The fully visible Boltzmann machine and the Senate of the 45th Australian Parliament in 2016

Bagnall, Jessica; Jones, Andrew; Karavarsamis, Natalie; Nguyen, Hien D.

doi:10.1007/s42001-019-00055-7

Cited by 1 publication

(3 citation statements)

References 25 publications

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“…Sundberg 2019, Chapter 6), such as when Y arises from a fully visible Boltzmann machine (cf. Hyvarinen 2006;Bagnall et al 2020), or when data arise from variance mixtures of normal distributions. In such cases, by (4), we can define θ (s) as the root of the first-order condition…”

Section: Introductionmentioning

confidence: 99%

“…Hyvarinen 2006; Bagnall et al . 2020), or when data arise from variance mixtures of normal distributions. In such cases, by 4), we can define

\bar{θ} (bold-italics)

as the root of the first‐order condition

J_{bold-italicϕ} false(bold-italicθ false) bold-italics - \frac{\partial ψ}{\partial θ} false(bold-italicθ false) = bold0,

where

J_{bold-italicϕ} false(bold-italicθ false) = \partial ϕ / \partial θ

is the Jacobian of ϕ , with respect to θ , as a function of θ .…”

Section: Introductionmentioning

confidence: 99%

“…= a 1 ,…, a d , b 12 , b 13 ,…, b d −1,d ∈ R d (d +1)=2 and ζ = 1 ,…, d , which has an exponential family form with h(y) = 1, (θ ) = log (θ ), s(y) = (y 1 ,…, y d , y 1 y 2 , y 1 y 3 ,…, y d −1 y d ) and φ(θ ) = θ. Models of form (26) are often referred to as fully visible Boltzmann machines in the machine learning literature (see, e.g Bagnall et al 2020)…”

mentioning

confidence: 99%

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Global implicit function theorems and the online expectation–maximisation algorithm

Nguyen¹,

Forbes

2022

Aus NZ J of Statistics

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Summary The expectation–maximisation (EM) algorithm framework is an important tool for statistical computation. Due to the changing nature of data, online and mini‐batch variants of EM and EM‐like algorithms have become increasingly popular. The consistency of the estimator sequences that are produced by these EM variants often rely on an assumption regarding the continuous differentiability of a parameter update function. In many cases, the parameter update function is not in closed form and may only be defined implicitly, which makes the verification of the continuous differentiability property difficult. We demonstrate how a global implicit function theorem can be used to verify such properties in the cases of finite mixtures of distributions in the exponential family, and more generally, when the component‐specific distributions admit data augmentation schemes, within the exponential family. We then illustrate the use of such a theorem in the cases of mixtures of beta distributions, gamma distributions, fully visible Boltzmann machines and Student distributions. Via numerical simulations, we provide empirical evidence towards the consistency of the online EM algorithm parameter estimates in such cases.

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Section: Introductionmentioning

confidence: 99%

“…Hyvarinen 2006; Bagnall et al . 2020), or when data arise from variance mixtures of normal distributions. In such cases, by 4), we can define