Related papers: Algorithmic Randomness, Exchangeability, and the P…
Algorithmic approach is based on the assumption that any quantum evolution of many particle system can be simulated on a classical computer with the polynomial time and memory cost. Algorithms play the central role here but not the…
It is well-known that Choice and Regularity are independent of each other but have important common consequences of logical character (reflection principles, representations of classes by sets, etc.). We explain this phenomenon by isolating…
In inference problems involving a multi-dimensional parameter $\theta$, it is often natural to consider decision rules that have a risk which is invariant under some group $G$ of permutations of $\theta$. We show that this implies that the…
We develop a correspondence between the theory of sequential algorithms and classical reasoning, via Kreisel's no-counterexample interpretation. Our framework views realizers of the no-counterexample interpretation as dynamic processes…
The algorithmic theory of randomness is well developed when the underlying space is the set of finite or infinite sequences and the underlying probability distribution is the uniform distribution or a computable distribution. These…
Starting from the guiding principles of spacetime locality and operationalism, a general framework for a probabilistic description of nature is proposed. Crucially, no notion of time or metric is assumed, neither any specific physical…
We study efficient distributed algorithms for the fundamental problem of principal component analysis and leading eigenvector computation on the sphere, when the data are randomly distributed among a set of computational nodes. We propose a…
In this paper, we examine the fundamental performance limits of prediction, with or without side information. More specifically, we derive generic lower bounds on the $\mathcal{L}_p$ norms of the prediction errors that are valid for any…
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…
Statistical inference from data is a foundational task in science. Recently, it has received growing attention for its central role in inference systems of primary interest in data sciences and machine learning. However, the understanding…
While probability theory is normally applied to external environments, there has been some recent interest in probabilistic modeling of the outputs of computations that are too expensive to run. Since mathematical logic is a powerful tool…
In this article we demonstrate how algorithmic probability theory is applied to situations that involve uncertainty. When people are unsure of their model of reality, then the outcome they observe will cause them to update their beliefs. We…
The principle of maximum entropy (Maxent) is often used to obtain prior probability distributions as a method to obtain a Gibbs measure under some restriction giving the probability that a system will be in a certain state compared to the…
The results of the study provide guidelines for the development and applications of algorithms. When the number of steps for calculating an assumption tends to infinity, probability theory can be applied to predict whether the assumption…
In algorithmic randomness, when one wants to define a randomness notion with respect to some non-computable measure $\lambda $, a choice needs to be made. One approach is to allow randomness tests to access the measure $\lambda $ as an…
The notion of probability plays a crucial role in quantum mechanics. It appears in quantum mechanics as the Born rule. In modern mathematics which describes quantum mechanics, however, probability theory means nothing other than measure…
Previously referred to as `miraculous' in the scientific literature because of its powerful properties and its wide application as optimal solution to the problem of induction/inference, (approximations to) Algorithmic Probability (AP) and…
In this work, we develop a formal system of inductive logic. It uses an infinitary language that allows for countable conjunctions and disjunctions. It is based on a set of nine syntactic rules of inductive inference, and contains classical…
The problem of combining p-values is an old and fundamental one, and the classic assumption of independence is often violated or unverifiable in many applications. There are many well-known rules that can combine a set of arbitrarily…
We review some of our recent results (with collaborators) on information processing in an ordered linear spaces framework for probabilistic theories. These include demonstrations that many "inherently quantum" phenomena are in reality quite…