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Sep 23, 2013
09/13

by
Vladimir Vovk

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This paper suggests a perfect-information game, along the lines of Levy's characterization of Brownian motion, that formalizes the process of Brownian motion in game-theoretic probability. This is perhaps the simplest situation where probability emerges in a non-stochastic environment.

Source: http://arxiv.org/abs/0801.1309v1

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Sep 20, 2013
09/13

by
Vladimir Vovk

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We start from a simple asymptotic result for the problem of on-line regression with the quadratic loss function: the class of continuous limited-memory prediction strategies admits a "leading prediction strategy", which not only asymptotically performs at least as well as any continuous limited-memory strategy but also satisfies the property that the excess loss of any continuous limited-memory strategy is determined by how closely it imitates the leading strategy. More specifically,...

Source: http://arxiv.org/abs/cs/0607134v1

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4.0

Jun 29, 2018
06/18

by
Vladimir Vovk

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This paper makes a small step towards a non-stochastic version of superhedging duality relations in the case of one traded security with a continuous price path. Namely, we prove the coincidence of game-theoretic and measure-theoretic expectation for lower semicontinuous positive functionals. We consider a new broad definition of game-theoretic probability, leaving the older narrower definitions for future work.

Topics: Mathematical Finance, Probability, Quantitative Finance, Mathematics

Source: http://arxiv.org/abs/1608.02706

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Jul 19, 2013
07/13

by
Vladimir Vovk

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This paper considers possible price paths of a financial security in an idealized market. Its main result is that the variation index of typical price paths is at most 2; in this sense, typical price paths are not rougher than typical paths of Brownian motion. We do not make any stochastic assumptions and only assume that the price path is positive and right-continuous. The qualification "typical" means that there is a trading strategy (constructed explicitly in the proof) that risks...

Source: http://arxiv.org/abs/1005.0279v2

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Jul 20, 2013
07/13

by
Vladimir Vovk

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The game-theoretic version of Kolmogorov's strong law of large numbers says that Skeptic has a strategy forcing the statement of the law in a game of prediction involving Reality, Forecaster, and Skeptic. This note describes a simple matching strategy for Reality.

Source: http://arxiv.org/abs/1304.1074v1

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Sep 21, 2013
09/13

by
Vladimir Vovk

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In 1952 Lucien Le Cam announced his celebrated result that, for regular univariate statistical models, sets of points of superefficiency have Lebesgue measure zero. After reviewing the turbulent history of early studies of superefficiency, I suggest using the notion of computability as a tool for exploring the phenomenon of superefficiency. It turns out that only computable parameter points can be points of superefficiency for computable estimators. This algorithmic version of Le Cam's result...

Source: http://arxiv.org/abs/0808.2266v2

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Sep 21, 2013
09/13

by
Vladimir Vovk

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A new definition of events of game-theoretic probability zero in continuous time is proposed and used to prove results suggesting that trading in financial markets results in the emergence of properties usually associated with randomness. This paper concentrates on "qualitative" results, stated in terms of order (or order topology) rather than in terms of the precise values taken by the price processes (assumed continuous).

Source: http://arxiv.org/abs/0712.1275v2

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8.0

Jun 29, 2018
06/18

by
Vladimir Vovk

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The first part of this paper is another English translation of a 1986 note. It gives a natural definition of a finite Bernoulli sequence (i.e., a typical realization of a finite sequence of binary IID trials) and compares it with the Kolmogorov--Martin-Lof definition, which is interpreted as defining exchangeable sequences. The appendix gives the historical background and proofs.

Topics: Statistics, Statistics Theory, Mathematics

Source: http://arxiv.org/abs/1612.08859

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Sep 20, 2013
09/13

by
Vladimir Vovk

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We analyze a new algorithm for probability forecasting of binary observations on the basis of the available data, without making any assumptions about the way the observations are generated. The algorithm is shown to be well calibrated and to have good resolution for long enough sequences of observations and for a suitable choice of its parameter, a kernel on the Cartesian product of the forecast space $[0,1]$ and the data space. Our main results are non-asymptotic: we establish explicit...

