On the accuracy of Poisson approximation

Novak, Serguei

doi:10.1007/s10687-019-00350-6

Cited by 10 publications

(9 citation statements)

References 38 publications

(61 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then X n is the sum of n independent {0, 1}-valued random variables, each one with probability 1/(n − 1) of being 1. By a crude version of the Poisson approximation (see for example equation (1) in [13]) of n independent and identically distributed (with probability 1/(n − 1)) Bernoulli trials we get (for large enough n)…”

Section: Admissibility and The Main Resultsmentioning

confidence: 99%

Asymptotic elimination of partially continuous aggregation functions in directed graphical models

Koponen¹,

Weitkämper²

2021

Preprint

View full text Add to dashboard Cite

In Statistical Relational Artificial Intelligence, a branch of AI and machine learning which combines the logical and statistical schools of AI, one uses the concept parametrized probabilistic graphical model (PPGM) to model (conditional) dependencies between random variables and to make probabilistic inferences about events on a space of "possible worlds". The set of possible worlds with underlying domain D (a set of objects) can be represented by the set WD of all first-order structures (for a suitable signature) with domain D. Using a formal logic we can describe events on WD. By combining a logic and a PPGM we can also define a probability distribution PD on WD and use it to compute the probability of an event. We consider a logic, denoted P LA, with truth values in the unit interval, which uses aggregation functions instead of quantifiers. This is motivated by the fact that aggregation functions such as arithmetic mean, geometric mean, maximum and minimum are important tools in analysis of data and also by the fact that aggregation functions can play the role of quantifiers.However we face the problem of computational efficiency and this problem is an obstacle to the wider use of methods from Statistical Relational AI in practical applications. The brute force way of computing, for S ⊆ [0, 1] and a P LA-sentence ϕ, the probability that the value of ϕ belongs to S needs an amount of time which grows exponentially in the size of D. We address this problem by proving that the described probability will, under certain assumptions on the PPGM and the sentence ϕ, converge as the size of D tends to infinity. The convergence result is obtained by showing that every formula ϕ(x1, . . . , x k ) which contains only "admissible" aggregation functions (e.g. arithmetic and geometric mean, max and min) is asymptotically equivalent to a formula ψ(x1, . . . , x k ) without aggregation functions. This means that for every ε > 0 the probability that, for some parameters a1, . . . , a k ∈ D, the values of ϕ(a1, . . . , a k ) and ψ(a1, . . . , a k ) differ by more than ε approaches 0 as the domain size tends to infinity. The proof provides a method of computing the probability that the value of ϕ(a1, . . . , a k ) belongs to S with an amount of time which is independent of D and only depends on ϕ and the PPGM.

show abstract

Section: Admissibility and The Main Resultsmentioning

confidence: 99%

Asymptotic elimination of partially continuous aggregation functions in directed graphical models

Koponen¹,

Weitkämper²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Shifted Poisson approximation. Shifted (translated) Poisson approximation to B(n, p) has been considered by a number of authors (see [16,19,29,62,80] and references therein). The accuracy of shifted Poisson approximation can be sharper than that of pure Poisson approximation.…”

Section: Independent Bernoulli Rvsmentioning

confidence: 99%

“…A generalisation of (32) to the case of independent integer-valued r.v.s has been given by Novak [80].…”

Section: Independent Integer-valued Rvsmentioning

confidence: 99%

See 1 more Smart Citation

Poisson approximation

Novak¹

2019

Preprint

Self Cite

View full text Add to dashboard Cite

We overview results on the topic of Poisson approximation that are missed in existing surveys. The main attention is paid to the problem of Poisson approximation to the distribution of a sum of Bernoulli and, more generally, non-negative integervalued random variables.We do not restrict ourselves to a particular method, and overview the whole range of issues including the general limit theorem, estimates of the accuracy of approximation, asymptotic expansions, etc. Related results on the accuracy of compound Poisson approximation are presented as well.We indicate a number of open problems and discuss directions of further research.

show abstract

“…Regarding estimation of the total variation distance we refer the reader to the references in the book by Barbour et al (1992) and the recent review by Novak (2019b). Notice that there is a number of papers devoted to the Poisson approximation in terms of other probability distances like d(z, ·, ·) (e.g., see Ruzankin, 2001), the information divergence (e.g., see Harremoës and Ruzankin, 2004), χ 2 -distance (e.g., see Borisov and Vorozheikin, 2008), and of other distances (e.g., see Novak, 2019a;Ruzankin, 2004Ruzankin, , 2010.…”

mentioning

confidence: 99%