arXiv subjects

Vladimir Vovk

Publications and source records attributed to Vladimir Vovk.

At least 73 records · Page 4Linked to original sources

arXiv2017

Universal probability-free prediction

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.

cs.LG↗

arXiv2017

Towards a probability-free theory of continuous martingales

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 Itô integral. We also establish the existence of an equity premium and a CAPM relationship in this probability-free setting.

q-fin.MF↗

arXiv2017

Getting rich quick with the Axiom of Choice

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 that point (whereas reaching a point in time such that the variation of the log price is infinite in any left neighbourhood of that point is not sufficient). The practical value of these schemes is tempered by their use of the Axiom of Choice.

q-fin.MF↗

arXiv2016

On the concept of Bernoulliness

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.

math.ST↗

arXiv2016

Rough paths in idealized financial markets

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 only one monetary unit but brings infinite capital when the variation index of the realized price path exceeds 2. The paper also reviews some known results for continuous price paths and lists several open problems.

q-fin.GN↗

arXiv2016

Criteria of efficiency for conformal prediction

We study optimal conformity measures for various criteria of efficiency of classification in an idealised setting. This leads to an important class of criteria of efficiency that we call probabilistic; it turns out that the most standard criteria of efficiency used in literature on conformal prediction are not probabilistic unless the problem of classification is binary. We consider both unconditional and label-conditional conformal prediction.

cs.LG↗

arXiv2016

Another example of duality between game-theoretic and measure-theoretic probability

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.

q-fin.MF↗

arXiv2016

A probability-free and continuous-time explanation of the equity premium and CAPM

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.

q-fin.MF↗

arXiv2016

Purely pathwise probability-free Ito integral

This paper gives several simple constructions of the pathwise Ito integral $\int_0^tϕdω$ for an integrand $ϕ$ and a price path $ω$ as integrator, with $ϕ$ and $ω$ satisfying various topological and analytical conditions. The definitions are purely pathwise in that neither $ϕ$ nor $ω$ 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 existence of $\int_0^tϕdω$ for a cadlag integrand $ϕ$ and a cadlag integrator $ω$ with jumps bounded in a predictable manner.

q-fin.MF↗

arXiv2016

Fundamentals of p-values: Introduction

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.

math.ST↗

arXiv2015

Large-scale probabilistic predictors with and without guarantees of validity

This paper studies theoretically and empirically a method of turning machine-learning algorithms into probabilistic predictors that automatically enjoys a property of validity (perfect calibration) and is computationally efficient. The price to pay for perfect calibration is that these probabilistic predictors produce imprecise (in practice, almost precise for large data sets) probabilities. When these imprecise probabilities are merged into precise probabilities, the resulting predictors, while losing the theoretical property of perfect calibration, are consistently more accurate than the existing methods in empirical studies.

cs.LG↗

arXiv2015

The fundamental nature of the log loss function

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 under any computable proper mixable loss function; on the other hand, there is a data sequence and a prediction algorithm that is optimal for that sequence under either of the two other standard loss functions but not under the log loss function.

cs.LG↗

arXiv2015

Continuous-time trading and the emergence of probability

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 without risking more than one monetary unit if the process of quadratic variation does not exist. Replacing time by the quadratic variation process, we show that the price path becomes Brownian motion. This is essentially the same conclusion as in the Dubins-Schwarz result, except that the probabilities (constituting the Wiener measure) emerge instead of being postulated. We also give an elegant statement, inspired by Peter McCullagh's unpublished work, of this result in terms of game-theoretic probability theory.

math.PR↗

arXiv2014

Ito calculus without probability in idealized financial markets

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 to typical price paths; we give a brief overview of such results.

q-fin.TR↗

arXiv2014

Prediction with Advice of Unknown Number of Experts

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 Normal- Hedge bound, which mainly depends on the effective number of experts but 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 multivalued supermartingales.

cs.LG↗

arXiv2014

Venn-Abers predictors

This paper continues study, both theoretical and empirical, of the method of Venn prediction, concentrating on binary prediction problems. Venn predictors produce probability-type predictions for the labels of test objects which are guaranteed to be well calibrated under the standard assumption that the observations are generated independently from the same distribution. We give a simple formalization and proof of this property. We also introduce Venn-Abers predictors, a new class of Venn predictors based on the idea of isotonic regression, and report promising empirical results both for Venn-Abers predictors and for their more computationally efficient simplified version.

cs.LG↗

arXiv2014

From conformal to probabilistic prediction

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.

cs.LG↗

arXiv2014

Efficiency of conformalized ridge regression

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 loses little in efficiency as compared with the underlying algorithm (whereas being a conformal predictor, it has the stronger guarantee of validity). In this paper we explore the degree to which this additional requirement of efficiency is satisfied in the case of Bayesian ridge regression; we find that asymptotically conformal prediction sets differ little from ridge regression prediction intervals when the standard Bayesian assumptions are satisfied.

cs.LG↗