Related papers: Computable randomness is about more than probabili…
We consider the computability of entropy and information in classical Hamiltonian systems. We define the information part and total information capacity part of entropy in classical Hamiltonian systems using relative information under a…
Solomonoff unified Occam's razor and Epicurus' principle of multiple explanations to one elegant, formal, universal theory of inductive inference, which initiated the field of algorithmic information theory. His central result is that the…
Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not…
Conformal predictive systems are a recent modification of conformal predictors that output, in regression problems, probability distributions for labels of test observations rather than set predictions. The extra information provided by…
Probabilistic inference procedures are usually coded painstakingly from scratch, for each target model and each inference algorithm. We reduce this effort by generating inference procedures from models automatically. We make this code…
In this work we present a model for computation of random processes in digital computers which solves the problem of periodic sequences and hidden errors produced by correlations. We show that systems with non-invertible non-linearities can…
In computability theory and computable analysis, finite programs can compute infinite objects. Presenting a computable object via any program for it, provides at least as much information as presenting the object itself, written on an…
We investigate the connection between measure, capacity and algorithmic randomness for the space of closed sets. For any computable measure m, a computable capacity T may be defined by letting T(Q) be the measure of the family of closed…
The causal (belief) network is a well-known graphical structure for representing independencies in a joint probability distribution. The exact methods and the approximation methods, which perform probabilistic inference in causal networks,…
The difficulty of explaining non-local correlations in a fixed causal structure sheds new light on the old debate on whether space and time are to be seen as fundamental. Refraining from assuming space-time as given a priori has a number of…
Suppose that we are given an infinite binary sequence which is random for a Bernoulli measure of parameter $p$. By the law of large numbers, the frequency of zeros in the sequence tends to~$p$, and thus we can get better and better…
We introduce the notion of finitary computable reducibility on equivalence relations on the natural numbers. This is a weakening of the usual notion of computable reducibility, and we show it to be distinct in several ways. In particular,…
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…
We consider the product of infinitely many copies of a spin-$1\over 2$ system. We construct projection operators on the corresponding nonseparable Hilbert space which measure whether the outcome of an infinite sequence of $\sigma^x$…
Contrary to popular misconception, the question in the title is far from simple. It involves sets of numbers on the first level, sets of sets of numbers on the second level, and so on, endlessly. The infinite hierarchy of the levels…
In many applications, accurate class probability estimates are required, but many types of models produce poor quality probability estimates despite achieving acceptable classification accuracy. Even though probability calibration has been…
The notion of probability plays an important role in almost all areas of science and technology. In modern mathematics, however, probability theory means nothing other than measure theory, and the operational characterization of the notion…
This paper concerns instruction sequences that contain probabilistic instructions, i.e. instructions that are themselves probabilistic by nature. We propose several kinds of probabilistic instructions, provide an informal operational…
In this introductory article we present the basics of an approach to implementing computational interpreting of natural language aiming to model the meanings of words and phrases. Unlike other approaches, we attempt to define the meanings…
Probabilistic programming is becoming increasingly popular thanks to its ability to specify problems with a certain degree of uncertainty. In this work, we focus on term rewriting, a well-known computational formalism. In particular, we…