim n→∞ logWn W0 n . (3.27)
Proof:The investor’s wealth aftern independent, successive bets is
Wn=W0 n
Y
i=1
byi|xiOyi|xi , (3.28) where (xi, yi) is the realizations of (X, Y) in theith bet. So we have
lim n→∞ logWn W0 n = limn→∞ Pn i=1log(byi|xiOyi|xi) n
88 Utility-Based Learning from Data
The lemma follows then from Definition 3.12.2
Conditional Kelly investor
We generalize Definition 3.6 as follows.
Definition 3.13 (Conditional Kelly investor) A conditional Kelly investor is a conditional investor (in the sense of Definition 3.11) who allocates his wealth so as to maximize his expected wealth growth rate according to the model he believes.
Below we shall often, for the sake of brevity, refer to the conditional Kelly investor simply as Kelly investor.
Generalizing Theorem 3.1, the following theorem explicitly states the in- vestment strategy chosen by a Kelly investor.
Theorem 3.5 A conditional Kelly investor who believes the conditional prob- ability measurepY|X allocates his assets to the horse race according to
b∗y|x
pY|X=pYy||xX. (3.30)
Proof:It follows from Definition 3.13 and Definition 3.12 that
b∗pY|X= arg max {b:P y∈Yby|x=1} X y pYy||xXlog(by|xOy|x). (3.31)
In order to solve this optimization problem, which is convex, we write down its Lagrangian: L(b, λ) =X y pYy||xXlog(by|xOy|x)−λx X y by|x−1 ! . (3.32)
The optimal allocation,b∗, is the solution of
0 = ∂L(b, λ) ∂by|x b=b∗ = pYy||xX b∗ y|x −λx pXx , which is b∗y|x= py|x λx . (3.33) In order to find the value,λ∗
x, that corresponds to the solution of our convex
problem, we have to solve
1 =X
y
The Horse Race 89
forλx. We findλ∗x= 1. So we haveb∗y|x=p Y|X y|x .2
Entropy and wealth growth rate
As we have done for the case of the (unconditional) horse race, we relate the entropy to the expected conditional wealth growth rate of a conditional Kelly investor.
Theorem 3.6 A conditional Kelly investor who knows that the horses in a conditional horse race win with the probabilities given by the measure pX,Y
has the expected wealth growth rate
W∗ pX,Y pY|X=W∗∗ pX,Y −HY|X pX,Y , (3.35) where
Wp∗∗X,Y =EpX,Y [logO] =
X
y∈Y,x∈X
pX,Yx,y logOy|x (3.36)
is the wealth growth rate of a clairvoyant investor, i.e., of an investor who wins every bet.
Proof:It follows from Theorem 3.5 that a Kelly investor who believes the probability measure pY|X allocates according to
b∗
y|x
pY|X=pY|X
y|x . (3.37)
Inserting this equation into Definition 3.12, we obtain the pX,Y-expected
wealth growth rate of the Kelly investor as
Wp∗X,Y
pY|X= X
y∈Y,x∈X
pX,Yx,y log
pYy||xXOy|x
= X
y∈Y,x∈X
pX,Yx,y logp Y|X y|x +
X
y∈Y,x∈X
pX,Yx,y logOy|x
=−HY|X pX,Y
+Wp∗∗X,Y (from (3.36) and Definition 2.8).2 In Section 7.2 we shall use Theorem 3.6 as the starting point for a general- ization of the conditional entropy.
The following theorem, which is a generalization of Theorem 3.3, relates the relative entropy to the expected conditional wealth growth rate of a Kelly investor.
