Results 1  10
of
24
Universal Portfolios
, 1996
"... We exhibit an algorithm for portfolio selection that asymptotically outperforms the best stock in the market. Let x i = (x i1 ; x i2 ; : : : ; x im ) t denote the performance of the stock market on day i ; where x ij is the factor by which the jth stock increases on day i : Let b i = (b i1 ; b i2 ..."
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Cited by 207 (5 self)
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We exhibit an algorithm for portfolio selection that asymptotically outperforms the best stock in the market. Let x i = (x i1 ; x i2 ; : : : ; x im ) t denote the performance of the stock market on day i ; where x ij is the factor by which the jth stock increases on day i : Let b i = (b i1 ; b i2 ; : : : ; b im ) t ; b ij 0; P j b ij = 1 ; denote the proportion b ij of wealth invested in the jth stock on day i : Then S n = Q n i=1 b t i x i is the factor by which wealth is increased in n trading days. Consider as a goal the wealth S n = max b Q n i=1 b t x i that can be achieved by the best constant rebalanced portfolio chosen after the stock outcomes are revealed. It can be shown that S n exceeds the best stock, the Dow Jones average, and the value line index at time n: In fact, S n usually exceeds these quantities by an exponential factor. Let x 1 ; x 2 ; : : : ; be an arbitrary sequence of market vectors. It will be shown that the nonanticipating sequence ...
Informationtheoretic asymptotics of Bayes methods
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 1990
"... In the absence of knowledge of the true density function, Bayesian models take the joint density function for a sequence of n random variables to be an average of densities with respect to a prior. We examine the relative entropy distance D,, between the true density and the Bayesian density and sh ..."
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Cited by 140 (12 self)
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In the absence of knowledge of the true density function, Bayesian models take the joint density function for a sequence of n random variables to be an average of densities with respect to a prior. We examine the relative entropy distance D,, between the true density and the Bayesian density and show that the asymptotic distance is (d/2Xlogn)+ c, where d is the dimension of the parameter vector. Therefore, the relative entropy rate D,,/n converges to zero at rate (logn)/n. The constant c, which we explicitly identify, depends only on the prior density function and the Fisher information matrix evaluated at the true parameter value. Consequences are given for density estimation, universal data compression, composite hypothesis testing, and stockmarket portfolio selection.
Online portfolio selection using multiplicative updates
 Mathematical Finance
, 1998
"... We present an online investment algorithm which achieves almost the same wealth as the best constantrebalanced portfolio determined in hindsight from the actual market outcomes. The algorithm employs a multiplicative update rule derived using a framework introduced by Kivinen and Warmuth. Our algo ..."
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Cited by 94 (10 self)
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We present an online investment algorithm which achieves almost the same wealth as the best constantrebalanced portfolio determined in hindsight from the actual market outcomes. The algorithm employs a multiplicative update rule derived using a framework introduced by Kivinen and Warmuth. Our algorithm is very simple to implement and requires only constant storage and computing time per stock ineach trading period. We tested the performance of our algorithm on real stock data from the New York Stock Exchange accumulated during a 22year period. On this data, our algorithm clearly outperforms the best single stock aswell as Cover's universal portfolio selection algorithm. We also present results for the situation in which the We present an online investment algorithm which achieves almost the same wealth as the best constantrebalanced portfolio investment strategy. The algorithm employsamultiplicative update rule derived using a framework introduced by Kivinen and Warmuth [20]. Our algorithm is very simple to implement and its time and storage requirements grow linearly in the number of stocks.
Mutual Information, Metric Entropy, and Cumulative Relative Entropy Risk
 Annals of Statistics
, 1996
"... Assume fP ` : ` 2 \Thetag is a set of probability distributions with a common dominating measure on a complete separable metric space Y . A state ` 2 \Theta is chosen by Nature. A statistician gets n independent observations Y 1 ; : : : ; Y n from Y distributed according to P ` . For each time ..."
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Cited by 52 (2 self)
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Assume fP ` : ` 2 \Thetag is a set of probability distributions with a common dominating measure on a complete separable metric space Y . A state ` 2 \Theta is chosen by Nature. A statistician gets n independent observations Y 1 ; : : : ; Y n from Y distributed according to P ` . For each time t between 1 and n, based on the observations Y 1 ; : : : ; Y t\Gamma1 , the statistician produces an estimated distribution P t for P ` , and suffers a loss L(P ` ; P t ). The cumulative risk for the statistician is the average total loss up to time n. Of special interest in information theory, data compression, mathematical finance, computational learning theory and statistical mechanics is the special case when the loss L(P ` ; P t ) is the relative entropy between the true distribution P ` and the estimated distribution P t . Here the cumulative Bayes risk from time 1 to n is the mutual information between the random parameter \Theta and the observations Y 1 ; : : : ;...
A General Minimax Result for Relative Entropy
 IEEE Trans. Inform. Theory
, 1996
"... : Suppose Nature picks a probability measure P ` on a complete separable metric space X at random from a measurable set P \Theta = fP ` : ` 2 \Thetag. Then, without knowing `, a statistician picks a measure Q on X. Finally, the statistician suffers a loss D(P ` jjQ), the relative entropy between P ..."
