Conditional Densities and Likelihoods for Hypertoroidal Densities Based on Trigonometric Polynomials

Recently, trigonometric polynomials have been used to approximate densities or their square roots in the context of Bayesian estimation. Trigonometric polynomials were also used to interpolate function values on grids on hypertori. In this paper, we derive formulae for conditional densities and likelihoods for multivariate densities parameterized by grid values or Fourier coefficients. Efficient formulae are proposed for both representations that involve no more than
$O(n \log n)$
operations. The conditional densities can be described using a single parameter vector. For the likelihoods, formulae are given that allow for a precise evaluation using two parameter vectors. Furthermore, formulae involving only a single parameter vector are provided for approximations of the likelihoods.