Related papers: Algorithm, probability, and prime numbers
A fruitful way of obtaining meaningful, possibly concrete, algorithmically random numbers is to consider a potential behaviour of a Turing machine and its probability with respect to a measure (or semi-measure) on the input space of binary…
Reasoning under uncertainty is a fundamental challenge in Artificial Intelligence. As with most of these challenges, there is a harsh dilemma between the expressive power of the language used, and the tractability of the computational…
It is shown that the psychometric test reliability, based on any true-score model with randomly sampled items and conditionally independent errors, converges to 1 as the test length goes to infinity, assuming some fairly general regularity…
A number of algorithms have been developed to solve probabilistic inference problems on belief networks. These algorithms can be divided into two main groups: exact techniques which exploit the conditional independence revealed when the…
An equivalence is proven between the Riemann Hypothesis and the speed of convergence to 1/zeta(2) of the probability that two independent random variables following the same geometric distribution are coprime integers, when the parameter of…
Random probabilities are a key component to many nonparametric methods in Statistics and Machine Learning. To quantify comparisons between different laws of random probabilities several works are starting to use the elegant Wasserstein over…
When there exists uncertainty, AI machines are designed to make decisions so as to reach the best expected outcomes. Expectations are based on true facts about the objective environment the machines interact with, and those facts can be…
A significant barrier to the widespread adoption of Bayesian inference is the specification of prior distributions and likelihoods, which often requires specialized statistical expertise. This paper investigates the feasibility of using a…
Both statistics and quantum theory deal with prediction using probability. We will show that there can be established a connection between these two areas. This will at the same time suggest a new, less formalistic way of looking upon basic…
Predictive mean matching (PMM) is a popular imputation strategy that imputes missing values by borrowing observed values from other cases with similar expectations. We show that, unlike other imputation strategies, PMM is not guaranteed to…
Estimating a large alphabet probability distribution from a limited number of samples is a fundamental problem in machine learning and statistics. A variety of estimation schemes have been proposed over the years, mostly inspired by the…
Natural numbers can be divided in two non-overlapping infinite sets, primes and composites, with composites factorizing into primes. Despite their apparent simplicity, the elucidation of the architecture of natural numbers with primes as…
A classic result of Paul, Pippenger, Szemer\'edi and Trotter states that DTIME(n) is strictly contained in NTIME(n). The natural question then arises: could DTIME(t(n)) be contained in NTIME(n) for some superlinear time-constructible…
The singular values of a product of $M$ independent Ginibre matrices of size $N\times N$ form a determinantal point process. Near the soft edge, as both $M$ and $N$ go to infinity in such a way that $M/N\to \alpha$, $\alpha>0$, a scaling…
Let $P$ be the set of all prime numbers, ${q_1},{q_2}, \cdots ,{q_m} \in P$, $P_k$ be the k-th $(k = 1,2, \cdots m)$ element of $P$ in ascending order of size, ${\alpha _1},{\alpha _2}, \cdots ,{\alpha _m}$ be positive integers, and ${\beta…
Some statistical models are specified via a data generating process for which the likelihood function cannot be computed in closed form. Standard likelihood-based inference is then not feasible but the model parameters can be inferred by…
Recently, there has been considerable progress on designing algorithms with provable guarantees -- typically using linear algebraic methods -- for parameter learning in latent variable models. But designing provable algorithms for inference…
We are now witnessing a rapid growth of a new part of group theory which has become known as "statistical group theory". A typical result in this area would say something like ``a random element (or a tuple of elements) of a group G has a…
We calculate the probability that random polynomial matrices over a finite field with certain structures are right prime or left prime, respectively. In particular, we give an asymptotic formula for the probability that finitely many…
A long noted difficulty when assessing the reliability (or calibration) of forecasting systems is that reliability, in general, is a hypothesis not about a finite dimensional parameter but about an entire functional relationship. A…