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Cited by 44 (2 self)
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: Suppose Nature picks a probability measure P ` on a complete separable metric space X at random from a measurable set P \Theta = fP ` : ` 2 \Thetag. Then, without knowing `, a statistician picks a measure Q on X. Finally, the statistician suffers a loss D(P ` jjQ), the relative entropy between P ` and Q. We show that the minimax and maximin values of this game are always equal, and there is always a minimax strategy in the closure of the set of all Bayes strategies. This generalizes previous results of Gallager, and Davisson and LeonGarcia. Index terms: minimax theorem, minimax redundancy, minimax risk, Bayes risk, relative entropy, KullbackLeibler divergence, density estimation, source coding, channel capacity, computational learning theory 1 Introduction Consider a sequential estimation game in which a statistician is given n independent observations Y 1 ; : : : ; Yn distributed according to an unknown distribution ~ P ` chosen at random by Nature from the set f ~ P ` : ` 2 \...
Portfolio choice and the Bayesian Kelly criterion
 Adv. Appl. Prob
, 1994
"... We derive optimal gambling and investment policies for cases in which the underlying stochastic process has parameter values that are unobserved random variables. For the objective of maximizing logarithmic utility when the underlying stochastic process is a simple random walk in a random environmen ..."
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Cited by 24 (4 self)
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We derive optimal gambling and investment policies for cases in which the underlying stochastic process has parameter values that are unobserved random variables. For the objective of maximizing logarithmic utility when the underlying stochastic process is a simple random walk in a random environment, we show that a statedependent control is optimal, which is a generalization of the celebrated Kelly strategy: The optimal strategy is to bet a fraction of current wealth equal to a linear function of the posterior mean increment. To approximate more general stochastic processes, we consider a continuoustime analog involving Brownian motion. To analyze the continuoustime problem, we study the di usion limit of random walks in a random environment. We prove that they converge weakly to a Kiefer process, or tieddown Brownian sheet. We then nd conditions under which the discretetime process converges to a di usion, and analyze the resulting process. We analyze in detail the case of the natural conjugate prior, where the success probability has a beta distribution, and show that the resulting limiting di usion can be viewed as a rescaled Brownian motion. These results allow explicit computation of the optimal control policies for the continuoustime gambling and investment problems without resorting to continuoustime stochasticcontrol procedures. Moreover they also allow an explicit quantitative evaluation of the nancial value
Interpretations of Directed Information in Portfolio Theory
 Data Compression, and Hypothesis Testing”, IEEE Trans. on Inf. Th
"... Abstract—We investigate the role of directed information in portfolio theory, data compression, and statistics with causality constraints. In particular, we show that directed information is an upper bound on the increment in growth rates of optimal portfolios in a stock market due to causal side in ..."
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Cited by 23 (9 self)
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Abstract—We investigate the role of directed information in portfolio theory, data compression, and statistics with causality constraints. In particular, we show that directed information is an upper bound on the increment in growth rates of optimal portfolios in a stock market due to causal side information. This upper bound is tight for gambling in a horse race, which is an extreme case of stock markets. Directed information also characterizes the value of causal side information in instantaneous compression and quantifies the benefit of causal inference in joint compression of two stochastic processes. In hypothesis testing, directed information evaluates the best error exponent for testing whether a random process Y causally influences another process X or not. These results lead to a natural interpretation of directed information I(Y n! X n) as the amount of information that a random sequence Y n = (Y1;Y2;...;Yn) causally provides about another random sequence X n = (X1;X2;...;Xn). A new measure, directed lautum information, is also introduced and interpreted in portfolio theory, data compression, and hypothesis testing. Index Terms—Causal conditioning, causal side information, directed information, hypothesis testing, instantaneous compression, Kelly gambling, lautum information, portfolio theory. I.
General bounds on the mutual information between a parameter and n conditionally independent observations
 In Proceedings of the Seventh Annual ACM Workshop on Computational Learning Theory
, 1995
"... Each parameter in an abstract parameter space is associated with a di erent probability distribution on a set Y. A parameter is chosen at random from according to some a priori distribution on, and n conditionally independent random variables Y n = Y1�:::Yn are observed with common distribution dete ..."
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Cited by 16 (5 self)
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Each parameter in an abstract parameter space is associated with a di erent probability distribution on a set Y. A parameter is chosen at random from according to some a priori distribution on, and n conditionally independent random variables Y n = Y1�:::Yn are observed with common distribution determined by. We obtain bounds on the mutual information between the random variable, giving the choice of parameter, and the random variable Y n, giving the sequence of observations. We also bound the supremum of the mutual information, over choices of the prior distribution on. These quantities have applications in density estimation, computational learning theory, universal coding, hypothesis testing, and portfolio selection theory. The bounds are given in terms of the metric and information dimensions of the parameter space with respect to the Hellinger distance. 1
The efficiency of investment information
 IEEE Trans. Inform. Theory
, 1998
"... We investigate how the description of a correlated information V improves the investment in the stock market X. The objective is to maximize the growth rate of wealth in repeated investments. We find a singleletter characterization of the incremental growth rate Δ(R), the maximum increas ..."
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Cited by 14 (0 self)
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We investigate how the description of a correlated information V improves the investment in the stock market X. The objective is to maximize the growth rate of wealth in repeated investments. We find a singleletter characterization of the incremental growth rate &Delta;(R), the maximum increase in growth rate when V is described to the investor at rate R. The incremental growth rate specialized to the horse race market is related to source coding with side information of Wyner and Ahlswede–Körner. We provide two horse race examples: jointly binary and jointly Gaussian. The initial efficiency 1 H