Related papers: Predictability and Randomness
Mutual information I in infinite sequences (and in their finite prefixes) is essential in theoretical analysis of many situations. Yet its right definition has been elusive for a long time. I address it by generalizing Kolmogorov Complexity…
Conformal predictions make it possible to define reliable and robust learning algorithms. But they are essentially a method for evaluating whether an algorithm is good enough to be used in practice. To define a reliable learning framework…
We introduce new definitions of universal and superuniversal computable codes, which are based on a code's ability to approximate Kolmogorov complexity within the prescribed margin for all individual sequences from a given set. Such sets of…
We use a decision-theoretic framework to study the problem of forecasting discrete outcomes when the forecaster is unable to discriminate among a set of plausible forecast distributions because of partial identification or concerns about…
A central paradigm behind process semantics based on observability and testing is that the exact moment of occurring of an internal nondeterministic choice is unobservable. It is natural, therefore, for this property to hold when the…
Strict frequentism defines probability as the limiting relative frequency in an infinite sequence. What if the limit does not exist? We present a broader theory, which is applicable also to random phenomena that exhibit diverging relative…
The need to condition distributional properties such as expectation, variance, and entropy arises in algorithmic fairness, model simplification, robustness and many other areas. At face value however, distributional properties are not…
The main result presented in this article is that probability can fundamentally be characterized as a subset of conditional expectation induced by a plausible preorder on random quantities. This is justified by the fact that probability is…
Probabilistic programming languages allow programmers to write down conditional probability distributions that represent statistical and machine learning models as programs that use observe statements. These programs are run by accumulating…
The Bayesian framework is ideally suited for induction problems. The probability of observing $x_t$ at time $t$, given past observations $x_1...x_{t-1}$ can be computed with Bayes' rule if the true distribution $\mu$ of the sequences…
Predicting the future is an important component of decision making. In most situations, however, there is not enough information to make accurate predictions. In this paper, we develop a theory of causal reasoning for predictive inference…
The best subset selection (or "best subsets") estimator is a classic tool for sparse regression, and developments in mathematical optimization over the past decade have made it more computationally tractable than ever. Notwithstanding its…
Analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are ``well-distributed'' in some appropriate sense. In various discrepancy problems, combinatorics researchers have analyzed limitations to…
We generalise the randomness test definitions in the literature for both the Martin-L\"of and Schnorr randomness of a series of binary outcomes, in order to allow for interval-valued rather than merely precise forecasts for these outcomes,…
Martin-Lof's definition of random sequences of cbits as those not belonging to any set of constructive zero Lebesgue measure is reformulated in the language of Algebraic Probability Theory. The adoption of the Pour-El Richards theory of…
We study Martin-L\"{o}f random (ML-random) points on computable probability measures on sample and parameter spaces (Bayes models). We consider variants of conditional randomness defined by ML-randomness on Bayes models and those of…
We give a finite-sample analysis of predictive inference procedures after model selection in regression with random design. The analysis is focused on a statistically challenging scenario where the number of potentially important…
Probabilistic programs encode stochastic models as ordinary-looking programs with primitives for sampling numbers from predefined distributions and conditioning. Their applications include, among many others, machine learning and modeling…
As an alternative to the well-known methods of "chaining" and "bracketing" that have been developed in the study of random fields, a new method, which is based on a {\em stochastic maximal inequality} derived by using the formula for…
In this paper, we consider the sublinear expectation on bounded random variables. With the notion of uncorrelatedness for random variables under the sublinear expectation, a weak law of large numbers is obtained. With the notion of…