The matrix Wishart distribution on positive definite matrices

Source: R/distributions.R

This distribution is defined by a scalar number of degrees of freedom df and an instance of LinearOperator, which provides matrix-free access to a symmetric positive definite operator, which defines the scale matrix.

tfd_wishart(
  df,
  scale = NULL,
  scale_tril = NULL,
  input_output_cholesky = FALSE,
  validate_args = FALSE,
  allow_nan_stats = TRUE,
  name = "Wishart"
)

Arguments

df

float or double tensor, the degrees of freedom of the distribution(s). df must be greater than or equal to k.
scale

float or double Tensor. The symmetric positive definite scale matrix of the distribution. Exactly one of scale and 'scale_tril must be passed.
scale_tril

float or double Tensor. The Cholesky factorization of the symmetric positive definite scale matrix of the distribution. Exactly one of scale and 'scale_tril must be passed.
input_output_cholesky

Logical. If TRUE, functions whose input or output have the semantics of samples assume inputs are in Cholesky form and return outputs in Cholesky form. In particular, if this flag is TRUE, input to log_prob is presumed of Cholesky form and output from sample, mean, and mode are of Cholesky form. Setting this argument to TRUE is purely a computational optimization and does not change the underlying distribution; for instance, mean returns the Cholesky of the mean, not the mean of Cholesky factors. The variance and stddev methods are unaffected by this flag. Default value: FALSE (i.e., input/output does not have Cholesky semantics).
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; df, scale) = det(X)**(0.5 (df-k-1)) exp(-0.5 tr[inv(scale) X]) / Z
Z = 2**(0.5 df k) |det(scale)|**(0.5 df) Gamma_k(0.5 df)

where:

  • df >= k denotes the degrees of freedom,

  • scale is a symmetric, positive definite, k x k matrix,

  • Z is the normalizing constant, and,

  • Gamma_k is the multivariate Gamma function.

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