Theorem 3.7 In a conditional horse race where horses win with the probabil- ities given by the measure pX,Y, the difference in expected conditional wealth
growth rates between a conditional Kelly investor who knows the probability measurepY|X and a conditional Kelly investor who believes the (misspecified)
probability measureqY|X is given by
Wp∗X,Y
pY|X−Wp∗X,Y
90 Utility-Based Learning from Data
Proof: It follows from Theorem 3.5 that a conditional Kelly investor who believes the probability measure pallocates according to
b∗y|x
pY|X=pYy||xX, (3.39)
and a conditional Kelly investor who believes the probability measure qX,Y
allocates according to
b∗y|xqY|X=qYy||xX. (3.40)
Inserting the above two equations into Definition 3.12, we obtain the pX,Y-
expected wealth growth rate difference
Wp∗X,Y pY|X−Wp∗X,Y qY|X= X y∈Y,x∈X
pX,Yx,y log(p Y|X y|x Oy|x)
− X
y∈Y,x∈X
pX,Yx,y log(q Y|X y|x Oy|x)
= X
y∈Y,x∈X
pX,Yx,y log
pYy||xX qYy||xX =DY|XpX,YkqY|X (3.41) (from Definition 2.9).2
In Section 7.3, where we consider investors with more general risk- preferences, we will use Theorem 3.7 as the starting point for a generalization of the conditional relative entropy.
Next, we generalize Theorem 3.4.
Theorem 3.8 In a conditional horse race where horses win with the proba- bilities given by the measurepX,Y, the expected conditional wealth growth rate
of a conditional Kelly investor who believes the measureqY|X can be computed
as Wp∗X,Y qX,Y =EpX[logB] +DY|X pX,YkpY|X(h)−DY|XpX,YkqY|X . (3.42)
The Horse Race 91
investor who believes the measureqY|X is
Wp∗X,Y
qY|X=Wb∗qY|X, pX,Y
= X
y∈Y,x∈X
pX,Yx,y log
b∗y|xqY|XOy|x
= X
y∈Y,x∈X
pX,Yx,y log
qYy||xXOy|x (by Theorem 3.5) = X y∈Y,x∈X pX,Y x,y log pYy||xX pYy||xX(h) BxqyY||xX pYy||xX (by Definition 3.10) =EpX[logB] +DY|X pX,YkpY|X(h)−DY|XpX,YkqY|X (by Definition 2.9).2
It follows from Theorem 3.8 that the conditional Kelly investor has an expected conditional wealth growth rate in excess of the bank account growth rate only if the measure he believes is closer (in the relative entropy sense) to the measurepY|X than the homogeneous measure is.
Increase of expected wealth growth rate due to side information
Theorem 3.9 The increase in the expected wealth growth rate for a condi- tional Kelly investor in a conditional horse race due to the information pro- vided by the variable X is
∆W =I pX,Y
+W∗∗
pX,Y −Wp∗∗X×pY , (3.43)
whereI pX,Y
is the mutual information,W∗∗
pX,Y is the expected wealth growth
rate of a clairvoyant investor, andW∗∗
pX×pY is the expected wealth growth rate
of a clairvoyant investor in a horse race whereX and Y are independent. Proof:We compare the Kelly investor who knows the information provided byX, and therefore knows the measurepY|X, to an investor who doesn’t know
the information provided byX, and therefore believes the measure ˆpY|Xwith
ˆ pYy||xX =pY y , ∀x∈ X , (3.44) i.e., ˆ pX,Y =pX×pY . (3.45)
92 Utility-Based Learning from Data
The difference in expected wealth growth rate between these two investors is
∆W =Wp∗X,Y pY|X−Wp∗X,Y ˆ pY|X =−HY|X pX,Y +Wp∗∗X,Y +HY|X pˆX,Y −Wpˆ∗∗X,Y (by Theorem 3.6) =I pX,Y +Wp∗∗X,Y −Wp∗∗X×pY
(by Definitions 2.8 and 2.10, (3.44) and (3.45)).2
As a straightforward consequence of Theorem 3.9 and (3.36), we have the following corollary.
Corollary 3.1 In a conditional horse race where the odds ratios are indepen- dent ofX, i.e., whereOy|x=Oy, ∀x∈ X, the increase in the expected wealth
growth rate for a conditional Kelly investor in a conditional horse race due to the information provided by the variableX is
∆W =I pX,Y
. (3.46)