  This paper proposes an efficient numerical integration formula to compute the
normalizing constant of Fisher--Bingham distributions. This formula uses a
numerical integration formula with the continuous Euler transform to a
Fourier-type integral representation of the normalizing constant. As this
method is fast and accurate, it can be applied to the calculation of the
normalizing constant of high-dimensional Fisher--Bingham distributions. More
precisely, the error decays exponentially with an increase in the integration
points, and the computation cost increases linearly with the dimensions. In
addition, this formula is useful for calculating the gradient and Hessian
matrix of the normalizing constant. Therefore, we apply this formula to
efficiently calculate the maximum likelihood estimation (MLE) of
high-dimensional data. Finally, we apply the MLE to the hyperspherical
variational auto-encoder (S-VAE), a deep-learning-based generative model that
restricts the latent space to a unit hypersphere. We use the S-VAE trained with
images of handwritten numbers to estimate the distributions of each label. This
application is useful for adding new labels to the models.
