The Basel Problem with an Extension

Sofo, Anthony

doi:10.1002/9781119414421.ch18

Cited by 1 publication

(2 citation statements)

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“…Harmonic series and convergent Geometric series have been extensively studied (Sofo 2018;. However, approximations of these series that include a second parameter in the denominator have, to our knowledge, not been published.…”

Section: A Preprintmentioning

confidence: 99%

“…where f2 is only accurate for N > 200, and all computations that did not meet this criterion were calculated exactly (not using this approximation). Readers familiar with the harmonic and geometric series will recognize that ln(N ) + γ + 1 2N − 1 12N 2 is the finite series approximation for the harmonic series, and π 2 6 is the limit of the geometric series (Sofo 2018;. Examples of the above approximations are implemented in the open source code included with this manuscript (see Data Availability).…”

Section: Finite Limit Approximationsmentioning

confidence: 99%

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Fast Maximum Likelihood Estimation and Supervised Classification for the Beta-Liouville Multinomial

Lakin,

Abdo

2020

Preprint

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The multinomial and related distributions have long been used to model categorical, countbased data in fields ranging from bioinformatics to natural language processing. Commonly utilized variants include the standard multinomial and the Dirichlet multinomial distributions due to their computational efficiency and straightforward parameter estimation process. However, these distributions make strict assumptions about the mean, variance, and covariance between the categorical features being modeled. If these assumptions are not met by the data, it may result in poor parameter estimates and loss in accuracy for downstream applications like classification. Here, we explore efficient parameter estimation and supervised classification methods using an alternative distribution, called the Beta-Liouville multinomial, which relaxes some of the multinomial assumptions. We show that the Beta-Liouville multinomial is comparable in efficiency to the Dirichlet multinomial for Newton-Raphson maximum likelihood estimation, and that its performance on simulated data matches or exceeds that of the multinomial and Dirichlet multinomial distributions. Finally, we demonstrate that the Beta-Liouville multinomial outperforms the multinomial and Dirichlet multinomial on two out of four gold standard datasets, supporting its use in modeling data with low to medium class overlap in a supervised classification context.

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Section: A Preprintmentioning

confidence: 99%

Section: Finite Limit Approximationsmentioning

confidence: 99%