Source: http://arxiv.org/abs/cs/0506004v4

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Sep 18, 2013
09/13

by
Vladimir Vovk

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Prediction is a complex notion, and different predictors (such as people, computer programs, and probabilistic theories) can pursue very different goals. In this paper I will review some popular kinds of prediction and argue that the theory of competitive on-line learning can benefit from the kinds of prediction that are now foreign to it.

Source: http://arxiv.org/abs/cs/0606093v1

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Jun 26, 2018
06/18

by
Vladimir Vovk

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The standard loss functions used in the literature on probabilistic prediction are the log loss function, the Brier loss function, and the spherical loss function; however, any computable proper loss function can be used for comparison of prediction algorithms. This note shows that the log loss function is most selective in that any prediction algorithm that is optimal for a given data sequence (in the sense of the algorithmic theory of randomness) under the log loss function will be optimal...

Topics: Learning, Statistics, Computing Research Repository, Methodology

Source: http://arxiv.org/abs/1502.06254

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12

Jun 29, 2018
06/18

by
Vladimir Vovk

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This paper proposes new get-rich-quick schemes that involve trading in a financial security with a non-degenerate price path. For simplicity the interest rate is assumed zero. If the price path is assumed continuous, the trader can become infinitely rich immediately after it becomes non-constant (if it ever does). If it is assumed positive, he can become infinitely rich immediately after reaching a point in time such that the variation of the log price is infinite in any right neighbourhood of...

Topics: Mathematical Finance, Trading and Market Microstructure, Quantitative Finance

Source: http://arxiv.org/abs/1604.00596

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Sep 23, 2013
09/13

by
Vladimir Vovk

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This paper establishes a non-stochastic analogue of the celebrated result by Dubins and Schwarz about reduction of continuous martingales to Brownian motion via time change. We consider an idealized financial security with continuous price path, without making any stochastic assumptions. It is shown that typical price paths possess quadratic variation, where "typical" is understood in the following game-theoretic sense: there exists a trading strategy that earns infinite capital...

Source: http://arxiv.org/abs/0904.4364v3

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6.0

Jun 28, 2018
06/18

by
Vladimir Vovk

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This paper gives several simple constructions of the pathwise Ito integral $\int_0^t\phi d\omega$ for an integrand $\phi$ and a price path $\omega$ as integrator, with $\phi$ and $\omega$ satisfying various topological and analytical conditions. The definitions are purely pathwise in that neither $\phi$ nor $\omega$ are assumed to be paths of stochastic processes, and the Ito integral exists almost surely in a non-probabilistic financial sense. For example, one of the results shows the...

Topics: Mathematical Finance, Quantitative Finance

Source: http://arxiv.org/abs/1512.01698

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Sep 19, 2013
09/13

by
Vladimir Vovk

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This note makes the obvious observation that Hoeffding's original proof of his inequality remains valid in the game-theoretic framework. All details are spelled out for the convenience of future reference.

Source: http://arxiv.org/abs/0708.2502v1

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Sep 21, 2013
09/13

by
Vladimir Vovk

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This note continues investigation of randomness-type properties emerging in idealized financial markets with continuous price processes. It is shown, without making any probabilistic assumptions, that the strong variation exponent of non-constant price processes has to be 2, as in the case of continuous martingales.

Source: http://arxiv.org/abs/0712.1483v2

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Sep 23, 2013
09/13

by
Vladimir Vovk

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We consider the problem of on-line prediction of real-valued labels, assumed bounded in absolute value by a known constant, of new objects from known labeled objects. The prediction algorithm's performance is measured by the squared deviation of the predictions from the actual labels. No stochastic assumptions are made about the way the labels and objects are generated. Instead, we are given a benchmark class of prediction rules some of which are hoped to produce good predictions. We show that...

