Hamiltonian Monte Carlo is a Markov Chain Monte Carlo method that has been widely applied to numerous posterior inference problems within the machine learning literature. Markov Chain Monte Carlo estimators have higher variance than classical Monte Carlo estimators due to autocorrelations present between the generated samples. In this work we present three new methods for tackling the high variance problem in Hamiltonian Monte Carlo based estimators: 1) We combine antithetic and importance sampling techniques where the importance sampler is based on sampling from a modified or shadow Hamiltonian using Separable Shadow Hamiltonian Hybrid Monte Carlo, 2) We present the antithetic Magnetic Hamiltonian Monte Carlo algorithm that is based on performing antithetic sampling on the Magnetic Hamiltonian Monte Carlo algorithm and 3) We propose the antithetic Magnetic Momentum Hamiltonian Monte Carlo algorithm based on performing antithetic sampling on the Magnetic Momentum Hamiltonian Monte Carlo method. We find that the antithetic Separable Shadow Hamiltonian Hybrid Monte Carlo and antithetic Magnetic Momentum Hamiltonian Monte Carlo algorithms produce effective sample sizes that are higher than antithetic Hamiltonian Monte Carlo on all the benchmark datasets. We further find that antithetic Separable Shadow Hamiltonian Hybrid Monte Carlo and antithetic Magnetic Hamiltonian Monte Carlo produce higher effective sample sizes normalised by execution time in higher dimensions than antithetic Hamiltonian Monte Carlo. In addition, the antithetic versions of all the algorithms have higher effective sample sizes than their non-antithetic counterparts, indicating the usefulness of adding antithetic sampling to Markov Chain Monte Carlo algorithms. The methods are assessed on benchmark datasets using Bayesian logistic regression and Bayesian neural network models.INDEX TERMS Antithetic sampling, bayesian logistic regression, bayesian neural network, hamiltonian monte carlo, machine learning, magnetic hamiltonian monte carlo, shadow hamiltonian, variance reduction

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Financial statement fraud has been on the increase in the past two decades and includes prominent scandals such as Enron, WorldCom and more recently in South Africa, Steinhoff. These scandals have led to billions of dollars being lost in the form of market capitalisation from different stock exchanges across the world. During this time, there has been an increase in the literature on applying automated methods to detecting financial statement fraud using publicly available data. This paper provides a survey of the literature on automated financial statement fraud detection and identifies current gaps in the literature. The paper highlights a number of important considerations in the implementation of financial statement fraud detection decision support systems, including 1) the definition of fraud, 2) features used for detecting fraud, 3) region of the case study, dataset size and imbalance, 4) algorithms used for detection, 5) approach to feature selection / feature engineering, 6) treatment of missing data, and 7) performance measure used. The current study discusses how these and other implementation factors could be approached within the South African context.

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