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Probabilistic forecasts in the form of probability distributions over future events have become popular in several fields of statistical science. The dissimilarity between a probability forecast and an outcome is measured by a loss function…
We consider the task of forecasting an infinite sequence of future observations based on some number of past observations, where the probability measure generating the observations is "suspected" to satisfy one or more of a set of…
We use the martingale-theoretic approach of game-theoretic probability to incorporate imprecision into the study of randomness. In particular, we define several notions of randomness associated with interval, rather than precise,…
Prediction is a well-studied machine learning task, and prediction algorithms are core ingredients in online products and services. Despite their centrality in the competition between online companies who offer prediction-based products,…
In the framework of the game-theoretic probability of Shafer and Vovk (2001) it is of basic importance to construct an explicit strategy weakly forcing the strong law of large numbers (SLLN) in the bounded forecasting game. We present a…
We propose to smooth out the calibration score, which measures how good a forecaster is, by combining nearby forecasts. While regular calibration can be guaranteed only by randomized forecasting procedures, we show that smooth calibration…
Bayesian statistics has gained popularity in psychological research due to its intuitive uncertainty quantification and convenient information-updating rules. In many applications, however, prior distributions are introduced merely as…
In this paper we focus on noncooperative games with uncertain constraints coupling the agents' decisions. We consider a setting where bounded deviations of agents' decisions from the equilibrium are possible, and uncertain constraints are…
Schervish (1985b) showed that every forecasting system is noncalibrated for uncountably many data sequences that it might see. This result is strengthened here: from a topological point of view, failure of calibration is typical and…
When we test a theory using data, it is common to focus on correctness: do the predictions of the theory match what we see in the data? But we also care about completeness: how much of the predictable variation in the data is captured by…
Can a probabilistic gambler get arbitrarily rich when all deterministic gamblers fail? We study this problem in the context of algorithmic randomness, introducing a new notion -- almost everywhere computable randomness. A binary sequence…
We provide a unifying framework for the design and analysis of multicalibrated predictors. By placing the multicalibration problem in the general setting of multi-objective learning -- where learning guarantees must hold simultaneously over…
In game theory, players have continuous expected payoff functions and can use fixed point theorems to locate equilibria. This optimization method requires that players adopt a particular type of probability measure space. Here, we introduce…
We use the martingale-theoretic approach of game-theoretic probability to incorporate imprecision into the study of randomness. In particular, we define a notion of computable randomness associated with interval, rather than precise,…
Game-theoretic probability uses the structure of gambles to define a concept like probability, but which is more flexible and robust. We show that results in game-theoretic probability can be thought of as minimax theorems for specific…
There are only limited classes of multi-player stochastic games in which independent learning is guaranteed to converge to a Nash equilibrium. Markov potential games are a key example of such classes. Prior work has outlined sets of…
We introduce an evolutionary game with feedback between perception and reality, which we call the reality game. It is a game of chance in which the probabilities for different objective outcomes (e.g., heads or tails in a coin toss) depend…
Machine learning applications often require calibrated predictions, e.g. a 90\% credible interval should contain the true outcome 90\% of the times. However, typical definitions of calibration only require this to hold on average, and offer…
Vapnik--Chervonenkis' theorem is a seminal result in machine learning. It establishes sufficient conditions for empirical probabilities to converge to theoretical probabilities, uniformly over families of events. It also provides an…
Football forecasting models traditionally rate teams on past match results, that is based on the number of goals scored. Goals, however, involve a high element of chance and thus past results often do not reflect the performances of the…