Source: http://arxiv.org/abs/cs/0511058v2

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Sep 23, 2013
09/13

by
Vladimir Vovk

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This note discusses the problem of multiple testing of a single hypothesis, with a standard goal of combining a number of p-values without making any assumptions about their dependence structure. An old result by Rueschendorf shows that the p-values can be combined by scaling up their average by a factor of 2 (but no smaller factor is sufficient in general).

Source: http://arxiv.org/abs/1212.4966v2

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Sep 20, 2013
09/13

by
Vladimir Vovk

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Assuming that the loss function is convex in the prediction, we construct a prediction strategy universal for the class of Markov prediction strategies, not necessarily continuous. Allowing randomization, we remove the requirement of convexity.

Source: http://arxiv.org/abs/cs/0607136v1

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45

Sep 20, 2013
09/13

by
Vladimir Vovk

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In this paper we introduce the class of stationary prediction strategies and construct a prediction algorithm that asymptotically performs as well as the best continuous stationary strategy. We make mild compactness assumptions but no stochastic assumptions about the environment. In particular, no assumption of stationarity is made about the environment, and the stationarity of the considered strategies only means that they do not depend explicitly on time; we argue that it is natural to...

Source: http://arxiv.org/abs/cs/0607067v1

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Sep 23, 2013
09/13

by
Vladimir Vovk

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This note studies the behavior of an index I_t which is assumed to be a tradable security, to satisfy the BSM model dI_t/I_t = \mu dt + \sigma dW_t, and to be efficient in the following sense: we do not expect a prespecified trading strategy whose value is almost surely always nonnegative to outperform the index greatly. The efficiency of the index imposes severe restrictions on its growth rate; in particular, for a long investment horizon we should have \mu\approx r+\sigma^2, where r is the...

Source: http://arxiv.org/abs/1109.2327v1

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51

Sep 23, 2013
09/13

by
Vladimir Vovk

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A simple example shows that losing all money is compatible with a very high Sharpe ratio (as computed after losing all money). However, the only way that the Sharpe ratio can be high while losing money is that there is a period in which all or almost all money is lost. This note explores the best achievable Sharpe and Sortino ratios for investors who lose money but whose one-period returns are bounded below (or both below and above) by a known constant.

Source: http://arxiv.org/abs/1109.0706v1

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Sep 18, 2013
09/13

by
Vladimir Vovk

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This note introduces Venn-Abers predictors, a new class of Venn predictors based on the idea of isotonic regression. As all Venn predictors, Venn-Abers predictors are well calibrated under the exchangeability assumption.

Source: http://arxiv.org/abs/1211.0025v1

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Sep 20, 2013
09/13

by
Vladimir Vovk

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We consider the problem of sequential decision making under uncertainty in which the loss caused by a decision depends on the following binary observation. In competitive on-line learning, the goal is to design decision algorithms that are almost as good as the best decision rules in a wide benchmark class, without making any assumptions about the way the observations are generated. However, standard algorithms in this area can only deal with finite-dimensional (often countable) benchmark...

Source: http://arxiv.org/abs/cs/0506041v3

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Sep 19, 2013
09/13

by
Vladimir Vovk

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The method of defensive forecasting is applied to the problem of prediction with expert advice for binary outcomes. It turns out that defensive forecasting is not only competitive with the Aggregating Algorithm but also handles the case of "second-guessing" experts, whose advice depends on the learner's prediction; this paper assumes that the dependence on the learner's prediction is continuous.

Source: http://arxiv.org/abs/0708.1503v1

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Sep 21, 2013
09/13

by
Vladimir Vovk

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We consider idealized financial markets in which price paths of the traded securities are cadlag functions, imposing mild restrictions on the allowed size of jumps. We prove the existence of quadratic variation for typical price paths, where the qualification "typical" means that there is a trading strategy that risks only one monetary unit and brings infinite capital if quadratic variation does not exist. This result allows one to apply numerous known results in pathwise Ito calculus...

