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Directional Metropolis–Hastings algorithms on hyperplanes

By Hugo Hammer and Håkon Tjelmel


In this paper we define and study new directional Metropolis–Hastings algorithms that propose states in hyperplanes. Each iteration in directional Metropolis–Hastings algorithms consist of three steps. First a direction is sampled by an auxiliary variable. Then a potential new state is proposed in the subspace defined by this direction and the current state. Lastly, the potential new state is accepted or rejected according to the Metropolis–Hastings acceptance probability. Traditional directional Metropolis–Hastings algorithms define the direction by one vector and so the corresponding subspace in which the potential new state is sampled is a line. In this paper we let the direction be defined by two or more vectors and so the corresponding subspace becomes a hyperplane. We compare the performance of directional Metropolis–Hastings algorithms defined on hyperplanes with other frequently used Metropolis–Hastings schemes. The experience is that hyperplane algorithms on average produce larger jumps in the sample space and thereby have better mixing properties per iteration. However, with our implementations the hyperplane algorithms is more computation intensive per iteration and so the simpler algorithms in most cases are better when run for the same amount of computer time. An interesting area for future research is therefore to find variants of our directional Metropolis–Hastings algorithms that require less computation time per iteration

Topics: Angular Gaussian distribution, Directional Metropolis–Hastings, Markov chain Monte Carlo, Hyperplane algorithms, Subspace
Year: 2011
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