arXiv subjects
Publications and source records attributed to Vladimir Vovk.
We prove game-theoretic generalizations of some well known zero-one laws. Our proofs make the martingales behind the laws explicit, and our results illustrate how martingale arguments can have implications going beyond measure-theoretic probability.
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 frequency of error is close to $\varepsilon$; (2) it is not suitable for prediction of complex systems as it assumes that the number of observations exceeds the number of parameters. We state a general result showing that in the on-line protocol the frequency of error for the classical prediction intervals does equal the nominal significance level, up to statistical fluctuations. We also describe alternative regression models in which informative prediction intervals can be found before the number of observations exceeds the number of parameters. One of these models, which only assumes that the observations are independent and identically distributed, is popular in machine learning but greatly underused in the statistical theory of regression.
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 result of this article is that, in the case of predicting binary outcomes, the two notions of probability in fact coincide on the analytic sets (in particular, on the Borel sets).
In this paper we apply computer learning methods to diagnosing ovarian cancer using the level of the standard biomarker CA125 in conjunction with information provided by mass-spectrometry. We are working with a new data set collected over a period of 7 years. Using the level of CA125 and mass-spectrometry peaks, our algorithm gives probability predictions for the disease. To estimate classification accuracy we convert probability predictions into strict predictions. Our algorithm makes fewer errors than almost any linear combination of the CA125 level and one peak's intensity (taken on the log scale). To check the power of our algorithm we use it to test the hypothesis that CA125 and the peaks do not contain useful information for the prediction of the disease at a particular time before the diagnosis. Our algorithm produces $p$-values that are better than those produced by the algorithm that has been previously applied to this data set. Our conclusion is that the proposed algorithm is more reliable for prediction on new data.
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 defensive forecasting algorithm enjoys the same performance guarantee as that attainable by the Aggregating Algorithm in the standard setting and known to be optimal. This result is also applied to the case of "specialist" (or "sleeping") experts. In this case, the defensive forecasting algorithm reduces to a simple modification of the Aggregating Algorithm.
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.
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.
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.
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).
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 are allowed to be randomized. This note shows that the randomized variety can be obtained from the continuous variety by smearing Sceptic's moves to make them continuous.
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.
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.
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.
Conformal prediction uses past experience to determine precise levels of confidence in new predictions. Given an error probability $ε$, 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 $1-ε$. Conformal prediction can be applied to any method for producing $\hat{y}$: a nearest-neighbor method, a support-vector machine, ridge regression, etc. Conformal prediction is designed for an on-line setting in which labels are predicted successively, each one being revealed before the next is predicted. The most novel and valuable feature of conformal prediction is that if the successive examples are sampled independently from the same distribution, then the successive predictions will be right $1-ε$ of the time, even though they are based on an accumulating dataset rather than on independent datasets. In addition to the model under which successive examples are sampled independently, other on-line compression models can also use conformal prediction. The widely used Gaussian linear model is one of these. This tutorial presents a self-contained account of the theory of conformal prediction and works through several numerical examples. A more comprehensive treatment of the topic is provided in "Algorithmic Learning in a Random World", by Vladimir Vovk, Alex Gammerman, and Glenn Shafer (Springer, 2005).
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 in a game of prediction.
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 reliability. These measures are provably valid under the assumption of randomness, traditional in machine learning: the objects and their labels are assumed to be generated independently from the same probability distribution. In particular, it becomes possible to control (up to statistical fluctuations) the number of erroneous predictions by selecting a suitable confidence level. Validity being achieved automatically, the remaining goal of hedged prediction is efficiency: taking full account of the new objects' features and other available information to produce as accurate predictions as possible. This can be done successfully using the powerful machinery of modern machine learning.
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 function spaces readily imply bounds on the performance of on-line prediction strategies. This paper discusses strengths and limitations of the direct approach to competitive on-line prediction via metric entropy, including comparisons to other approaches.
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.