The von Mises-Fisher distribution over unit vectors on S^{n-1}

The von Mises-Fisher distribution is a directional distribution over vectors on the unit hypersphere S^{n-1} embedded in n dimensions (R^n).

tfd_von_mises_fisher(
  mean_direction,
  concentration,
  validate_args = FALSE,
  allow_nan_stats = TRUE,
  name = "VonMisesFisher"
)

Arguments

mean_direction	Floating-point Tensor with shape [B1, ... Bn, D]. A unit vector indicating the mode of the distribution, or the unit-normalized direction of the mean. (This is not in general the mean of the distribution; the mean is not generally in the support of the distribution.) NOTE: D is currently restricted to <= 5.
concentration	Floating-point Tensor having batch shape [B1, ... Bn] broadcastable with mean_direction. The level of concentration of samples around the mean_direction. concentration=0 indicates a uniform distribution over the unit hypersphere, and concentration=+inf indicates a Deterministic distribution (delta function) at mean_direction.
validate_args	Logical, default FALSE. When TRUE distribution parameters are checked for validity despite possibly degrading runtime performance. When FALSE invalid inputs may silently render incorrect outputs. Default value: FALSE.
allow_nan_stats	Logical, default TRUE. When TRUE, statistics (e.g., mean, mode, variance) use the value NaN to indicate the result is undefined. When FALSE, an exception is raised if one or more of the statistic's batch members are undefined.
name	name prefixed to Ops created by this class.

Value

a distribution instance.

Details

Mathematical details The probability density function (pdf) is,

pdf(x; mu, kappa) = C(kappa) exp(kappa * mu^T x)
where,
C(kappa) = (2 pi)^{-n/2} kappa^{n/2-1} / I_{n/2-1}(kappa),
I_v(z) being the modified Bessel function of the first kind of order v

where:

• mean_direction = mu; a unit vector in R^k,

• concentration = kappa; scalar real >= 0, concentration of samples around mean_direction, where 0 pertains to the uniform distribution on the hypersphere, and inf indicates a delta function at mean_direction.

NOTE: Currently only n in 2, 3, 4, 5 are supported. For n=5 some numerical instability can occur for low concentrations (<.01).

See also

For usage examples see e.g. tfd_sample(), tfd_log_prob(), tfd_mean().

Other distributions: tfd_autoregressive(), tfd_batch_reshape(), tfd_bates(), tfd_bernoulli(), tfd_beta_binomial(), tfd_beta(), tfd_binomial(), tfd_categorical(), tfd_cauchy(), tfd_chi2(), tfd_chi(), tfd_cholesky_lkj(), tfd_continuous_bernoulli(), tfd_deterministic(), tfd_dirichlet_multinomial(), tfd_dirichlet(), tfd_empirical(), tfd_exp_gamma(), tfd_exp_inverse_gamma(), tfd_exponential(), tfd_gamma_gamma(), tfd_gamma(), tfd_gaussian_process_regression_model(), tfd_gaussian_process(), tfd_generalized_normal(), tfd_geometric(), tfd_gumbel(), tfd_half_cauchy(), tfd_half_normal(), tfd_hidden_markov_model(), tfd_horseshoe(), tfd_independent(), tfd_inverse_gamma(), tfd_inverse_gaussian(), tfd_johnson_s_u(), tfd_joint_distribution_named_auto_batched(), tfd_joint_distribution_named(), tfd_joint_distribution_sequential_auto_batched(), tfd_joint_distribution_sequential(), tfd_kumaraswamy(), tfd_laplace(), tfd_linear_gaussian_state_space_model(), tfd_lkj(), tfd_log_logistic(), tfd_log_normal(), tfd_logistic(), tfd_mixture_same_family(), tfd_mixture(), tfd_multinomial(), tfd_multivariate_normal_diag_plus_low_rank(), tfd_multivariate_normal_diag(), tfd_multivariate_normal_full_covariance(), tfd_multivariate_normal_linear_operator(), tfd_multivariate_normal_tri_l(), tfd_multivariate_student_t_linear_operator(), tfd_negative_binomial(), tfd_normal(), tfd_one_hot_categorical(), tfd_pareto(), tfd_pixel_cnn(), tfd_poisson_log_normal_quadrature_compound(), tfd_poisson(), tfd_power_spherical(), tfd_probit_bernoulli(), tfd_quantized(), tfd_relaxed_bernoulli(), tfd_relaxed_one_hot_categorical(), tfd_sample_distribution(), tfd_sinh_arcsinh(), tfd_skellam(), tfd_spherical_uniform(), tfd_student_t_process(), tfd_student_t(), tfd_transformed_distribution(), tfd_triangular(), tfd_truncated_cauchy(), tfd_truncated_normal(), tfd_uniform(), tfd_variational_gaussian_process(), tfd_vector_diffeomixture(), tfd_vector_exponential_diag(), tfd_vector_exponential_linear_operator(), tfd_vector_laplace_diag(), tfd_vector_laplace_linear_operator(), tfd_vector_sinh_arcsinh_diag(), tfd_von_mises(), tfd_weibull(), tfd_wishart_linear_operator(), tfd_wishart_tri_l(), tfd_wishart(), tfd_zipf()