Source: http://arxiv.org/abs/1108.0799v1

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118

Jul 20, 2013
07/13

by
Vladimir Vovk

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Competitive on-line prediction (also known as universal prediction of individual sequences) is a strand of learning theory avoiding making any stochastic assumptions about the way the observations are generated. The predictor's goal is to compete with a benchmark class of prediction rules, which is often a proper Banach function space. Metric entropy provides a unifying framework for competitive on-line prediction: the numerous known upper bounds on the metric entropy of various compact sets in...

Source: http://arxiv.org/abs/cs/0609045v1

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52

Sep 18, 2013
09/13

by
Vladimir Vovk

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This paper gives game-theoretic versions of several results on "merging of opinions" obtained in measure-theoretic probability and algorithmic randomness theory. An advantage of the game-theoretic versions over the measure-theoretic results is that they are pointwise, their advantage over the algorithmic randomness results is that they are non-asymptotic, but the most important advantage over both is that they are very constructive, giving explicit and efficient strategies for players...

Source: http://arxiv.org/abs/0705.0372v1

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45

Sep 19, 2013
09/13

by
Vladimir Vovk

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Defensive forecasting is a method of transforming laws of probability (stated in game-theoretic terms as strategies for Sceptic) into forecasting algorithms. There are two known varieties of defensive forecasting: "continuous", in which Sceptic's moves are assumed to depend on the forecasts in a (semi)continuous manner and which produces deterministic forecasts, and "randomized", in which the dependence of Sceptic's moves on the forecasts is arbitrary and Forecaster's moves...

Source: http://arxiv.org/abs/0708.2353v2

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59

Sep 21, 2013
09/13

by
Vladimir Vovk

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This article continues study of the prequential framework for evaluating a probability forecaster. Testing the hypothesis that the sequence of forecasts issued by the forecaster is in agreement with the observed outcomes can be done using prequential notions of probability. It turns out that there are two natural notions of probability in the prequential framework: game-theoretic, whose idea goes back to von Mises and Ville, and measure-theoretic, whose idea goes back to Kolmogorov. The main...

Source: http://arxiv.org/abs/0905.1673v1

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50

Sep 23, 2013
09/13

by
Vladimir Vovk

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We consider a financial market in which two securities are traded: a stock and an index. Their prices are assumed to satisfy the Black-Scholes model. Besides assuming that the index is a tradable security, we also assume that it is efficient, in the following sense: we do not expect a prespecified self-financing trading strategy whose wealth is almost surely nonnegative at all times to outperform the index greatly. We show that, for a long investment horizon, the appreciation rate of the stock...

Source: http://arxiv.org/abs/1109.5144v1

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63

Sep 18, 2013
09/13

by
Vladimir Vovk

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In this short note I apply the methodology of game-theoretic probability to calculating non-asymptotic confidence intervals for the coefficient of a simple first order scalar autoregressive model. The most distinctive feature of the proposed procedure is that with high probability it produces confidence intervals that always cover the true parameter value when applied sequentially.

Source: http://arxiv.org/abs/0707.0660v1

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115

Sep 23, 2013
09/13

by
Vladimir Vovk

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We consider a Black-Scholes market in which a number of stocks and an index are traded. The simplified Capital Asset Pricing Model is the conjunction of the usual Capital Asset Pricing Model, or CAPM, and the statement that the appreciation rate of the index is equal to its squared volatility plus the interest rate. (The mathematical statement of the conjunction is simpler than that of the usual CAPM.) Our main result is that either we can outperform the index or the simplified CAPM holds.

