The Power Spherical distribution over unit vectors on S^{n-1}.

The Power Spherical distribution is a distribution over vectors on the unit hypersphere S^{n-1} embedded in n dimensions (R^n). It serves as an alternative to the von Mises-Fisher distribution with a simpler (faster) log_prob calculation, as well as a reparameterizable sampler. In contrast, the Power Spherical distribution does have -mean_direction as a point with zero density (and hence a neighborhood around that having arbitrarily small density), in contrast with the von Mises-Fisher distribution which has non-zero density everywhere. NOTE: mean_direction is not in general the mean of the distribution. For spherical distributions, the mean is generally not in the support of the distribution.

tfd_power_spherical(
  mean_direction,
  concentration,
  validate_args = FALSE,
  allow_nan_stats = TRUE,
  name = "PowerSpherical"
)

Arguments

mean_direction	Floating-point Tensor with shape [B1, ... Bn, N]. A unit vector indicating the mode of the distribution, or the unit-normalized direction of the mean.
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) (1 + mu^T x) ** k
  where,
     C(kappa) = 2**(a + b) pi**b Gamma(a) / Gamma(a + b)
     a = (n - 1) / 2. + k
     b = (n - 1) / 2.

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.

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_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_fisher(), tfd_von_mises(), tfd_weibull(), tfd_wishart_linear_operator(), tfd_wishart_tri_l(), tfd_wishart(), tfd_zipf()