Source: http://arxiv.org/abs/1111.2846v1

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58

Sep 18, 2013
09/13

by
Vladimir Vovk

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Conformal predictors are set predictors that are automatically valid in the sense of having coverage probability equal to or exceeding a given confidence level. Inductive conformal predictors are a computationally efficient version of conformal predictors satisfying the same property of validity. However, inductive conformal predictors have been only known to control unconditional coverage probability. This paper explores various versions of conditional validity and various ways to achieve them...

Source: http://arxiv.org/abs/1209.2673v2

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Aug 11, 2020
08/20

by
Glenn Shafer; Vladimir Vovk

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Source: http://academictorrents.com/details/903a5ecb011e8d9af24933ebe07cbe50e1f07be2

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7.0

Jun 30, 2018
06/18

by
Vladimir Vovk; Glenn Shafer

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Without probability theory, we define classes of supermartingales, martingales, and semimartingales in idealized financial markets with continuous price paths. This allows us to establish probability-free versions of a number of standard results in martingale theory, including the Dubins-Schwarz theorem, the Girsanov theorem, and results concerning the It\^o integral. We also establish the existence of an equity premium and a CAPM relationship in this probability-free setting.

Topics: Quantitative Finance, Mathematical Finance

Source: http://arxiv.org/abs/1703.08715

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72

Sep 21, 2013
09/13

by
Glenn Shafer; Vladimir Vovk

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Conformal prediction uses past experience to determine precise levels of confidence in new predictions. Given an error probability $\epsilon$, together with a method that makes a prediction $\hat{y}$ of a label $y$, it produces a set of labels, typically containing $\hat{y}$, that also contains $y$ with probability $1-\epsilon$. Conformal prediction can be applied to any method for producing $\hat{y}$: a nearest-neighbor method, a support-vector machine, ridge regression, etc. Conformal...

Source: http://arxiv.org/abs/0706.3188v1

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79

Sep 23, 2013
09/13

by
Alexey Chernov; Vladimir Vovk

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In the framework of prediction with expert advice, we consider a recently introduced kind of regret bounds: the bounds that depend on the effective instead of nominal number of experts. In contrast to the NormalHedge bound, which mainly depends on the effective number of experts and also weakly depends on the nominal one, we obtain a bound that does not contain the nominal number of experts at all. We use the defensive forecasting method and introduce an application of defensive forecasting to...

Source: http://arxiv.org/abs/1006.0475v1

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11

Jun 29, 2018
06/18

by
Yuri Gurevich; Vladimir Vovk

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We explain the concept of p-values presupposing only rudimentary probability theory. We also use the occasion to introduce the notion of p-function, so that p-values are values of a p-function. The explanation is restricted to the discrete case with no outcomes of zero probability. We are going to address the general case elsewhere.

Topics: Statistics, Statistics Theory, Mathematics

Source: http://arxiv.org/abs/1603.03952

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138

Sep 23, 2013
09/13

by
Vladimir Vovk; Fedor Zhdanov

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We show that the Brier game of prediction is mixable and find the optimal learning rate and substitution function for it. The resulting prediction algorithm is applied to predict results of football and tennis matches. The theoretical performance guarantee turns out to be rather tight on these data sets, especially in the case of the more extensive tennis data.

Source: http://arxiv.org/abs/0710.0485v2

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7.0

Jun 30, 2018
06/18

by
Evgeny Burnaev; Vladimir Vovk

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Conformal prediction is a method of producing prediction sets that can be applied on top of a wide range of prediction algorithms. The method has a guaranteed coverage probability under the standard IID assumption regardless of whether the assumptions (often considerably more restrictive) of the underlying algorithm are satisfied. However, for the method to be really useful it is desirable that in the case where the assumptions of the underlying algorithm are satisfied, the conformal predictor...

Topics: Machine Learning, Computing Research Repository, Statistics, Learning

Source: http://arxiv.org/abs/1404.2083

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7.0

Jun 29, 2018
06/18

by
Vladimir Vovk; Dusko Pavlovic

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We construct universal prediction systems in the spirit of Popper's falsifiability and Kolmogorov complexity and randomness. These prediction systems do not depend on any statistical assumptions (but under the IID assumption they dominate, to within the usual accuracy, conformal prediction). Our constructions give rise to a theory of algorithmic complexity and randomness of time containing analogues of several notions and results of the classical theory of Kolmogorov complexity and randomness.

Topics: Computing Research Repository, Learning

Source: http://arxiv.org/abs/1603.04283

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62

Sep 19, 2013
09/13

by
Fedor Zhdanov; Vladimir Vovk

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We study the problem of online regression. We prove a theoretical bound on the square loss of Ridge Regression. We do not make any assumptions about input vectors or outcomes. We also show that Bayesian Ridge Regression can be thought of as an online algorithm competing with all the Gaussian linear experts.

Source: http://arxiv.org/abs/0910.4683v2

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56

Sep 21, 2013
09/13

by
Alexey Chernov; Vladimir Vovk

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We introduce a new protocol for prediction with expert advice in which each expert evaluates the learner's and his own performance using a loss function that may change over time and may be different from the loss functions used by the other experts. The learner's goal is to perform better or not much worse than each expert, as evaluated by that expert, for all experts simultaneously. If the loss functions used by the experts are all proper scoring rules and all mixable, we show that the...

Source: http://arxiv.org/abs/0902.4127v2

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6.0

Jun 29, 2018
06/18

by
Vladimir Vovk; Glenn Shafer

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This paper gives yet another definition of game-theoretic probability in the context of continuous-time idealized financial markets. Without making any probabilistic assumptions (but assuming positive and continuous price paths), we obtain a simple expression for the equity premium and derive a version of the capital asset pricing model.

Topics: Mathematical Finance, Quantitative Finance

Source: http://arxiv.org/abs/1607.00830

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157

Sep 18, 2013
09/13

by
Alexander Gammerman; Vladimir Vovk

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Recent advances in machine learning make it possible to design efficient prediction algorithms for data sets with huge numbers of parameters. This paper describes a new technique for "hedging" the predictions output by many such algorithms, including support vector machines, kernel ridge regression, kernel nearest neighbours, and by many other state-of-the-art methods. The hedged predictions for the labels of new objects include quantitative measures of their own accuracy and...

Source: http://arxiv.org/abs/cs/0611011v1

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91

Sep 18, 2013
09/13

by
Glenn Shafer; Vladimir Vovk

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Andrei Kolmogorov's Grundbegriffe der Wahrscheinlichkeits-rechnung put probability's modern mathematical formalism in place. It also provided a philosophy of probability--an explanation of how the formalism can be connected to the world of experience. In this article, we examine the sources of these two aspects of the Grundbegriffe--the work of the earlier scholars whose ideas Kolmogorov synthesized.

Source: http://arxiv.org/abs/math/0606533v1

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6.0

Aug 11, 2020
08/20

by
Vladimir Vovk; Fedor Zhdanov

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Source: http://academictorrents.com/details/5839ddb77c57af03dd14e4b9da0cd7ce727664ae

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49

Sep 22, 2013
09/13

by
Vladimir Vovk; Ilia Nouretdinov; Alex Gammerman

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We consider the on-line predictive version of the standard problem of linear regression; the goal is to predict each consecutive response given the corresponding explanatory variables and all the previous observations. The standard treatment of prediction in linear regression analysis has two drawbacks: (1) the classical prediction intervals guarantee that the probability of error is equal to the nominal significance level $\varepsilon$, but this property per se does not imply that the long-run...

Source: http://arxiv.org/abs/0906.3123v1

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5.0

Jun 30, 2018
06/18

by
Vladimir Vovk; Ivan Petej; Valentina Fedorova

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This paper proposes a new method of probabilistic prediction, which is based on conformal prediction. The method is applied to the standard USPS data set and gives encouraging results.

Topics: Computing Research Repository, Learning

Source: http://arxiv.org/abs/1406.5600